Mc graw hill - data management

McGraw-Hili Ryerson

D�-J�
Mathematics of
J:;j�jJ�0ajSJajJ-J

This book was distributed by Jack Truong for use at William Lyon Mackenzie Collegiate Institute.

1
PT ER
Tools for Data Management
CHA

Specific Expectations Section

Locate data to answer questions of significance or personal interest, by 1.3
searching well-organized databases.

Use the Internet effectively as a source for databases. 1.3

Create database or spreadsheet templates that facilitate the manipulation 1.2, 1.3, 1.4
and retrieval of data from large bodies of information that have a variety
of characteristics.

Represent simple iterative processes, using diagrams that involve 1.1
branches and loops.

Represent complex tasks or issues, using diagrams. 1.1, 1.5

Solve network problems, using introductory graph theory. 1.5

Represent numerical data, using matrices, and demonstrate an 1.6, 1.7
understanding of terminology and notation related to matrices.

Demonstrate proficiency in matrix operations, including addition, scalar 1.6, 1.7
multiplication, matrix multiplication, the calculation of row sums, and the
calculation of column sums, as necessary to solve problems, with and
without the aid of technology.

Solve problems drawn from a variety of applications, using matrix 1.6, 1.7
methods.

Chapter Problem
VIA Rail Routes 1. a) List several routes you have
When travelling by bus, train, or airplane, travelled where you were able to
you usually want to reach your destination reach your destination directly.
without any stops or transfers. However, b) List a route where you had to
it is not always possible to reach your change vehicles exactly once before
destination by a non-stop route. The reaching your destination.
following map shows the VIA Rail routes 2. a) List all the possible routes from
for eight major cities. The arrows Montréal to Toronto by VIA Rail.
represent routes on which you do not have
b) Which route would you take to get
to change trains.
from Montréal to Toronto in the
Montréal least amount of time? Explain your
Sudbury reasoning.
3. a) List all the possible routes from
Ottawa
Kingston to London.
Kingston b) Give a possible reason why VIA Rail
chooses not to have a direct train
Toronto from Kingston to London.
This chapter introduces graph theory,
matrices, and technology that you can use
London
to model networks like the one shown. You
Niagara Falls will learn techniques for determining the
Windsor number of direct and indirect routes from
one city to another. The chapter also
discusses useful data-management tools
including iterative processes, databases,
software, and simulations.

Review of Prerequisite Skills

If you need help with any of the skills listed in purple below, refer to Appendix A.

1. Order of operations Evaluate each 5. Graphing data Organize the following set of
expression. data using a fully-labelled double-bar graph.
a) (−4)(5) + (2)(−3) City Snowfall (cm) Total Precipitation (cm)
b) (−2)(3) + (5)(−3) + (8)(7) St. John’s 322.1 148.2
Charlottetown 338.7 120.1
c) (1)(0) + (1)(1) + (0)(0) + (0)(1)
Halifax 261.4 147.4
12
d) (2)(4) + ᎏᎏ − (3)2 Fredericton 294.5 113.1
3 Québec City 337.0 120.8
2. Substituting into equations Given Montréal 214.2 94.0
f (x) = 3x2 − 5x + 2 and g(x) = 2x − 1, Ottawa 221.5 91.1
evaluate each expression. Toronto 135.0 81.9
Winnipeg 114.8 50.4
a) f (2)
Regina 107.4 36.4
b) g(2) Edmonton 129.6 46.1
c) f (g(−1)) Calgary 135.4 39.9
d) f ( g(1)) Vancouver 54.9 116.7
Victoria 46.9 85.8
e) f ( f (2))
Whitehorse 145.2 26.9
f) g( f (2))
Yellowknife 143.9 26.7
3. Solving equations Solve for x.
6. Graphing data The following table lists the
a) 2x − 3 = 7
average annual full-time earnings for males
b) 5x + 2 = −8 and females. Illustrate these data using a
x fully-labelled double-line graph.
c) ᎏ − 5 = 5
2
d) 4x − 3 = 2x − 1 Year Women ($) Men ($)
1989 28 219 42 767
e) x2 = 25
1990 29 050 42 913
f) x3 = 125 1991 29 654 42 575
g) 3(x + 1) = 2(x − 1) 1992 30 903 42 984
2x − 5 3x − 1 1993 30 466 42 161
h) ᎏ = ᎏ
2 4 1994 30 274 43 362
1995 30 959 42 338
4. Graphing data In a sample of 1000 1996 30 606 41 897
Canadians, 46% have type O blood, 43% 1997 30 484 43 804
have type A, 8% have type B, and 3% have 1998 32 553 45 070
type AB. Represent these data with a fully-
labelled circle graph.

4 MHR • Tools for Data Management

7. Using spreadsheets Refer to the spreadsheet 10. Ratios of areas Draw two squares on a sheet
section of Appendix B, if necessary. of grid paper, making the dimensions of the
a) Describe how to refer to a specific cell. second square half those of the first.
b) Describe how to refer to a range of cells a) Use algebra to calculate the ratio of the
in the same row. areas of the two squares.
c) Describe how to copy data into another b) Confirm this ratio by counting the
cell. number of grid units contained in each
square.
d) Describe how to move data from one
column to another. c) If you have access to The Geometer’s
Sketchpad or similar software, confirm
e) Describe how to expand the width of a
the area ratio by drawing a square,
column.
dilating it by a factor of 0.5, and
f) Describe how to add another column. measuring the areas of the two squares.
g) What symbol must precede a Refer to the help menu in the software,
mathematical expression? if necessary.

8. Similar triangles Determine which of the 11. Simplifying expressions Expand and simplify
following triangles are similar. Explain each expression.
your reasoning. a) (x – 1)2

B 3 A
D b) (2x + 1)(x – 4)
2 7 4 c) –5x(x – 2y)
4
C
E d) 3x(x – y)2
F
6
G e) (x – y)(3x)2
12 f) (a + b)(c – d)
6
12. Fractions, percents, decimals Express as a
J
9 decimal.
H
5 23 2
a) ᎏᎏ b) ᎏᎏ c) ᎏᎏ
9. Number patterns Describe each of the 20 50 3
following patterns. Show the next three 138 6
d) ᎏᎏ e) ᎏᎏ f) 73%
terms. 12 7
a) 65, 62, 59, … 13. Fractions, percents, decimals Express as a
b) 100, 50, 25, … percent.
1 1 1 4 1
c) 1, − ᎏ , ᎏ , − ᎏ , … a) 0.46 b) ᎏᎏ c) ᎏᎏ
2 4 8 5 30
d) a, b, aa, bb, aaa, bbbb, aaaa, bbbbbbbb, … 11
d) 2.25 e) ᎏᎏ
8

Review of Prerequisite Skills • MHR 5

1.1 The Iterative Process

If you look carefully at the branches of a tree,
you can see the same pattern repeated over
and over, but getting smaller toward the end
of each branch. A nautilus shell repeats the
same shape on a larger and larger scale from
its tip to its opening. You yourself repeat
many activities each day. These three
examples all involve an iterative process.
Iteration is a process of repeating the same
procedure over and over. The following
activities demonstrate this process.

I N V E S T I G AT E & I N Q U I R E : Developing a Sor t Algorithm

Often you need to sort data using one or more criteria, such as alphabetical
or numerical order. Work with a partner to develop an algorithm to sort the
members of your class in order of their birthdays. An algorithm is a
procedure or set of rules for solving a problem.

1. Select two people and compare their birthdays.
2. Rank the person with the later birthday second.
3. Now, compare the next person’s birthday with the last ranked birthday.
Rank the later birthday of those two last.
4. Describe the continuing process you will use to ﬁnd the classmate with
the latest birthday.
5. Describe the process you would use to ﬁnd the person with the second
latest birthday. With whom do you stop comparing?
6. Describe a process to rank all the remaining members of your class
by their birthdays.
7. Illustrate your process with a diagram.

The process you described is an iterative process because it involves repeating
the same set of steps throughout the algorithm. Computers can easily be
programmed to sort data using this process.


I N V E S T I G AT E & I N Q U I R E : T h e S i e r p i n s k i Tr i a n g l e

Method 1: Pencil and Paper
1. Using isometric dot paper, draw a large equilateral triangle with side
lengths of 32 units.
2. Divide this equilateral triangle into four smaller equilateral triangles.
3. Shade the middle triangle. What fraction of the original triangle is shaded?
4. For each of the unshaded triangles, repeat this process. What fraction of
the original triangle is shaded?
5. For each of the unshaded triangles, repeat this process again. What fraction
of the original triangle is shaded now?
6. Predict the fraction of the original triangle that would be shaded for the
fourth and fifth steps in this iterative process.
7. Predict the fraction of the original triangle that would be shaded if this
iterative process continued indefinitely.
Method 2: The Geometer’s Sketchpad®
1. Open a new sketch and a new script.
2. Position both windows side by side.
3. Click on REC in the script window.
4. In the sketch window, construct a triangle. Shift-click on each side of the
triangle. Under the Construct menu, choose Point at Midpoint and then
Polygon Interior of the midpoints.
5. Shift-click on one vertex and the two adjacent midpoints. Choose Loop
in your script.
6. Repeat step 5 for the other two vertices.
7. Shift-click on the three midpoints. From the Display menu,
choose Hide Midpoints.
8. Stop your script.
9. Open a new sketch. Construct a new triangle. Mark the
three vertices. Play your script at a recursion depth of at
least 3. You may increase the speed by clicking on Fast.
10. a) What fraction of the original triangle is shaded
i) after one recursion?
ii) after two recursions?
iii) after three recursions?
b) Predict what fraction would be shaded after four and five recursions.
c) Predict the fraction of the original triangle that would be shaded if
this iterative (recursion) process continued indefinitely.
11. Experiment with recursion scripts to design patterns with repeating shapes.

1.1 The Iterative Process • MHR 7

The Sierpinski triangle is named after the Polish
mathematician, Waclaw Sierpinski (1882−1924). It is
an example of a fractal, a geometric ﬁgure that is www.mcgrawhill.ca/links/MDM12
generally created using an iterative process. One
part of the process is that fractals are made of self- Visit the above web site and follow the links to
learn more about the Sierpinski triangle and
similar shapes. As the shapes become smaller and
fractals. Choose an interesting fractal and describe
smaller, they keep the same geometrical
how it is self-similar.
characteristics as the original larger shape. Fractal
geometry is a very rich area of study. Fractals can be
used to model plants, trees, economies, or the
honeycomb pattern in human bones.

Example 1 Modelling With a Fractal

Fractals can model the branching of a tree. Describe
the algorithm used to model the tree shown.

Solution
Begin with a 1-unit segment. Branch off at 60° with two
segments, each one half the length of the previous branch.
Repeat this process for a total of three iterations.

Arrow diagrams can illustrate iterations. Such diagrams show the sequence
of steps in the process.

Example 2 The Water Cycle

Illustrate the water cycle using an arrow diagram.

Solution The Water Cycle
The water, or hydrologic, cycle is Condensation
an iterative process. Although the
timing of the precipitation can Precipitation Transpiration
Evaporation
vary, the cycle will repeat itself Surface
indeﬁnitely. runoff

Lake
Percolation
Streams and Ocean
Water table Rivers

Groundwater


Example 3 Tree Diagram
a) Illustrate the results of a best-of-five hockey playoff series between
Ottawa and Toronto using a tree diagram.
b) How many different outcomes of the series are possible?

Solution
a) For each game, the tree diagram has two branches, O
one representing a win by Ottawa (O) and the other O
O
a win by Toronto (T). Each set of branches T O
T
represents a new game in the playoff round. As soon O T
O
as one team wins three games, the playoff round O O
T
ends, so the branch representing that sequence also T T
stops. O O
T T
T
b) By counting the endpoints of the branches, you can O
determine that there are 20 possible outcomes for O O
T T
this series. O O
O
T T
T T
O
O T
O
T
T
T

1st game 2nd 3rd 4th 5th

Example 4 Recursive Formula

The recursive formula tn = 3tn-1 − tn-2 defines a sequence of numbers.
Find the next five terms in the sequence given that the initial or seed
values are t1 = 1 and t2 = 3.

Solution

t3 = 3t2 − t1 t4 = 3t3 − t2 t5 = 3t4 − t3
= 3(3) − 1 = 3(8) − 3 = 3(21) − 8
=8 = 21 = 55

t6 = 3t5 − t4 t7 = 3t6 − t5
= 3(55) − 21 = 3(144) − 55
= 144 = 377

The next five terms are 8, 21, 55, 144, and 377.


Key Concepts

• Iteration occurs in many natural and mathematical processes. Iterative
processes can create fractals.

• A process that repeats itself can be illustrated using arrows and loops.

• A tree diagram can illustrate all the possible outcomes of a repeated process
involving two or more choices at each step.
• For recursive functions, the first step is calculated using initial or seed values,
then each successive term is calculated using the result of the preceding step.

Communicate Your Understanding

1. Describe how fractals have been used
to model the fern leaf shown on the
right.

2. Describe your daily routine as an iterative process.

Practise 2. The diagram below illustrates the carbon-
oxygen cycle. Draw arrows to show the
A gains and losses of carbon dioxide.
1. Which of the following involve an iterative
process? The Carbon-Oxygen Cycle
World atmospheric
CO2
a) the cycle of a washing machine carbon dioxide supply
CO
CO2
b) your reflections in two mirrors that face CO2 CO2 CO2 CO2 CO2 CO2 CO2 Industrial
activity (H20)

each other Soil
Animal Plant Photosynthesis Combustion
respir– respir–
respir– ation
c) the placement of the dials on an ation
ation

automobile dashboard Surface
exchange Ocean Land
d) a chart of sunrise and sunset times Photo– CO2 in seawater Peat
synthesis Molten
e) substituting a value for the variable in Respiration
Fossil
rocks
Plankton Marine Coal
a quadratic equation Animals
fuels

f) a tessellating pattern, such as paving Sediments Organic sediments
(hydrocarbons)
Petroleum
Natural gas
bricks that fit together without gaps
CaCO2 in rock (calcium carbonate)


3. Draw a tree diagram representing the 8. Application In 1904 the Swedish
playoffs of eight players in a singles tennis mathematician Helge von Koch
tournament. The tree diagram should show (1870−1924) developed a fractal based on
the winner of each game continuing to the an equilateral triangle. Using either paper
next round until a champion is decided. and pencil or a drawing program, such as
The Geometer’s Sketchpad, draw a large
Apply, Solve, Communicate equilateral triangle and trisect each side.
Replace each middle segment with two
B segments the same length as the middle
4. Draw a diagram to represent the food chain. segment, forming an equilateral triangle
with the base removed, as shown below.
5. Communication Describe how the tracing
of heartbeats on a cardiac monitor or
electrocardiogram is iterative. Illustrate
your description with a sketch.

6. In the first investigation, on page 6, you
developed a sort algorithm in which new
data were compared to the lowest ranked Repeat the process of trisection and
birthday until the latest birthday was found. replacement on each of the 12 smaller
Then, the second latest, third latest, and so segments. If you are using a computer
on were found in the same manner. program, continue this process for at least
a) Write a sort algorithm in which this two more iterations.
process is reversed so that the highest a) How many segments are there after
ranked item is found instead of the three iterations?
lowest. b) How many segments are there after
b) Write a sort algorithm in which you four iterations?
compare the first two data, then the c) What pattern can you use to predict the
second and third, then the third and number of segments after n iterations?
fourth, and so on, interchanging the
order of the data in any pair where the 9. The first two terms of a sequence are given
second item is ranked higher. as t1 = 2 and t2 = 4. The recursion formula is
tn = (tn−1 ) 2 − 3tn−2. Determine the next four
7. Application Sierpinski’s carpet is similar to terms in the sequence.
Sierpinski’s triangle, except that it begins
with a square. This square is divided into
nine smaller squares and the middle one is
shaded. Use paper and pencil or a drawing
program to construct Sierpinski’s carpet to
at least three stages. Predict what fraction of
the original square will be shaded after n
stages.


10. Each of the following fractal trees has a a) Select a starting point near the centre of
different algorithm. Assume that each tree a sheet of grid paper. Assign the numbers
begins with a segment 1 unit long. 1 to 4 to the directions north, south,
a) Illustrate or describe the algorithm for east, or west in any order. Now, generate
each fractal tree. random whole numbers between 1 and 4
using a die, coin, or graphing calculator.
i) ii)
Draw successive line segments one unit
long in the directions corresponding to
the random numbers until you reach an
edge of the paper.
b) How would a random walk be affected if
it were self-avoiding, that is, not allowed
iii) to intersect itself? Repeat part a) using
this extra rule.
c) Design your own random walk with a
different set of rules. After completing
the walk, trade drawings with a classmate
and see if you can deduce the rules for
each other’s walk.

www.mcgrawhill.ca/links/MDM12

To learn more about chaos theory, visit the above
web site and follow the links. Describe an aspect
of chaos theory that interests you.

b) What is the total length of the branches
in each tree?
c) An interesting shape on a fractal tree is 12. Use the given values for t1 to ﬁnd the
a spiral, which you can trace by tracing successive terms of the following recursive
a branch to its extremity. Are all spirals formulas. Continue until a pattern appears.
within a tree self-similar? Describe the pattern and make a prediction
d) Write your own set of rules for a fractal for the value of the nth term.
tree. Draw the tree using paper and a) tn = 2−tn−1; t1 = 0
pencil or a drawing program. b) tn = ͙tn − 1; t1 = 256
ෆ
1
11. Inquiry/Problem Solving Related to fractals c) tn = ᎏ ; t1 = 2
is the mathematical study of chaos, in which tn − 1
no accurate prediction of an outcome can
be made. A random walk can illustrate such
“chaotic” outcomes.


ACHIEVEMENT CHECK 15. Inquiry/Problem Solving The infinite series
S = cos θ + cos2 θ + cos3 θ + … can be
Knowledge/ Thinking/Inquiry/
Understanding Problem Solving
Communication Application illustrated by drawing a circle centred at the
origin, with radius of 1. Draw an angle θ
13. a) Given t1 = 1, list the next five terms for
and, on the x-axis, label the point (cos θ, 0)
the recursion formula tn = n × tn-1. as P1. Draw a new circle, centred at P1 ,
b) In this sequence, tk is a factorial number, with radius of cos θ. Continue this iterative
often written as k! Show that process. Predict the length of the line
tk = k! segment defined by the infinite series
= k(k − 1)(k − 2)…(2)(1). S = cos θ + cos2 θ + cos3 θ + ….
c) Explain what 8! means. Evaluate 8!
16. Communication Music can be written using
d) Explain why factorial numbers can be
fractal patterns. Look up this type of music
considered an iterative process. in a library or on the Internet. What
e) Note that characteristics does fractal music have?
(25 )(5!)
= (2 × 2 × 2 × 2 × 2)(5 × 4 × 3 × 2 × 1) 17. Computers use binary (base 2) code to
= (2 × 5)(2 × 4)(2 × 3)(2 × 2)(2 × 1) represent numbers as a series of ones and
= 10 × 8 × 6 × 4 × 2 zeros.
which is the product of the first five Base 10 Binary
even positive integers. Write a formula 0 0
for the product of the first n even 1 1
positive integers. Explain why your 2 10
formula is correct. 3 11
10!
f) Write ᎏ as a product of 4 100
(25)(5!)
Ӈ Ӈ
consecutive odd integers.
g) Write a factorial formula for the a) Describe an algorithm for converting
product of integers from base 10 to binary.
i) the first six odd positive integers b) Write each of the following numbers
in binary.
ii) the first ten odd positive integers
i) 16 ii) 21
iii) the first n odd positive integers
iii) 37 iv) 130
c) Convert the following binary numbers
C to base 10.
14. Inquiry/Problem Solving Recycling can be i) 1010 ii) 100000
considered an iterative process. Research iii) 111010 iv) 111111111
the recycling process for a material such as
newspaper, aluminum, or glass and illustrate
the process with an arrow diagram.


1.2 Data Management Software

I N V E S T I G AT E & I N Q U I R E : S o f t w a r e To o l s

1. List every computer program you can think of that can be used to
manage data.
2. Sort the programs into categories, such as word-processors and spreadsheets.
3. Indicate the types of data each category of software would be best suited
to handle.
4. List the advantages and disadvantages of each category of software.
5. Decide which of the programs on your list would be best for storing and
accessing the lists you have just made.

Most office and business software manage data of some kind. Schedulers and
organizers manage lists of appointments and contacts. E-mail programs allow
you to store, access, and sort your messages. Word-processors help you manage
your documents and often have sort and outline functions for organizing data
within a document. Although designed primarily for managing financial
information, spreadsheets can perform calculations related to the management
and analysis of a wide variety of data. Most of these programs can easily
transfer data to other applications.

Database programs, such as Microsoft® Access and Corel®
Paradox®, are powerful tools for handling large numbers of
records. These programs produce relational databases,
ones in which different sets of records can be linked and
sorted in complex ways based on the data contained in the
records. For example, many organizations use a relational
database to generate a monthly mailing of reminder letters
to people whose memberships are about to expire. However,
these complex relational database programs are difficult to
learn and can be frustrating to use until you are thoroughly
familiar with how they work. Partly for this reason, there
are thousands of simpler database programs designed for
specific types of data, such as book indexes or family trees.

Of particular interest for this course are programs that can
do statistical analysis of data. Such programs range from
modest but useful freeware to major data-analysis packages
costing thousands of dollars. The more commonly used
programs include MINITAB™, SAS, and SST (Statistical


Software Tools). To demonstrate statistical software, some examples in this book have
alternative solutions that use FathomTM, a statistical software package specifically
designed for use in schools.

Data management programs can perform complex calculations and link, search, sort, and
graph data. The examples in this section use a spreadsheet to illustrate these operations.
A spreadsheet is software that arranges data in rows and columns. For basic spreadsheet
instructions, please refer to the spreadsheet section of Appendix B. If you are not already
familiar with spreadsheets, you may find it helpful to try each of the examples yourself
before answering the Practise questions at the end of the section. The two most
commonly used spreadsheets are Corel Quattro® Pro and Microsoft Excel.

Formulas and Functions
A formula entered in a spreadsheet cell can perform calculations based on values or
formulas contained in other cells. Formulas retrieve data from other cells by using
cell references to indicate the rows and columns where the data are located. In the
formulas C2*0.05 and D5+E5, each reference is to an individual cell. In both
Microsoft® Excel and Corel® Quattro® Pro, it is good practice to begin a formula
with an equals sign. Although not always necessary, the equals sign ensures that a
formula is calculated rather than being interpreted as text.

Built-in formulas are called functions. Many functions, such as the SUM function or MAX
function use references to a range of cells. In Corel® Quattro Pro, precede a function
with an @ symbol. For example, to find the total of the values in cells A2 through A6, you
would enter
Corel® Quattro Pro: @SUM(A2..A6) Microsoft® Excel: SUM(A2:A6)
Similarly, to find the total for a block of cells from A2 through B6, enter
Corel® Quattro Pro: @SUM(A2..B6) Microsoft® Excel: SUM(A2:B6)

A list of formulas is available in the Insert menu by selecting Function…. You may select
from a list of functions in categories such as Financial, Math & Trig, and Database.

Example 1 Using Formulas and Functions

The first three columns of the spreadsheet on
the right list a student’s marks on tests and
assignments for the first half of a course.
Determine the percent mark for each test or
assignment and calculate an overall midterm mark.

1.2 Data Management Software • MHR 15

Solution

In column D, enter formulas with cell referencing to find the percent for
each individual mark. For example, in cell D2, you could use the formula
B2/C2*100.

Use the SUM function to find totals for columns B and C, and then
convert to percent in cell D12 to find the midterm mark.

Relative and Absolute Cell References
Spreadsheets automatically adjust cell references whenever cells are copied,
moved, or sorted. For example, if you copy a SUM function, used to calculate the
sum of cells A3 to E3, from cell F3 to cell F4, the spreadsheet will change the cell
references in the copy to A4 and E4. Thus, the value in cell F4 will be the sum of
those in cells A4 to E4, rather than being the same as the value in F3.

Because the cell references are relative to a location, this automatic adjustment
is known as relative cell referencing. If the formula references need to be kept
exactly as written, use absolute cell referencing. Enter dollar signs before the
row and column references to block automatic adjustment of the references.

Fill and Series Features
When a formula or function is to be copied to several adjoining cells, as for the
percent calculations in Example 1, you can use the Fill feature instead of Copy.
Click once on the cell to be copied, then click and drag across or down through
the range of cells into which the formula is to be copied.

To create a sequence of numbers, enter the first two values in adjoining cells,
then select Edit/Fill/Series to continue the sequence.


Example 2 Using the Fill Feature

The relationship between Celsius and Fahrenheit temperatures is given by
the formula Fahrenheit = 1.8 × Celsius + 32. Use a spreadsheet to produce
a conversion table for temperatures from 1°C to 15ºC.

Solution
Enter 1 into cell E2 and 2 into cell E3.
Use the Fill feature to put the numbers 3
through 15 into cells E4 to E16. Enter the
conversion formula E2*1.8+32 into cell
F2. Then, use the Fill feature to copy the
formula into cells F3 through F16. Note
that the values in these cells show that the
cell references in the formulas did change
when copied. These changes are an
example of relative cell referencing.

Charting
Another important feature of spreadsheets is the ability to display numerical
data in the form of charts or graphs, thereby making the data easier to
understand. The ﬁrst step is to select the range of cells to be graphed. For
non-adjoining ﬁelds, hold down the Ctrl key while highlighting the cells.
Then, use the Chart feature to specify how you want the chart to appear.

You can produce legends and a title for your graph as well as labels for the
axes. Various two- and three-dimensional versions of bar, line, and circle
graphs are available in the menus.

Example 3 Charting

The results and standings of a hockey
league are listed in this spreadsheet.
Produce a two-dimensional bar chart
using the TEAM and POINTS columns.


Solution

Holding down the Ctrl key, highlight
cells A1 to A7 and then G1 to G7.
Use the Chart feature and follow the
on-screen instructions to customize
your graph. You will see a version of
the bar graph as shown here.

Sorting
Spreadsheets have the capability to sort data alphabetically, numerically, by date,
and so on. The sort can use multiple criteria in sequence. Cell references will
adjust to the new locations of the sorted data. To sort, select the range of cells
to be sorted. Then, use the Sort feature.

Select the criteria under which the data are to be sorted. A sort may be made in
ascending or descending order based on the data in any given column. A sort
with multiple criteria can include a primary sort, a secondary sort within it, and
a tertiary sort within the secondary sort.

Example 4 Sorting

Rank the hockey teams in Example 3, counting points first (in descending
order), then wins (in descending order), and finally losses (in ascending order).

Solution

When you select the Sort feature, the pop-up window asks if there is a header
row. Confirming that there is a header row excludes the column headings from
the sort so that they are left in place. Next, set up a three-stage sort:
• a primary sort in descending order, using the points column
• then, a secondary sort in descending order, using the wins column
• finally, a tertiary sort in ascending order, using the losses column


Searching
To search for data in individual cells, select Find and Replace.

Then, in the dialogue box, enter the data and the criteria under which you are
searching. You have the option to search or to search and replace.

A filtered search allows you to search for rows containing the data for which
you are searching.

Arrows will appear at the top of each column containing data. Clicking on an
arrow opens a pull-down menu where you can select the data you wish to find.
The filter will then display only the rows containing these data. You can filter
for a specific value or select custom… to use criteria such as greater than, begins
with, and does not contain. To specify multiple criteria, click the And or Or
options. You can set different filter criteria for each column.

Example 5 Filtered Search

In the hockey-league spreadsheet from Example 3, use a filtered search to list
only those teams with fewer than 16 points.


Solution

In Microsoft® Excel, select
Data/Filter/Autofilter to begin the
filter process. Click on the arrow in
the POINTS column and select
custom… In the dialogue window,
select is less than and key in 16.

In Corel® Quattro® Pro, you use
Tools/Quickfilter/custom….

Now, the filter shows only the rows
representing teams with fewer than
16 points.

Adding and Referencing Worksheets
To add worksheets within your spreadsheet file, click on one of the sheet tabs
at the bottom of the data area. You can enter data onto the additional
worksheet using any of the methods described above or you can copy and Project
Prep
paste data from the first worksheet or from another file.

To reference data from cells in another worksheet, preface the cell reference The calculation,
with the worksheet number for the cells. sorting, and
charting
Such references allow data entered in sheet A or sheet 1 to be manipulated in capabilities of
another sheet without changing the values or order of the original data. Data spreadsheets could
edited in the original sheet will be automatically updated in the other sheets be particularly
that refer to it. Any sort performed in the original sheet will carry through to useful for your
any references in other sheets, but any other data in the secondary sheets will tools for data
not be affected. Therefore, it is usually best to either reference all the data in management
the secondary sheets or to sort the data only in the secondary sheets. project.


Example 6 Sheet Referencing

Reference the goals for (GF) and goals against (GA) for the hockey teams
in Example 3 on a separate sheet and rank the teams by their goals scored.

Solution

Sheet 2 needs only the data in the columns titled GF and GA
in sheet 1. Notice that cell C2 contains a cell reference to sheet 1.
This reference ensures the data in cell F2 of sheet 1 will carry
through to cell C2 of sheet 2 even if the data in sheet 1 is edited.
Although the referenced and sorted data on sheet 2 appear
as shown, the order of the teams on sheet 1 is unchanged.

Key Concepts

• Thousands of computer programs are available for managing data. These
programs range from general-purpose software, such as word-processors and
spreadsheets, to highly specialized applications for specific types of data.
• A spreadsheet is a software application that is used to enter, display, and
manipulate data in rows and columns. Spreadsheet formulas perform
calculations based on values or formulas contained in other cells.
• Spreadsheets normally use relative cell referencing, which automatically
adjusts cell references whenever cells are copied, moved, or sorted. Absolute
cell referencing keeps formula references exactly as written.
• Spreadsheets can produce a wide variety of charts and perform sophisticated
sorts and searches of data.
• You can add additional worksheets to a file and reference these sheets to cells
in another sheet.


1. Explain how you could use a word-processor as a data management tool.
2. Describe the advantages and drawbacks of relational database programs.
3. Explain what software you would choose if you wanted to determine
whether there was a relationship between class size and subject in your
school. Would you choose different software if you were going to look at
class sizes in all the schools in Ontario?
4. Give an example of a situation requiring relative cell referencing and one
requiring absolute cell referencing.
5. Briefly describe three advantages that spreadsheets have over hand-written
tables for storing and manipulating data.


Practise Apply, Solve, Communicate
A B
1. Application Set up a spreadsheet page in which 3. Set up a spreadsheet page that converts
you have entered the following lists of data. angles in degrees to radians using the
For the appropriate functions, look under the formula Radians = π×Degrees/180, for
Statistical category in the Function list. angles from 0° to 360° in steps of 5°. Use
Student marks: the series capabilities to build the data in the
65, 88, 56, 76, 74, 99, 43, 56, 72, 81, 80, Degrees column. Use π as defined by the
30, 92 spreadsheet. Calculations should be rounded
to the nearest hundredth.
Dentist appointment times in minutes:
45, 30, 40, 32, 60, 38, 41, 45, 40, 45 4. The first set of data below represents the
a) Sort each set of data from smallest to number of sales of three brands of CD
greatest. players at two branches of Mad Dog Music.
b) Calculate the mean (average) value for Enter the data into a spreadsheet using two
each set of data. rows and three columns.
c) Determine the median (middle) value Branch Brand A Brand B Brand C
for each set of data. Store P 12 4 8
Store Q 9 15 6
d) Determine the mode (most frequent) BRAND A BRAND B BRAND C
value for each set of data. The second set of data Brand Price
represents the prices for A $102
2. Using the formula features of the
these CD players. Enter B $89
spreadsheet available in your school, write
the data using one C $145
a formula for each of the following:
column into a second
a) the sum of the numbers stored in cells sheet of the same spreadsheet workbook.
A1 to A9 Set up a third sheet of the spreadsheet
b) the largest number stored in cells F3 to K3 workbook to reference the first two sets of
c) the smallest number in the block from data and calculate the total revenue from CD
A1 to K4 player sales at each Mad Dog Music store.
d) the sum of the cells A2, B5, C7, and D9 5. Application In section 1.1, question 12, you
e) the mean, median, and mode of the predicted the value of the nth term of the
numbers stored in the cells F5 to M5 recursion formulas listed below. Verify your
f) the square root of the number in cell A3 predictions by using a spreadsheet to
g) the cube of the number in cell B6
calculate the first ten terms for each
formula.
h) the number in cell D2 rounded off to
a) tn = 2−tn–1; t1 = 0
four decimal places
b) tn = ͙tn − 1; t1 = 256
ෆ
i) the number of cells between cells D3 and
1
M9 that contain data c) tn = ᎏ ; t1 = 2
tn − 1
j) the product of the values in cells A1, B3,
and C5 to C10
k) the value of π


6. a) Enter the data shown in the table below f) Perform a search in the second sheet to
into a spreadsheet and set up a second find the cereals containing less than 1 g of
sheet with relative cell references to the fat and more than 1.5 g of fibre. Make a
Name, Fat, and Fibre cells in the original three-dimensional bar graph of the results.
sheet.
C
Nutritional Content of 14 Breakfast Cereals
7. In section 1.1, question 10, you described
(amounts in grams)
the algorithm used to draw each fractal tree
Name Protein Fat Sugars Starch Fibre Other TOTALS
below. Assuming the initial segment is 4 cm
Alphabits 2.4 1.1 12.0 12.0 0.9 1.6 in each tree, use a spreadsheet to determine
Bran Flakes 4.4 1.2 6.3 4.7 11.0 2.4 the total length of a spiral in each tree,
Cheerios 4.0 2.3 0.8 18.7 2.2 2.0 calculated to 12 iterative stages.
Crispix 2.2 0.3 3.2 22.0 0.5 1.8 a)
Froot Loops 1.3 0.8 14.0 12.0 0.5 1.4
Frosted Flakes 1.4 0.2 12.0 15.0 0.5 0.9
Just Right 2.2 0.8 6.6 17.0 1.4 2.0
Lucky Charms 2.1 1.0 13.0 11.0 1.4 1.5
Nuts ’n Crunch 2.3 1.6 7.1 16.5 0.7 1.8
Rice Krispies 2.1 0.4 2.9 22.0 0.3 2.3
Shreddies 2.9 0.6 5.0 16.0 3.5 2.0 b)
Special K 5.1 0.4 2.5 20.0 0.4 1.6
Sugar Crisp 2.0 0.7 14.0 11.0 1.1 1.2
Trix 0.9 1.6 13.0 12.0 1.1 1.4
AVERAGES
MAXIMUM
MINIMUM

b) On the first sheet, calculate the values for
the TOTALS column and AVERAGES
row.
c) Determine the maximum and minimum
values in each column.
d) Rank the cereals using fibre content in
decreasing order as a primary criterion, 8. Communication Describe how to lock column
protein content in decreasing order as a and row headings in your spreadsheet
secondary criterion, and sugar content in software so that they remain visible when
increasing order as a tertiary criterion. you scroll through a spreadsheet.
e) Make three circle graphs or pie charts:
9. Inquiry/Problem Solving Outline a
one for the averages row in part b), one
for the cereal at the top of the list in part spreadsheet algorithm to calculate
d), and one for the cereal at the bottom n × (n − 1) × (n − 2) … 3 × 2 × 1 for any
of the list in part d). natural number n without using the built-in
factorial function.


TE C H N OL OG Y E X T EN S I O N

Introduction to Fathom™
Fathom™ is a statistics software package that offers a variety of powerful data-
analysis tools in an easy-to-use format. This section introduces the most basic
features of Fathom™: entering, displaying, sorting, and ﬁltering data. A
Appendix B includes
complete guide is available on the Fathom™ CD. The real power of this
details on all the
software will be demonstrated in later chapters with examples that apply its Fathom™ functions
sophisticated tools to statistical analysis and simulations. used in this text.
When you enter data into Fathom™, it creates a collection, an object that
contains the data. Fathom™ can then use the data from the collection to
produce other objects, such as a graph, table, or statistical test. These secondary
objects display and analyse the data from the collection, but they do not actually
contain the data themselves. If you delete a graph, table, or statistical test, the
data still remains in the collection.

Fathom™ displays a collection as a rectangular window
with gold balls in it. The gold balls of the collection
represent the original or “raw” data. Each of the gold balls
represents a case. Each case in a collection can have a
number of attributes. For example the cases in a collection
of medical records could have attributes such as the
patient’s name, age, sex, height, weight, blood pressure, and
so on. There are two basic types of attributes, categorical
(such as male/female) and continuous (such as height or
mass). The case table feature displays the cases in a
collection in a format similar to a spreadsheet, with a row
for each case and a column for each attribute. You can add,
modify, and delete cases using a case table.

Example 1 Tables and Graphs

a) Set up a collection for the hockey league standings from section 1.2,
Example 3 on page 17.
b) Graph the Team and Points attributes.

Solution

a) To enter the data, start Fathom™ and drag
the case table icon from the menu bar
down onto the work area.


Click on the attribute <new>, type the heading Team, and press Enter. Fathom™
will automatically create a blank cell for data under the heading and start a new
column to the right of the ﬁrst. Enter the heading Wins at the top of the new
column, and continue this process to enter the rest of the headings. You can type
entries into the cells under the headings in much the same way as you would enter
data into the cells of a spreadsheet.

Note that Fathom™ has stored your data as Collection 1, which will remain intact
even if you delete the case table used to enter the data. To give the collection a
more descriptive name, double-click on Collection 1 and type in HockeyStats.

b) Drag the graph icon onto the work area. Now, drag the Team attribute from
the case table to the x-axis of the graph and the Points attribute to the y-axis of the
graph.

➔

Your graph should look like this:

Technology Extension: Introduction to Fathom™ • MHR 25

Fathom™ can easily sort or filter data using the various attributes.

Example 2 Sorting and Filtering

a) Rank the hockey teams in Example 1 by points first, then by wins if two teams
have the same number of points, and finally by losses if two teams have the
same number of points and wins.
b) List only those teams with fewer than 16 points.
c) Set up a separate table showing only the goals for (GF) and goals against (GA)
data for the teams and rank the teams by their goals scored.

Solution

a) To Sort the data, right-click on the Points attribute and choose Sort Descending.
Fathom™ will list the team with the most points first, with the others
following in descending order by their point totals. To set the secondary sort,
right-click on the Wins attribute and choose Sort Descending. Similarly, right-
click on the Losses attribute and choose Sort Ascending for the final sort, giving
the result below.

b) To Filter the data, from the Data menu, choose Add Filter. Click on the
plus sign beside Attributes.
Now, double-click on the Points attribute, choose the less-than button , and
type 16. Click the Apply button and then OK.

The results should look like this:


The Filter is listed at the bottom as Points < 16.

c) Click on HockeyStats, and then drag a new table onto the work area.
Click on the Wins attribute. From the Display menu, choose Hide
Attribute. Use the same method to hide the Losses, Ties, and Points
attributes. Right-click the GF attribute and use Sort Descending to
rank the teams.

1. Enter the data from Example 1 into Fathom™. Use the built-in For details on functions
functions in Fathom™ to ﬁnd the following. in Fathom™, see the
Fathom™ section of
a) the mean of goals against (GA)
Appendix B or
b) the largest value of goals for (GF) consult the Fathom™
c) the smallest value of GF Help screen or manual.
d) the sum of GA
e) the sum of GA and GF for each case

2. a) Set up a new collection with the following student marks:
65, 88, 56, 76, 74, 99, 43, 56, 72, 81, 80, 30, 92
b) Sort the marks from lowest to highest.
c) Calculate the mean mark.
d) Determine the median (middle) mark.

3. Explain how you would create a graph of class size versus subjects in
your school using Fathom™.

4. Brieﬂy compare the advantages and disadvantages
of using Fathom™ and spreadsheets for
storing and manipulating data.

For more examples, data, and information on how
to use Fathom™, visit the above web site and
follow the links.

Technology Extension: Introduction to Fathom™ • MHR 27

1.3 Databases

A database is an organized
store of records. Databases
may contain information
about almost any subject—
incomes, shopping habits,
demographics, features of
cars, and so on.

I N V E S T I G AT E & I N Q U I R E : D a t a b a s e s i n a L i b r a r y

In your school or local public library, log on to the library catalogue.

1. Describe the types of ﬁelds under which a search can be conducted
(e.g., subject).
2. Conduct a search for a speciﬁc topic of your choice.
3. Describe the results of your search. How is the information
presented to the user?

I N V E S T I G AT E & I N Q U I R E : T h e E - S TAT D a t a b a s e

1. Connect to the Statistics Canada web site and go to the E-STAT database.
Your school may have a direct link to this database. If not, you can follow
the Web Connection links shown here. You may need to get a password
from your teacher to log in.
2. Locate the database showing the educational attainment data for Canada
by following these steps:
a) Click on Data.
b) Under the heading People, click on
Education.
c) Click on Educational Attainment, To connect to E-STAT visit the above web site and
then under the heading Census follow the links.
databases, select Educational
Attainment again.
d) Select Education, Mobility and
Migration for the latest census.


3. Scroll down to the heading University, pop. 15 years and over by highest level of
schooling, hold down the Ctrl key, and select all four subcategories under this
heading. View the data in each of the following formats:
a) table b) bar graph c) map
4. Describe how the data are presented in each instance. What are the
advantages and disadvantages of each format? Which format do you
think is the most effective for displaying this data? Explain why.
5. Compare the data for the different provinces and territories. What
conclusions could you draw from this data?

A database record is a set of data that is treated as a unit. A record is usually
divided into fields that are reserved for specific types of information. For
example, the record for each person in a telephone book has four fields: last
name, first name or initial, address, and telephone number. This database is
sorted in alphabetical order using the data in the first two fields. You search this
database by finding the page with the initial letters of a person’s name and then
simply reading down the list.

A music store will likely keep its inventory records on a computerized database.
The record for each different CD could have fields for information, such as title,
artist, publisher, music type, price, number in stock, and a product code (for
example, the bar code number). The computer can search such databases for
matches in any of the data fields. The staff of the music store would be able to
quickly check if a particular CD was in stock and tell the customer the price and
whether the store had any other CDs by the same artist.

Databases in a Library
A library catalogue is a database. In the past, library databases were accessed
through a card catalogue. Most libraries are now computerized, with books listed
by title, author, publisher, subject, a Dewey Decimal or Library of Congress
catalogue number, and an international standard book number (ISBN). Records
can be sorted and searched using the information in any of the fields.

Such catalogues are examples of a well-organized database because they Project
are easy to access using keywords and searches in multiple fields, many of Prep
which are cross-referenced. Often, school libraries are linked to other
Skills in researching
libraries. Students have access to a variety of print and online databases in
library and on-line
the library. One powerful online database is Electric Library Canada, a
databases will help you
database of books, newspapers, magazines, and television and radio
find the information
transcripts. Your school probably has access to it or a similar library
needed for your tools
database. Your local public library may also have online access to Electric
for data management
Library Canada.
project.

1.3 Databases • MHR 29

Statistics Canada
Statistics Canada is the federal government department responsible for
collecting, summarizing, analysing, and storing data relevant to Canadian
demographics, education, health, and so on. Statistics Canada maintains a
number of large databases using data collected from a variety of sources
including its own research and a nation-wide census. One such database is
CANSIM II (the updated version of the Canadian Socio-economic Information
Management System), which proﬁles the Canadian people, economy, and
industries. Although Statistics Canada charges a fee for access to some of its
data, a variety of CANSIM II data is available to the public for free on Statistics
Canada’s web site.

Statistics Canada also has a free educational database, called E-STAT. It gives Data in Action
access to many of Statistics Canada’s extensive, well-organized databases, By law, Statistics
including CANSIM II. E-STAT can display data in a variety of formats and Canada is required
allows students to download data into a spreadsheet or statistical software to conduct a census
program. of Canada’s
population and
agriculture every ﬁve
years. For the 2001
census, Statistics
Canada needed
about 37 000 people
to distribute the
questionnaires.
Entering the data
from the
approximately
13.2 million
questionnaires will
take about 5 billion
keystrokes.


Key Concepts

• A database is an organized store of records. A well-organized database can be
easily accessed through searches in multiple fields that are cross-referenced.

• Although most databases are computerized, many are available in print form.


1. For a typical textbook, describe how the table of contents and the index are
sorted. Why are they sorted differently?

2. Describe the steps you need to take in order to access the 1860−61 census
results through E-STAT.

Practise 3. a) Describe how you would locate a
database showing the ethnic makeup of
A your municipality. List several possible
1. Which of the following would be considered sources.
databases? Explain your reasoning. b) If you have Internet access, log onto
a) a dictionary E-STAT and go to the data on ethnic
b) stock-market listings origins of people in large urban centres:
c) a catalogue of automobile specifications i) Select Data on the Table of Contents
and prices page.
d) credit card records of customers’ ii) Select Population and Demography.
spending habits iii) Under Census, select Ethnic Origin.
e) an essay on Shakespeare’s Macbeth
iv) Select Ethnic Origin and Visible
f) a teacher’s mark book Minorities for the latest census in
g) the Guinness World Records book large urban centres.
h) a list of books on your bookshelf v) Enter a postal code for an urban area
and select two or more ethnic origins
Apply, Solve, Communicate while holding down the Ctrl key.
B vi) View table, bar graph, and map in
2. Describe each field you would include in a turn and describe how the data are
database of presented in each instance.
a) a person’s CD collection c) Compare these results with the data you
b) a computer store’s software inventory
get if you leave the postal code section
line blank. What conclusions could you
c) a school’s textbook inventory
draw from the two sets of data?
d) the backgrounds of the students in a
school
e) a business’s employee records

1.3 Databases • MHR 31

4. Application 6. Application The Internet is a link between
a) Describe how you could find data to many databases. Search engines, such as
compare employment for males and Yahoo Canada, Lycos, Google, and Canoe,
females. List several possible sources. are large databases of web sites. Each search
engine organizes its database differently.
b) If you have Internet access, log onto
E-STAT and go to the data on a) Use three different search engines to
employment and work activity: conduct a search using the keyword
automobile. Describe how each search
i) Under the People heading, select
engine presents its data.
Labour.
b) Compare the results of searches with
ii) Under the Census databases heading,
three different search engines using the
select Salaries and Wages.
following keywords:
iii) Select Sources of Income (Latest
i) computer monitors
census, Provinces, Census Divisions,
Municipalities). ii) computer+monitors

iv) While holding down the Ctrl key, iii) computer or monitors
click on All persons with iv) “computer monitors”
employment income by work activity,
Males with employment income by 7. Use the Internet to check whether the map
pte
ha of VIA Rail routes at the start of this chapter
work activity, and Females with
C

r

employment income by work activity. is up-to-date. Are there still no trains that
m
P

r
oble
go from Montréal or Kingston right
v) Download this data as a spreadsheet
through to Windsor?
file. Record the path and file name
for the downloaded data. 8. Communication Log on to the Electric
c) Open the data file with a spreadsheet. Library Canada web site or a similar
You may have to convert the format to database available in your school library.
match your spreadsheet software. Use Enter your school’s username and password.
your spreadsheet to Perform a search for magazine articles,
i) calculate the percentage difference newspaper articles, and radio transcripts
between male and female about the “brain drain” or another issue of
employment interest to you. Describe the results of your
search. How many articles are listed? How
ii) display all fields as a bar graph
are the articles described? What other
5. Communication Go to the reference area of information is provided?
your school or local library and find a
published database in print form.
a) Briefly describe how the database is
organized.
b) Describe how to search the database.
c) Make a list of five books that are set up
as databases. Explain why they would be
considered databases.


1.4 Simulations

A simulation is an experiment,
model, or activity that imitates
real or hypothetical conditions.
The newspaper article shown here
describes how astrophysicists used
computers to simulate a collision
between Earth and a planet the
size of Mars, an event that would
be impossible to measure directly. The simulation showed that such a collision
could have caused both the formation of the moon and the rotation of Earth,
strengthening an astronomical theory put forward in the 1970s.

I N V E S T I G AT E & I N Q U I R E : Simulations

For each of the following, describe what is being simulated, the
advantages of using a simulation, and any drawbacks.

a) crash test dummies b) aircraft simulators
c) wind tunnels d) zero-gravity simulator
e) 3-D movies f) paint-ball games
g) movie stunt actors h) grow lights
i) architectural scale models

In some situations, especially those with many variables, it can be difﬁcult to
calculate an exact value for a quantity. In such cases, simulations often can provide
a good estimate. Simulations can also help verify theoretical calculations.

Example 1 Simulating a Multiple-Choice Test

When writing a multiple-choice test, you may have wondered “What are my
chances of passing just by guessing?” Suppose that you make random guesses on a
test with 20 questions, each having a choice of 5 answers. Intuitively, you would
assume that your mark will be somewhere around 4 out of 20 since there is a 1 in 5
chance of guessing right on each question. However, it is possible that you could
get any number of the questions right—anywhere from zero to a perfect score.
a) Devise a simulation for making guesses on the multiple-choice test.
b) Run the simulation 100 times and use the results to estimate the mark
you are likely to get, on average.
c) Would it be practical to run your simulation 1000 times or more?

1.4 Simulations • MHR 33

Solution 1 Using Pencil and Paper

a) Select any five cards from a deck of cards. Designate one of these cards to
represent guessing the correct answer on a question. Shuffle the five cards
and choose one at random. If it is the designated card, then you got the
first question right. If one of the other four cards is chosen, then you got
the question wrong.

Put the chosen card back with the others and repeat the process 19 times
to simulate answering the rest of the questions on the test. Keep track of
the number of right answers you obtained.

b) You could run 100 simulations by repeating the process in part a) over and
over. However, you would have to choose a card 2000 times, which would
be quite tedious. Instead, form a group with some of your classmates and
pool your results, so that each student has to run only 10 to 20 repetitions
of the simulation.

Make a table of the scores on the 100 simulated tests and calculate the
mean score. You will usually find that this average is fairly close to the
intuitive estimate of a score around 4 out of 20. However, a mean does not
tell the whole story. Tally up the number of times each score appears in
your table. Now, construct a bar graph showing the frequency for each
score. Your graph will look something like the one shown.
25

20

15

10

5

0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

This graph gives a much more detailed picture of the results you could
expect. Although 4 is the most likely score, there is also a good chance of
getting 2, 3, 5, or 6, but the chance of guessing all 20 questions correctly is
quite small.

c) Running the simulation 1000 times would require shuffling the five cards
and picking one 20 000 times—possible, but not practical.


Solution 2 Using a Graphing Calculator See Appendix B for
more details on how to
a) You can use random numbers as the basis for a simulation. If you
use the graphing
generate random integers from 1 to 5, you can have 1 correspond to calculator and
a correct guess and 2 through 5 correspond to wrong answers. software functions in
Solutions 2 to 4.
Use the STAT EDIT menu to view lists L1 and L2. Make sure both lists
are empty. Scroll to the top of L1 and enter the randInt function from
the MATH PRB menu. This function produces random integers.

Enter 1 for the lower limit, 5 for the upper limit, and 20 for the
number of trials. L1 will now contain 20 random integers between 1
and 5. Next, sort the list with the SortA function on the LIST OPS menu.
Press 2nd 1 to enter L1 into the sort function. When you return to L1,
the numbers in it will appear in ascending order. Now, you can easily
scroll down the list to determine how many correct answers there were
in this simulation.

b) The simplest way to simulate 100 tests is to repeat the procedure in
part a) and keep track of the results by entering the number of correct
answers in L2. Again, you may want to pool results with your classmates
to reduce the number of times you have to enter the same formula over
and over. If you know how to program your calculator, you can set it to
re-enter the formulas for you automatically. However, unless you are
experienced in programming the calculator, it will probably be faster for
you to just re-key the formulas.

Once you have the scores from 100 simulations in L2, calculate the
average using the mean function on the LIST MATH menu. To see which
scores occur most frequently, plot L2 using STAT PLOT.
i) Turn off all plots except Plot1.
ii) For Type, choose the bar-graph icon and enter L2 for Xlist.
Freq should be 1, the default value.
iii) Use ZOOM/ZoomStat to set the window for the data. Press
WINDOW to check the window settings. Set Xscl to 1 so that the
bars correspond to integers.
iv) Press GRAPH to display the bar graph.

c) It is possible to program the calculator to run a large number of
simulations automatically. However, the maximum list length on the
TI-83 Plus is 999, so you would have to use at least two lists to run
the simulation a 1000 times or more.


Solution 3 Using a Spreadsheet

a) Spreadsheets have built-in functions that you can use to generate and count
the random numbers for the simulation.

The RAND() function produces a random real number that is equal to or
greater than zero and less than one. The INT function rounds a real number
down to the nearest integer. Combine these functions to generate a random
integer between 1 and 5. In Microsoft® Excel,
you can use
Enter the formula INT(RAND( )*5)+1 or RANDBETWEEN(1,5) in A1 and copy RANDBETWEEN only if
it down to A20. Next, use the COUNTIF function to count the number of 1s you have the Analysis
in column A. Record this score in cell A22. Toolpak installed.

b) To run 100 simulations, copy A1:A22 into columns B through CV using the
Fill feature. Then, use the average function to find the mean score for the
100 simulated tests. Record this average in cell B23.

Next, use the COUNTIF function to find the number of times each possible
score occurs in cells A22 to CV22. Enter the headings SUMMARY, Score, and
Frequency in cells A25, A26, and A27, respectively. Then, enter 0 in cell B26
and highlight cells B26 through V26. Use the Fill feature to enter the
integers 0 through 20 in cells B26 through V26. In B27, enter the formula
for the number of zero scores; in C27, the number of 1s; in D27, the
number of 2s; and so on, finishing with V27 having the number of perfect


scores. Note that by using absolute cell referencing you can simply copy
the COUNTIF function from B27 to the other 20 cells.

Finally, use the Chart feature to plot frequency versus score. Highlight
cells A26 through V27, then select Insert/Chart/XY.

c) The method in part b) can easily handle 1000 simulations or more.

Solution 4 Using FathomTM

a) FathomTM also has built-in functions to generate
random numbers and count the scores in the
simulations.

Launch FathomTM and open a new document if
necessary. Drag a new collection box to the
document and rename it MCTest. Right-click on
the box and create 20 new cases.

Drag a case table to the work area. You should
see your 20 cases listed. Expand the table if you
cannot see them all on the screen.

Rename the <new> column Guess. Right-click on
Guess and select Edit Formula, Expand Functions,
then Random Numbers. Enter 1,5 into the
randomInteger() function and click OK to ﬁll the
Guess column with random integers between 1
and 5. Scroll down the column to see how many
correct guesses there are in this simulation.


b) You can run a new simulation just by pressing Ctrl-Y, which will fill the
Guess column with a new set of random numbers. Better still, you can set
FathomTM to automatically repeat the simulation 100 times automatically
and keep track of the number of correct guesses.

First, set up the count function. Right–click on the collection box and select
Inspect Collection. Select the Measures tab and rename the <new> column
Score. Then, right-click the column below Formula and select Edit Formula,
Functions, Statistical, then One Attribute. Select count, enter Guess = 1
between the brackets, and click OK to count the number of correct guesses
in your case table.

Click on the MCTest collection box. Now, select Analyse, Collect Measures
from the main menu bar, which creates a new collection box called Measures
from MCTest. Click on this box and drag a new case table to the document.
FathomTM will automatically run five simulations of the multiple-choice test
and show the results in this case table .

To simulate 100 tests, right-click on the Measures from MCTest collection box
and select Inspect Collection. Turn off the animation in order to speed up the
simulation. Change the number of measures to 100. Then, click on the
Collect More Measures button. You should now have 100 measures in the
case table for Measures from MCTest.

Next, use the mean function to find the average score for these simulations.
Go back to the Inspect Measures from MCTest collection box and change the
column heading <new> to Average. Right-click on Formula and select Edit
Formula, Functions, Statistical, then One Attribute. Select mean, enter Score
between the brackets, and select OK to display the mean mark on the 100
tests.

Finally, plot a histogram of the scores from the simulations. Drag the graph
icon onto the work area. Then, drag the Score column from the Measures
from MCTest case table to the horizontal axis of the graph. FathomTM then
automatically produces a dot plot of your data. To display a histogram
instead, simply click the menu in the upper right hand corner of the graph
and choose Histogram.


c) FathomTM can easily run this simulation 1000 times or more. All you have
to do is change the number of measures.

Key Concepts

• Simulations can be useful tools for estimating quantities that are difﬁcult to
calculate and for verifying theoretical calculations.

• A variety of simulation methods are available, ranging from simple manual
models to advanced technology that makes large-scale simulations feasible.


1. Make a table summarizing the pros and cons of the four simulation methods
used in Example 1.

2. A manufacturer of electric motors has a failure rate of 0.2% on one of its
products. A quality-control inspector needs to know the range of the number
of failures likely to occur in a batch of 1000 of these motors. Which tool
would you use to simulate this situation? Give reasons for your choice.


Practise 7. Inquiry/Problem Solving Consider a random
walk in which a coin toss determines the
A direction of each step. On the odd-
1. Write a graphing calculator formula for numbered tosses, walk one step north for
a) generating 100 random integers between heads and one step south for tails. On even-
1 and 25 numbered tosses, walk one step east for
heads and one step west for tails.
b) generating 24 random integers between
−20 and 20 a) Beginning at position (0, 0) on a
Cartesian graph, simulate this random
2. Write a spreadsheet formula for walk for 100 steps. Note the coordinates
a) generating 100 random numbers where you ﬁnish.
between 1 and 25 b) Repeat your simulation 10 times and
b) generating 100 random integers between record the results.
1 and 25 c) Use these results to formulate a hypothesis
c) generating 16 random integers between about the endpoints for this random walk.
−40 and 40 d) Change the rules of the random walk and
d) counting the number of entries that investigate the effect on the end points.
equal 42.5 in the range C10 to V40
ACHIEVEMENT CHECK
Apply, Solve, Communicate
Communication Application
B
3. Communication Identify two simulations you 8. a) Use technology to simulate rolling two
use in everyday life and list the advantages dice 100 times and record the sum of
of using each simulation. the two dice each time. Make a
histogram of the sums.
4. Describe three other manual methods you b) Which sum occurs most often? Explain
could use to simulate the multiple-choice why this sum is likely to occur more
test in Example 1. often than the other sums.
5. Communication c) Which sum or sums occur least often?
Explain this result.
a) Describe a calculation or mechanical
process you could use to produce d) Suppose three dice are rolled 100 times
random integers. and the sums are recorded. What sums
would you expect to be the most
b) Could you use a telephone book to
frequent and least frequent? Give
generate random numbers? Explain
reasons for your answers.
why or why not.

6. Application A brother and sister each tell the
truth two thirds of the time. The brother C
stated that he owned the car he was driving. 9. Communication Describe a quantity that
The sister said he was telling the truth. would be difﬁcult to calculate or to measure
Develop a simulation to show whether you in real life. Outline a simulation procedure
should believe them. you could use to determine this quantity.


1.5 Graph Theory

Graph theory is a branch of mathematics in which graphs or networks are used
to solve problems in many ﬁelds. Graph theory has many applications, such as
• setting examination timetables
• colouring maps
• modelling chemical compounds
• designing circuit boards
• building computer, communication, or transportation networks
• determining optimal paths

In graph theory, a graph is unlike the traditional Cartesian graph used for
graphing functions and relations. A graph (also known as a network) is a
collection of line segments and nodes. Mathematicians usually call the nodes
vertices and the line segments edges. Networks can illustrate the relationships
among a great variety of objects or sets.

This network is an illustration of the subway system in Toronto. In order to
show the connections between subway stations, this map is not to scale. In fact,
networks are rarely drawn to scale.

I N V E S T I G AT E & I N Q U I R E : Map Colouring

In each of the following diagrams the lines represent borders between
countries. Countries joined by a line segment are considered neighbours,
but countries joining at only a single point are not.

1. Determine the smallest number of colours needed for each map such
that all neighbouring countries have different colours.
a) b)

1.5 Graph Theory • MHR 41

c) d) e)

2. Make a conjecture regarding the maximum number of colours needed
to colour a map. Why do you think your conjecture is correct?

Although the above activity is based on maps, it is very mathematical.
It is about solving problems involving connectivity. Each country could
be represented as a node or vertex. Each border could be represented
by a segment or edge.

Example 1 Representing Maps With Networks

Represent each of the following maps with a network.
a) b)
A B
A
B
D C
D
C F E

Solution

a) Let A, B, C, and D be vertices representing countries A, B, C, and A
D, respectively. A shares a border with both B and D but not with
C, so A should be connected by edges to B and D only. Similarly, B D B
is connected to only A and C; C, to only B and D; and D, to only A
and C.
C

b) Let A, B, C, D, E, and F be vertices representing countries A, B, C, A B
D, E, and F, respectively. Note that the positions of the vertices are
not important, but their interconnections are. A shares borders with C
F
B, C, and F, but not with D or E. Connect A with edges to B, C,
and F only. Use the same process to draw the rest of the edges. E D


As components of networks, edges could represent connections such as roads,
wires, pipes, or air lanes, while vertices could represent cities, switches, airports,
computers, or pumping stations. The networks could be used to carry vehicles,
power, messages, ﬂuid, airplanes, and so on.
If two vertices are connected by an edge, they are considered to be adjacent. A B
In the network on the right, A and B are adjacent, as are B and C. A and C are
not adjacent.
The number of edges that begin or end at a vertex is called the degree of the
C
vertex. In the network, A has degree 1, B has degree 2, and C has degree 3. The
loop counts as both an edge beginning at C and an edge ending at C.
Any connected sequence of vertices is called a path. If the path begins and ends
at the same vertex, the path is called a circuit. A circuit is independent of the
starting point. Instead, the circuit depends on the route taken.

Example 2 Circuits

Determine if each path is a circuit.
a) A B b) A B c) A B

D C D C D C

Solution
a) Path: BC to CD to DA
Since this path begins at B and ends at A, it is not a circuit.

b) Path: BC to CD to DA to AB
This path begins at B and ends at B, so it is a circuit.

c) Path: CA to AB to BC to CD to DA
Since this path begins at C and ends at A, it is not a circuit.

A network is connected if and only if there is at least one path connecting each pair of
vertices. A complete network is a network with an edge between every pair of vertices.

Connected but not complete: Not Connected and complete: All vertices Neither connected nor complete:
all vertices are joined directly. are joined to each other by edges. Not all vertices are joined.


In a traceable network all the vertices are connected to at least one other
vertex and all the edges can be travelled exactly once in a continuous path.
B B P P

A C A C S Q S Q

D D R R
Traceable: All vertices are connected to at least Non-traceable: No continuous path can
one other vertex, and the path from A to B to C travel all the edges only once.
to D to A to C includes all the edges without
repeating any of them.

Example 3 The Seven Bridges of Koenigsberg

The eighteenth-century German town of Koenigsberg (now the Russian
city of Kaliningrad) was situated on two islands and the banks of the
Pregel River. Koenigsberg had seven bridges as shown in the map.

People of the town believed—but could not prove—that it was
impossible to tour the town, crossing each bridge exactly once,
regardless of where the tour started or ﬁnished. Were they right?

Solution

Reduce the map to a simple network of vertices and edges. Let vertices A
A and C represent the mainland, with B and D representing the islands.
Each edge represents a bridge joining two parts of the town.
B D

C
If, for example, you begin at vertex D, you will
Begin Pass through
leave and eventually return but, because D has and leave
a degree of 3, you will have to leave again. Return Return and end
D D

Leave again

Conversely, if you begin elsewhere, you will pass through vertex D at some
point, entering by one edge and leaving by another. But, because D has
degree 3, you must return in order to trace the third edge and, therefore,


must end at D. So, your path must either begin or end at vertex D. Because
all the vertices are of odd degree, the same argument applies to all the other
vertices. Since you cannot begin or end at more than two vertices, the network
is non-traceable. Therefore, it is indeed impossible to traverse all the town’s
bridges without crossing one twice.

Leonhard Euler developed this proof of Example 3 in 1735. He laid the foundations
for the branch of mathematics now called graph theory. Among other discoveries,
Euler found the following general conditions about the traceability of networks.
• A network is traceable if it has only vertices of even degree (even vertices)
or exactly two vertices of odd degree (odd vertices).

• If the network has two vertices of odd degree, the tracing path must begin
at one vertex of odd degree and end at the other vertex of odd degree.

Example 4 Traceability and Degree

For each of the following networks,
a) list the number of vertices with odd degree and with even degree
b) determine if the network is traceable
i) ii) iii) iv)

Solution
i) a) 3 even vertices ii) a) 0 even vertices iii) a) 3 even vertices iv) a) 1 even vertex
0 odd vertices 4 odd vertices 2 odd vertices 4 odd vertices
b) traceable b) non-traceable b) traceable b) non-traceable

If it is possible for a network to be drawn on a two-dimensional surface so
that the edges do not cross anywhere except at vertices, it is planar.

Example 5 Planar Networks

Determine whether each of the following networks is planar.
a) b) c) d) e)


Solution

a) Planar

b) Planar

c) Planar

d)

can be redrawn as

Therefore, the network is planar.

e)

cannot be redrawn as a planar network:

Therefore, the network is non-planar.

Example 6 Map Colouring (The Four-Colour Problem)

A graphic designer is working on a logo representing the different tourist
regions in Ontario. What is the minimum number of colours required for D
the design shown on the right to have all adjacent areas coloured B
A
E
differently?
C

Solution

Because the logo is two-dimensional, you can redraw it as a planar network B D
as shown on the right. This network diagram can help you see the
relationships between the regions. The vertices represent the regions and A
the edges show which regions are adjacent. Vertices A and E both connect
to the three other vertices but not to each other. Therefore, A and E can C
E
have the same colour, but it must be different from the colours for B, C,
and D. Vertices B, C, and D all connect to each other, so they require
three different colours. Thus, a minimum of four colours is necessary for
the logo.


This example is a specific case of a famous problem in graph
theory called the four-colour problem. As you probably
conjectured in the investigation at the start of this www.mcgrawhill.ca/links/MDM12
section, the maximum number of colours required in
any planar map is four. This fact had been suspected Visit the above web site and follow the links to find
out more about the four-colour problem. Write a
for centuries but was not proven until 1976. The
short report on the history of the four-colour
proof by Wolfgang Haken and Kenneth Appel at
problem.
the University of Illinois required a supercomputer to
break the proof down into cases and many years of
verification by other mathematicians. Non-planar maps
can require more colours.

Example 7 Scheduling

The mathematics department has five committees. Each of these committees
meets once a month. Membership on these committees is as follows:

Committee A: Szczachor, Large, Ellis
Committee B: Ellis, Wegrynowski, Ho, Khan
Committee C: Wegrynowski, Large
Committee D: Andrew, Large, Szczachor
Committee E: Bates, Card, Khan, Szczachor

What are the minimum number of time slots needed to schedule the
committee meetings with no conflicts?

Solution

Draw the schedule as a network, with each vertex representing a different A B
committee and each edge representing a potential conflict between
committees (a person on two or more committees). Analyse the network
as if you were colouring a map. E
C
The network can be drawn as a planar graph. Therefore, a maximum
of four time slots is necessary to “colour” this graph. Because Committee A D
is connected to the four other committees (degree 4), at least two
time slots are necessary: one for committee A and at least one for Project
all the other committees. Because each of the other nodes has degree 3, Prep
at least one more time slot is necessary. In fact, three time slots are
sufficient since B is not connected to D and C is not connected to E. Graph theory provides
problem-solving
Time Slot Committees
techniques that will be
1 A
useful in your tools for
2 B, D data management
3 C, E project.


Key Concepts

• In graph theory, a graph is also known as a network and is a collection of line
segments (edges) and nodes (vertices).

• If two vertices are connected by an edge, they are adjacent. The degree of a
vertex is equal to the number of edges that begin or end at the vertex.

• A path is a connected sequence of vertices. A path is a circuit if it begins and
ends at the same vertex.

• A connected network has at least one path connecting each pair of vertices.
A complete network has an edge connecting every pair of vertices.

• A connected network is traceable if it has only vertices of even degree (even
vertices) or exactly two vertices of odd degree (odd vertices). If the network
has two vertices of odd degree, the tracing must begin at one of the odd
vertices and end at the other.

• A network is planar if its edges do not cross anywhere except at the vertices.

• The maximum number of colours required to colour any planar map is four.


1. Describe how to convert a map into a network. Use an example to aid in
your description.

2. A network has ﬁve vertices of even degree and three vertices of odd degree.
Using a diagram, show why this graph cannot be traceable.

3. A modern zoo contains natural habitats for its animals. However, many of
the animals are natural enemies and cannot be placed in the same habitat.
Describe how to use graph theory to determine the number of different
habitats required.


Practise 5. Is it possible to add one bridge to the
Koenigsberg map to make it traceable?
A Provide evidence for your answer.
1. For each network,
6. Inquiry/Problem Solving The following chart
i) find the degree of each vertex
indicates the subjects studied by five students.
ii) state whether the network is traceable
C. Powell B. Bates G. Farouk
a) A b) P
E English Calculus Calculus
S French French French
History Geometry Geography
Q U
B Music Physics Music
D
T E. Ho N. Khan
C R
Calculus English
2. Draw a network diagram representing the English Geography
maps in questions 1d) and 1e) of the Geometry Mathematics of Data
investigation on pages 41 and 42. Mathematics of Data Management
Management Physics
3. a) Look at a map of Canada. How many
colours are needed to colour the ten a) Draw a network to illustrate the overlap
provinces and three territories of Canada? of subjects these students study.
b) How many colours are needed if the map b) Use your network to design an
includes the U.S.A. coloured with a examination timetable without conflicts.
single colour? (Hint: Consider each subject to be one
vertex of a network.)
7. A highway inspector wants to travel each
B road shown once and only once to inspect
4. The following map is made up of curved for winter damage. Determine whether it is
lines that cross each other and stop only at possible to do so for each map shown below.
the boundary of the map. Draw three other
a)
maps using similar lines. Investigate the four
maps and make a conjecture of how many
colours are needed for this type of map.

b)


8. Inquiry/Problem Solving 10. Application
a) Find the degree of each vertex in the a) Three houses are located at positions A,
network shown. B, and C, respectively. Water, gas, and
A electrical utilities are located at positions
D, E, and F, respectively. Determine
whether the houses can each be
B connected to all three utilities without
D any of the connections crossing. Provide
evidence for your decision. Is it necessary
to reposition any of the utilities? Explain.
C
b) Find the sum of the degrees of the A D
vertices.
B E
c) Compare this sum with the number of
edges in the network. Investigate other
networks and determine the sum of the C F
degrees of their vertices.
b) Show that a network representing two
d) Make a conjecture from your
observations. houses attached to n utilities is planar.

11. The four Anderson sisters live near each
9. a) The following network diagram of the
main floor of a large house uses vertices other and have connected their houses by
to represent rooms and edges to a network of paths such that each house has
represent doorways. The exterior of the a path leading directly to each of the other
house can be treated as one room. Sketch three houses. None of these paths intersect.
a floor plan based on this network. Can their brother Warren add paths from
his house to each of his sisters’ houses
Library Conservatory without crossing any of the existing paths?

12. In the diagram below, a sheet of paper with
Kitchen
Dining a circular hole cut out partially covers a
Room
Family
drawing of a closed figure. Given that point
Hallway
Room A is inside the closed figure, determine
Living
Room whether point B is inside or outside. Provide
Tea
Room reasons for your answer.
Parlour

Exterior

b) Draw a floor plan and a network diagram A
for your own home. B


13. Application A communications network 15. In a communications network, the optimal
between offices of a company needs to path is the one that provides the fastest link.
provide a back-up link in case one part of a In the network shown, all link times are in
path breaks down. For each network below, seconds.
determine which links need to be backed up. Thunder Bay
Describe how to back up the links. 2.7
a) Thunder Bay Sudbury
Sudbury 4.5 1.7
North Bay
2.3 2.0
North Bay
0.5 Ottawa
Kitchener
Kitchener Ottawa 0.8
1.2 0.6
1.2
Windsor 1.4 Hamilton

Hamilton Determine the optimal path from
Windsor a) Thunder Bay to Windsor
b) b) Hamilton to Sudbury
Charlottetown
c) Describe the method you used to
Halifax
estimate the optimal path.
Toronto
Montréal 16. A salesperson must travel by air to all of the
Kingston
cities shown in the diagram below. The
Winnipeg
Saskatoon diagram shows the cheapest one-way fare for
Edmonton flights between the cities. Determine the
Vancouver least expensive travel route beginning and
ending in Toronto.
14. During an election campaign, a politician
Thunder Bay
will visit each of the cities on the map below. $319
$150
Sudbury
Waterloo
55 Guelph Vancouver $225
Stratford 31 60 $378 $175
63 23 $111 $349
41 23 Toronto
Cambridge Orangeville $378 $213
46 51 Calgary $349
Woodstock 52 116
25 Halifax
$119 $218
38 $321
Brantford 45 Hamilton
$109
Windsor Montréal
a) Is it possible to visit each city only once?
$399
b) Is it possible to begin and end in the
same city?
c) Find the shortest route for visiting all
the cities. (Hint: You can usually find the
shortest paths by considering the shortest
edge at each vertex.)


ACHIEVEMENT CHECK
19. Inquiry/Problem Solving Use graph theory
to determine if it is possible to draw the
Communication Application diagram below using only three strokes of
a pencil.
17. The diagram below shows the floor plan
of a house.

20. Communication
a) Find a route that passes through each a) Can a connected graph of six vertices
doorway of this house exactly once. be planar? Explain your answer.
b) Use graph theory to explain why such b) Can a complete graph of six vertices
a route is possible. be planar? Explain.
c) Where could you place two exterior 21. Can the graph below represent a map in two
doors so that it is possible to start dimensions. Explain.
outside the house, pass through each
doorway exactly once, and end up on B
the exterior again? Explain your
reasoning.
d) Is a similar route possible if you add A C
three exterior doors instead of two?
Explain your answer.
E D
C 22. Can a network have exactly one vertex with
18. a) Six people at a party are seated at a table. an odd degree? Provide evidence to support
No three people at the table know each your answer.
other. For example, if Aaron knows
Carmen and Carmen knows Allison, then 23. Communication A graph is regular if all its
Aaron and Allison do not know each vertices have the same degree. Consider
other. Show that at least three of the six graphs that do not have either loops
people seated at the table must be connecting a vertex back to itself or multiple
strangers to each other. (Hint: Model this edges connecting any pair of vertices.
situation using a network with six a) Draw the four regular planar graphs that
vertices.) have four vertices.
b) Show that, among five people, it is b) How many regular planar graphs with
possible that no three all know each five vertices are there?
other and that no three are all strangers.
c) Explain the difference between your
results in parts a) and b).


1.6 Modelling With Matrices

A matrix is a rectangular array of numbers used to manage and organize data,
somewhat like a table or a page in a spreadsheet. Matrices are made up of horizontal
rows and vertical columns and are usually enclosed in square brackets. Each number
appearing in the matrix is called an entry. For instance, A =
΄5
21 0 ΅
−2 3 is a matrix

with two rows and three columns, with entries 5, −2, and 3 in the first row and entries
2, 1, and 0 in the second row. The dimensions of this matrix are 2 × 3. A matrix with
m rows and n columns has dimensions of m × n.

I N V E S T I G AT E & I N Q U I R E : Olympic Medal Winners

At the 1998 Winter Olympic games in
Nagano, Japan, Germany won 12 gold,
9 silver, and 8 bronze medals; Norway
won 10 gold, 10 silver, and 5 bronze
medals; Russia won 9 gold, 6 silver, and
3 bronze medals; Austria won 3 gold,
5 silver, and 9 bronze medals; Canada
won 6 gold, 5 silver, and 4 bronze
medals; and the United States won
6 gold, 3 silver, and 4 bronze medals.

1. Organize the data using a matrix
with a row for each type of medal
and a column for each country.
2. State the dimensions of the matrix.
3. a) What is the meaning of the entry in row 3, column 1?
b) What is the meaning of the entry in row 2, column 4?
4. Find the sum of all the entries in the first row of the matrix. What is the
significance of this row sum? What would the column sum represent?
5. Use your matrix to estimate the number of medals each country would
win if the number of Olympic events were to be increased by 20%.
6. a) Interchange the rows and columns in your matrix by “reflecting” the
matrix in the diagonal line beginning at row 1, column 1.
b) Does this transpose matrix provide the same information? What
are its dimensions?
7. State one advantage of using matrices to represent data.

1.6 Modelling With Matrices • MHR 53

In general, use a capital letter as the symbol for a matrix and represent each
entry using the corresponding lowercase letter with two indices. For example,

΄ ΅
c11 c12 c13 … c1n
…

΄ ΅ ΄ ΅
a11 a12 a13 b11 b12 c21 c22 c23 c2n
A = a21 a22 a23 B = b21 b22 C = c31 c32 c33 … c3n
a31 a32 a33 b31 b32 Ӈ Ӈ Ӈ Ӈ Ӈ
cm1 cm2 cm3 … cmn

Here, ai j , bi j , and ci j represent the entries in row i and column j of these matrices.

The transpose of a matrix is indicated by a superscript t, so the transpose of A is
shown as At. A matrix with only one row is called a row matrix, and a matrix
with only one column is a column matrix. A matrix with the same number of
rows as columns is called a square matrix.

΄ ΅ ΄ ΅
−3 3 4 9
[1 −2 5 −9] 0 −1 0 2
5 5 −10 −3
a row matrix a column matrix a square matrix

Example 1 Representing Data With a Matrix

The number of seats in the House of Commons won by each party in the
federal election in 1988 were Bloc Québécois (BQ), 0; Progressive Conservative
Party (PC), 169; Liberal Party (LP), 83; New Democratic Party (NDP), 43;
Reform Party (RP), 0; Other, 0. In 1993, the number of seats won were BQ, 54;
PC, 2; LP, 177; NDP, 9; RP, 52; Other, 1. In 1997, the number of seats won
were BQ, 44; PC, 20; LP, 155; NDP, 21; RP, 60; Other, 1.
a) Organize the data using a matrix S with a row for each political party.
b) What are the dimensions of your matrix?
c) What does the entry s43 represent?
d) What entry has the value 52?
e) Write the transpose matrix for S. Does S t provide the same information
as S?
f) The results from the year 2000 federal election were Bloc Québécois, 38;
Progressive Conservative, 12; Liberal, 172; New Democratic Party, 13;
Canadian Alliance (formerly Reform Party), 66; Other, 0. Update your
matrix to include the results from the 2000 federal election.


Solution

a) 1988 1993 1997

΄ ΅
0 54 44 BQ
169 2 20 PC
83 177 155 LP
S = 43 9 21 NDP
0 52 60 RP
0 1 1 Other

Labelling the rows and columns in large matrices can help you keep track
of what the entries represent.

b) The dimensions of the matrix are 6 × 3.

c) The entry s43 shows that the NDP won 21 seats in 1997.

d) The entry s52 has the value 52.

e) The transpose matrix is
BQ PC LP NDP RP Other

΄ ΅
0 169 83 43 0 0 1988
S = t 54 2 177 9 52 1 1993
44 20 155 21 60 1 1997
Comparing the entries in the two matrices shows that they do contain
exactly the same information.

f) 1988 1993 1997 2000

΄ ΅
0 54 44 38 BQ
169 2 20 12 PC
83 177 155 172 LP
43 9 21 13 NDP
0 52 60 66 CA (RP)
0 1 1 0 Other

Two matrices are equal only if each entry in one matrix is equal to the
corresponding entry in the other.
1.5 4 −8

΄ ΅ ΄ ΅
3
ᎏᎏ ͙16
ෆ (−2)3
For example, 2 and 1 are equal matrices.
ᎏᎏ −4 2
5−1 −4 −(−2) 5


Two or more matrices can be added or subtracted, provided that their
dimensions are the same. To add or subtract matrices, add or subtract the
corresponding entries of each matrix. For example,

−1 5 5 −3
΄2
0 7 −8 ΅ + ΄ −2
0
4 −1 ΅ = ΄ −2
2 4 2
11 −9 ΅
Matrices can be multiplied by a scalar or constant. To multiply a matrix by a
scalar, multiply each entry of the matrix by the scalar. For example,

΄ ΅΄ ΅
4 5 −12 −15
−3 −6 0 = 18 0
3 −8 −9 24

Example 2 Inventory Problem

The owner of Lou’s ’Lectronics Limited has two stores. The manager takes
inventory of their top-selling items at the end of the week and notes that at the
eastern store, there are 5 video camcorders, 7 digital cameras, 4 CD players,
10 televisions, 3 VCRs, 2 stereo systems, 7 MP3 players, 4 clock radios, and
1 DVD player in stock. At the western store, there are 8 video camcorders,
9 digital cameras, 3 CD players, 8 televisions, 1 VCR, 3 stereo systems, 5 MP3
players, 10 clock radios, and 2 DVD players in stock. During the next week,
the eastern store sells 3 video camcorders, 2 digital cameras, 4 CD players,
3 televisions, 3 VCRs, 1 stereo system, 4 MP3 players, 1 clock radio, and no
DVD players. During the same week, the western store sells 5 video
camcorders, 3 digital cameras, 3 CD players, 8 televisions, no VCRs, 1 stereo
system, 2 MP3 players, 7 clock radios, and 1 DVD player. The warehouse then
sends each store 4 video camcorders, 3 digital cameras, 4 CD players,
4 televisions, 5 VCRs, 2 stereo systems, 2 MP3 players, 3 clock radios, and
1 DVD player.
a) Use matrices to determine how many of each item is in stock at the stores
after receiving the new stock from the warehouse.
b) Immediately after receiving the new stock, the manager phones the head
ofﬁce and requests an additional 25% of the items presently in stock in
anticipation of an upcoming one-day sale. How many of each item will be
in stock at each store?



a) Let matrix A represent the initial inventory, matrix B represent the number of items
sold, and matrix C represent the items in the ﬁrst shipment of new stock.
E W
5 8 camcorders 3 5 4 4
7 9 cameras 2 3 3 3
4 3 CD players 4 3 4 4
10 8 TVs 3 8 4 4
A= 3 1 VCRs B= 3 0 C= 5 5
2 3 stereos 1 1 2 2
7 5 MP3 players 4 2 2 2
4 10 clock radios 1 7 3 3
1 2 DVD players 0 1 1 1

Since the dimensions of matrices A, B, and C are the same, matrix addition and
subtraction can be performed. Then, the stock on hand before the extra shipment is
5 8 3 5 4 4 6 7
7 9 2 3 3 3 8 9
4 3 4 3 4 4 4 4
10 8 3 8 4 4 11 4
D=A−B+C= 3 1 − 3 0 + 5 5 = 5 6
2 3 1 1 2 2 3 4
7 5 4 2 2 2 5 5
4 10 1 7 3 3 6 6
1 2 0 1 1 1 2 2
Let E represent the stock in the stores after the extra shipment from the warehouse.
6 7 7.5 8.75
8 9 10 11.25
4 4 5 5
11 4 13.75 5
E = 125% × D = 1.25 5 6 = 6.25 7.5
3 4 3.75 5
5 5 6.25 6.25
6 6 7.5 7.5
2 2 2.5 2.5
Assuming the manager rounds to the nearest whole number, the stock at the eastern
store will be 8 video camcorders, 10 digital cameras, 5 CD players, 14 televisions,
6 VCRs, 4 stereo systems, 6 MP3 players, 8 clock radios, and 3 DVD players in
stock. At the western store, there will be 9 video camcorders, 11 digital cameras,
5 CD players, 5 televisions, 8 VCRs, 5 stereo systems, 6 MP3 players, 8 clock
radios, and 3 DVD players in stock.


Solution 2 Using a Graphing Calculator

a) As in the pencil-and-paper solution, let matrix A represent the initial
inventory, matrix B the items sold, and matrix C the first shipment of
new stock. Use the MATRX EDIT menu to store matrices. Press ENTER
to select a matrix name, then key in the dimensions and the entries.
The calculator will store the matrix until it is cleared or overwritten.
Matrix names and entries appear in square brackets on the calculator
screen.

Use the MATRX NAMES menu to copy the matrices into the expression
for D, the matrix representing the stock on hand before the extra
shipment. Just move the cursor to the matrix you need and press ENTER.

b) To find the stock on hand after the extra shipment for the one-day sale,
multiply matrix D by 1.25 and store the result in matrix E. Then, you
can use the round function in the MATH NUM menu to display the
closest whole numbers for the entries in matrix E.


a) You can easily perform matrix operations using a spreadsheet. It is also
easy to add headings and row labels to keep track of what the entries
represent. Enter each matrix using two adjacent columns: matrix A (initial
stock) in columns A and B, matrix B (sales) in columns C and D, and
matrix C (new stock) in columns E and F.

To find the amount of stock on hand after the first shipment from the
warehouse, enter the formula A3–C3+E3 in cell H3.

Then, use the Fill feature to copy this formula for the rest of the entries
in columns H and I.

b) Use the Fill feature in a similar way to copy the formula for the entries
in matrix E, the stock on hand after the extra shipment from the
warehouse. You can use the ROUND function to find the nearest whole
number automatically. The formula for cell J3, the first entry, is
ROUND(1.25*H3,0).


Key Concepts

• A matrix is used to manage and organize data.

• A matrix made up of m rows and n columns has dimensions m × n.

• Two matrices are equal if they have the same dimensions and all corresponding
entries are equal.

• The transpose matrix is found by interchanging rows with the corresponding
columns.

• To add or subtract matrices, add or subtract the corresponding entries of each
matrix. The dimensions of the matrices must be the same.
• To multiply a matrix by a scalar, multiply each entry of the matrix by the scalar.


1. Describe how to determine the dimensions of any matrix.

2. Describe how you know whether two matrices are equal. Use an example to
illustrate your answer.

3. Can transpose matrices ever be equal? Explain.

4. a) Describe how you would add two matrices. Give an example.
b) Explain why the dimensions of the two matrices need to be the same to add
or subtract them.

5. Describe how you would perform scalar multiplication on a matrix. Give an
example.


Practise 5. a) Give two examples of square matrices.
b) State the dimensions of each matrix in
A part a).
1. State the dimensions of each matrix.
6. a) Write a 3 × 4 matrix, A, with the
5 −1
a)
΄ 4
−2 3 8 ΅ b) [1 0 −7] property that entry aij = i + j.
b) Write a 4 × 4 matrix, B, with the property
3 if i = j
Άi × j if i
΄ ΅
3 −9 −6 that entry bij =
5 4 7 j
c)
1 0 8 7. Solve for w, x, y, and z.
8 −1 2
a)
΄ −2
x 4
4z − 2 ΅ ΄
= 3
w
y−1
6 ΅
΄ ΅
−5 3 2
6 0 −1
΄w ΅ = ΄ 8 −8 2y 9
΅
3
2. For the matrix A = , b) x2
4 8 −3 2y 3z 2z − 5
7 1 −4
a) state the value in entry

΄ ΅ ΄ ΅
2 −1 3 4
i) a21 ii) a43 iii) a13 3 9 −6 1
8. Let A = ,B= ,
b) state the entry with value 5 0 8 2
−4 1 −1 −5
i) 4 ii) −3 iii) 1
3 −2
and C =
΄ 6 5 .
΅
΄ ΅
a b c d e 1 4 0 −8
f g h i j
3. Let A = Calculate, if possible,
k l m n o
a) A + B b) B + A c) B − C
p q r s t 1
d) 3A e) −ᎏᎏB f) 2(B − A)

΄ ΅
u v 2
g) 3A − 2B
and B = w x .
y z

΄ ΅ ΄ ΅
8 −6 0 −1
For each of the following, replace aij or bij 9. Let A = 1 −2 , B = 2 4 ,
with its corresponding entry in the above −4 5 9 −3
matrices to reveal a secret message.

΄ ΅
2 3
a) a33a11a45a43a24a13a15a44 8 −6 .
and C =
a11a43a15 a21b11a34 4 1
b) a24 a32a35b12a15 a33a11a45a23 Show that
c) b21a35b21 a45a23a24a44 a) A + B = B + A
a24a44 a21b11a34 (commutative property)
4. a) Give two examples of row matrices and b) (A + B) + C = A + (B + C )
two examples of column matrices. (associative property)
b) State the dimensions of each matrix in c) 5(A + B) = 5A + 5B
part a). (distributive property)


10. Find the values of w, x, y, and z if sciences; U.K. with 21 Nobel prizes in

΄ ΅ ΄ ΅
5 −1 2 6 y 5 physics, 25 in chemistry, 24 in
4 x −8 + 2 −3 2 1 physiology/medicine, 8 in literature, 13 in
7 0 3 2 −3 z peace, and 7 in economic sciences; Germany
with 20 Nobel prizes in physics, 27 in

΄ ΅
34 10 24
1 chemistry, 16 in physiology/medicine, 7 in
= ᎏᎏ −4 24 −12 literature, 4 in peace, and 1 in economic
2 2w −12 42
sciences; France with 12 Nobel prizes in
11. Solve each equation. physics, 7 in chemistry, 7 in physiology/
medicine, 12 in literature, 9 in peace, and 1 in
a)
΄3
2 0 8 ΅
2 −5 + A = 7
−4 ΄ 0 1
3 −2 ΅ economic sciences; and Sweden with 4 Nobel
prizes in physics, 4 in chemistry, 7 in

΄ ΅ ΄ ΅΄ ΅
5 7 1 6 7 19 physiology/medicine, 7 in literature, 5 in
b) 4 0 +y 0 −4 = 4 −8 peace, and 2 in economic sciences.
−1 −3 2 5 3 7
a) Represent this data as a matrix, N. What
Apply, Solve, Communicate are the dimensions of N ?
b) Use row or column sums to calculate
B how many Nobel prizes have been
12. Application The map below shows driving awarded to citizens of each country.
distances between five cities in Ontario.
14. The numbers of university qualifications
Thunder Bay
(degrees, certificates, and diplomas) granted
in Canada for 1997 are as follows: social
sciences, 28 421 males and 38 244 females;
710 km education, 8036 males and 19 771 females;
humanities, 8034 males and 13 339 females;
North Bay health professions and occupations, 3460
Sault Ste. Ottawa males and 9613 females; engineering and
425 km 365 km
Marie
350 km applied sciences, 10 125 males and 2643
655 km 400 km
females; agriculture and biological sciences,
Toronto 4780 males and 6995 females; mathematics
and physical sciences, 6749 males and 2989
a) Represent the driving distances between females; fine and applied arts, 1706 males
each pair of cities with a matrix, A. and 3500 females; arts and sciences, 1730
b) Find the transpose matrix, At. males and 3802 females.
c) Explain how entry a23 in matrix A and
The numbers for 1998 are as follows: social
entry a32 in matrix At are related.
sciences, 27 993 males and 39 026 females;
13. Nobel prizes are awarded for physics, education, 7565 males and 18 391 females;
chemistry, physiology/medicine, literature, humanities, 7589 males and 13 227 females;
peace, and economic sciences. The top five health professions and occupations, 3514
Nobel prize-winning countries are U.S.A. males and 9144 females; engineering and
with 67 Nobel prizes in physics, 43 in applied sciences, 10 121 males and 2709
chemistry, 78 in physiology/medicine, 10 in females; agriculture and biological sciences,
literature, 18 in peace, and 25 in economic 4779 males and 7430 females;


mathematics and physical sciences, 6876 b) What is the total population for each age
males and 3116 females; fine and applied arts, group?
1735 males and 3521 females; arts and c) Suppose that Canada’s population grows
sciences, 1777 males and 3563 females. by 1.5% in all age groups. Calculate the
a) Enter two matrices in a graphing calculator anticipated totals for each age group.
or spreadsheet—one two-column matrix
for males and females receiving degrees in 16. a) Prepare a matrix showing the
pte
ha connections for the VIA Rail routes
1997 and a second two-column matrix for

C

r
the number of males and females receiving shown on page 3. Use a 1 to indicate a

m
P
r
oble
degrees in 1998. direct connection from one city to
another city. Use a 0 to indicate no direct
b) How many degrees were granted to males
connection from one city to another city.
in 1997 and 1998 for each field of study?
Also, use a 0 to indicate no direct
c) How many degrees were granted to connection from a city to itself.
females in 1997 and 1998 for each field
b) What does the entry in row 4, column 3
of study?
represent?
d) What is the average number of degrees
c) What does the entry in row 3, column 4
granted to females in 1997 and 1998 for
represent?
each field of study?
d) Explain the significance of the
15. Application The table below shows the relationship between your answers in
population of Canada by age and gender in parts b) and c).
the year 2000. e) Describe what the sum of the entries in
Age Group Number of Males Number of Females the first row represents.
0−4 911 028 866 302 f) Describe what the sum of the entries in
5−9 1 048 247 996 171 the first column represents.
10−14 1 051 525 997 615
g) Explain why your answers in parts e) and
15−19 1 063 983 1 007 631
f ) are the same.
20−24 1 063 620 1 017 566
25−29 1 067 870 1 041 900 C
30−34 1 154 071 1 129 095 17. Inquiry/Problem Solving Show that for any
35−39 1 359 796 1 335 765 m × n matrices, A and B
40−44 1 306 705 1 304 538
a) (At )t = A b) (A + B)t = At + B t
45−49 1 157 288 1 162 560
50−54 1 019 061 1 026 032 18. Communication Make a table to compare
55−59 769 591 785 657 matrix calculations with graphing calculators
60−64 614 659 641 914 and with spreadsheets. What are the
65−69 546 454 590 435 advantages, disadvantages, and limitations
70−74 454 269 544 008 of these technologies?
75−79 333 670 470 694
80−84 184 658 309 748 19. Inquiry/Problem Solving Search the
85−89 91 455 190 960 newspaper for data that could be organized
90+ 34 959 98 587 in a matrix. What calculations could you
perform with these data in matrix form? Is
a) Create two matrices using the above data,
there any advantage to using matrices for
one for males and another for females. these calculations?

1.7 Problem Solving With Matrices

The previous section demonstrated how to use matrices to model, organize, and
manipulate data. With multiplication techniques, matrices become a powerful
tool in a wide variety of applications.

I N V E S T I G AT E & I N Q U I R E : Matrix Multiplication

The National Hockey League standings on March 9, 2001 in the
Northeast Division are shown below along with the league’s point
system for a win, loss, tie, or overtime loss (OTL).
Team Win Loss Tie OTL Score Points
Ottawa 39 17 8 3 Win 2
Buffalo 36 25 5 1 Loss 0
Toronto 31 23 10 5 Tie 1
Boston 28 27 6 7 OTL 1
Montréal 23 36 5 4

1. Calculate the number of points
for each team in the Northeast
Division using the above tables.
Explain your method.
2. a) Represent the team standings
as a 5 × 4 matrix, A.
b) Represent the points system
as a column matrix, B.
3. Describe a procedure for
determining the total points for
Ottawa using the entries in row
1 of matrix A and column 1 of
matrix B.
4. How could you apply this
procedure to ﬁnd the points
totals for the other four teams?
5. Represent the total points for each team as a column matrix, C.
How are the dimensions of C related to those of A and B?
6. Would it make sense to deﬁne matrix multiplication using a
procedure such that A × B = C? Explain your reasoning.

1.7 Problem Solving With Matrices • MHR 63

In the above investigation, matrix A has dimensions 5 × 4 and A5x4 × B4x1 = C5x1
matrix B has dimensions 4 × 1. Two matrices can be multiplied
when their inner dimensions are equal. The outer dimensions same
are the dimensions of the resultant matrix when matrices A dimensions
and B are multiplied. outer dimensions give dimensions
of resultant matrix

Example 1 Multiplying Matrices

Matrix A represents the proportion of students at a high school who have part-time
jobs on Saturdays and the length of their shifts. Matrix B represents the number of
students at each grade level.

Gr 9 Gr 10 Gr 11 Gr 12 M F

΄ ΅
120 130 Gr 9

΄ ΅
0.20 0.10 0.20 0.15 ≤ 4 h
137 155 Gr 10
A = 0.25 0.30 0.25 0.45 4.1 − 6 h B=
103 110 Gr 11
0.05 0.25 0.15 0.10 > 6 h
95 92 Gr 12
a) Calculate AB. Interpret what each entry represents.
b) Calculate BA, if possible.

Solution

a) A and B have the same inner dimensions, so multiplication is possible
and their product will be a 3 × 2 matrix: A3×4 × B4×2 = C3×2

΅΄ ΅
120 130

΄
0.20 0.10 0.20 0.15
137 155
AB = 0.25 0.30 0.25 0.45
103 110
0.05 0.25 0.15 0.10
95 92

΄ ΅
(0.20)(120) + (0.10)(137) + (0.20)(103) + (0.15)(95) (0.20)(130) + (0.10)(155) + (0.20)(110) + (0.15)(92)
= (0.25)(120) + (0.30)(137) + (0.25)(103) + (0.45)(95) (0.25)(130) + (0.30)(155) + (0.25)(110) + (0.45)(92)
(0.05)(120) + (0.25)(137) + (0.15)(103) + (0.10)(95) (0.05)(130) + (0.25)(155) + (0.15)(110) + (0.10)(92)

΄ ΅
73 77
⋅
= 140 148
65 71
Approximately 73 males and 77 females work up to 4 h; 140 males and
148 females work 4− 6 h, and 65 males and 71 females work more than
6 h on Saturdays.

b) For B4×2 × A3×4, the inner dimensions are not the same, so BA cannot be calculated.


Technology is an invaluable tool for solving problems that involve large
amounts of data.

Example 2 Using Technology to Multiply Matrices

The following table shows the number and gender of full-time students
enrolled at each university in Ontario one year.
University Full-Time Students Males (%) Females (%)
Brock 6509 43 57
Carleton 12 376 55 45
Guelph 11 773 38 62
Lakehead 5308 48 52
Laurentian 3999 43 57
McMaster 13 797 46 54
Nipissing 1763 34 66
Ottawa 16 825 42 58
Queen’s 13 433 44 56
Ryerson 10 266 47 53
Toronto 40 420 44 56
Trent 3764 36 64
Waterloo 17 568 55 45
Western 21 778 46 54
Wilfred Laurier 6520 45 55
Windsor 9987 46 54
York 27 835 39 61

a) Set up two matrices, one listing the numbers of full-time students at
each university and the other the percents of males and females.
b) Determine the total number of full-time male students and the total
number of full-time female students enrolled in Ontario universities.


a) Use the MATRX EDIT menu to store matrices for a 1 × 17 matrix for the
numbers of full-time students and a 17 × 2 matrix for the percents of
males and females.

b) To multiply matrices, use the MATRX NAMES menu. Copy the matrices
into an expression such as [A]*[B] or [A][B].

There are 100 299 males and 123 622 females enrolled in Ontario
universities.


You can also enter matrices directly into an expression by using the square
brackets keys. This method is simpler for small matrices, but does not store
the matrix in the MATRX NAMES menu.


Enter the number of full-time students at each university as a 17 × 1 matrix in
cells B2 to B18. This placement leaves you the option of putting labels in the
first row and column. Enter the proportion of male and female students as a
2 × 17 matrix in cells D2 to T3.

Both Corel® Quattro Pro and Microsoft® Excel have built-in functions for
multiplying matrices, although the procedures in the two programs differ
somewhat.

Corel® Quattro Pro:
On the Tools menu, select Numeric Tools/Multiply. In the pop-up window,
enter the cell ranges for the two matrices you want to multiply and the cell
where you want the resulting matrix to start. Note that you must list the 2 × 17
matrix first.

Project
Prep

You can apply these
techniques for matrix
multiplication to the
calculations for your
tools for data
management project.

Microsoft® Excel:
The MMULT(matrix1,matrix2) function will calculate the product of the two
matrices but displays only the first entry of the resulting matrix. Use the INDEX
function to retrieve the entry for a specific row and column of the matrix.


΄ ΅
1 0 0 0 … 0
0 1 0 0 … 0
0 0 1 0 … 0
Identity matrices have the form I = with entries
0 0 0 1 … 0
Ӈ Ӈ Ӈ Ӈ Ӈ Ӈ
0 0 0 0 … 1
of 1 along the main diagonal and zeros for all other entries. The identity
matrix with dimensions n × n is represented by In. It can easily be shown that
Am×n In = Am×n for any m × n matrix A.

For most square matrices, there exists an inverse matrix A–1 with the
1
property that AA−1 = A−1A = I. Note that A−1 ᎏᎏ.
A

For 2 × 2 matrices, AA−1 = ΄ ac d ΅ ΄ w x ΅ = ΄ 1 0 ΅
b
y z 0 1
Multiplying the matrices gives four simultaneous equations for w, x, y, and z.
d −b
1
΄ ΅
Solving these equations yields A−1 = ᎏᎏ − c a . You can conﬁrm that
ad − bc
A A = I, also. If ad = bc, then A does not exist since it would require
–1 −1

dividing by zero.

The formulas for the inverses of larger matrices can be determined in the
same way as for 2 × 2 matrices, but the calculations become much more
involved. However, it is relatively easy to ﬁnd the inverses of larger matrices
with graphing calculators since they have the necessary formulas built in.


Example 3 Calculating the Inverse Matrix

Calculate, if possible, the inverse of
a) A =
΄ 3 7
4 −2 ΅ b) B = 6 8
΄
3 4 ΅

A−1 = ᎏ d − b ΄ ΅
1
a)
ad − bc − c a

= ᎏᎏ −2 −7 ΄ ΅
1
(3)( −2) − (7)(4) −4 3

= − ᎏᎏ −2 −7
΄ ΅
1
34 −4 3

΄ ΅
1 7
ᎏᎏ ᎏᎏ
= 17 34
2 3
ᎏᎏ −ᎏᎏ
17 34

b) For B, ad − bc = (6)(4) − (8)(3) = 0, so B −1 does not exist.


a) Use the MATRX EDIT menu to store the 2 × 2 matrix. Retrieve it with the
MATRX NAMES menu, then use x −1 to ﬁnd the inverse. To verify that the
decimal numbers shown are equal to the fractions in the pencil-and-paper
solution, use the ᭤Frac function from the MATH NUM menu.

b) For B, the calculator shows that the inverse cannot be calculated.



The spreadsheet functions for inverse matrices are similar to those for matrix
multiplication.
a) Enter the matrix in cells A1 to B2.

In Corel® Quattro Pro, use Tools/Numeric Tools/Invert… to enter the range
of cells for the matrix and the cell where you want the inverse matrix to start.
Use the Fraction feature to display the entries as fractions rather than
decimal numbers.

In Microsoft® Excel, use the MINVERSE function to produce the inverse
matrix and the INDEX function to access the entries in it. If you put absolute
cell references in the MINVERSE function for the ﬁrst entry, you can use the
Fill feature to generate the formulas for the other entries.
Use the Fraction feature to display the entries as fractions rather than
decimal numbers.


During the 1930s, Lester Hill, an American mathematician, developed methods
for using matrices to decode messages. The following example illustrates a
simpliﬁed version of Hill’s technique.

Example 4 Coding a Message Using Matrices

a) Encode the message PHONE ME TONIGHT using 2 × 2 matrices.
b) Determine the matrix key required to decode the message.

Solution

a) Write the message using 2 × 2 matrices. Fill in any missing entries with
the letter Z.

΄ O N ΅, ΄ E M ΅, ΄ OI N ΅, ΄ H T ΅
P H
E T G Z Z
Replace each letter with its corresponding number in the alphabet.

A B C D E F G H I J K L M
1 2 3 4 5 6 7 8 9 10 11 12 13

N O P Q R S T U V W X Y Z
14 15 16 17 18 19 20 21 22 23 24 25 26

΄ 16
15 14΅ ΄ 5 13 ΅, ΄ 15 14 ΅, ΄ 26 26 ΅
8 ,
5 20 9 7
8 20

Now, encode the message by multiplying with a coding matrix that only

΄ ΅
the sender and receiver know. Suppose that you chose C = 3 1 as your
coding matrix. 5 2

΄ 3 1 ΅΄ 16
5 2 15
8
14 ΅ = ΄ 110 68 ΅
63 38

΄ 3 1 ΅΄ 5 13 ΅ = ΄ 20 105 ΅
5 2 5 20 35
59

΄ 3 1 ΅΄ 15 14 ΅ = ΄ 54 49 ΅
5 2 9 7 93 84

΄ 3 1 ΅΄ 26 26 ΅ = ΄ 50 152 ΅
5 2
8 20
92
86

You would send the message as 63, 38, 110, 68, 20, 59, 35, 105, 54, 49, 93,
84, 50, 86, 92, 152.


b) First, rewrite the coded message as 2 × 2 matrices.

΄ 110 68 ΅, ΄ 20 105 ΅, ΄
63 38
35
59 54 49 ,
93 84 ΅ ΄ 92 152 ΅
50 86

You can decode the message with the inverse matrix for the coding matrix.
C −1 × CM = C −1C × M = IM = M
where M is the message matrix and C is the coding matrix.
Thus, the decoding matrix, or key, is the inverse matrix of the coding
matrix. For the coding matrix used in part a), the key is

= ᎏᎏ 2 −1 = 2 −1
−1

΄3 1΅ ΄ ΅ ΄ ΅
1
5 2 (3)(2) − (5)(1) −5 3 −5 3

Multiplying the coded message by this key gives
−1
΄ −5
2
3 ΅΄ 110 68 ΅ = ΄ 15 14 ΅
63 38 16 8

−1 20 59 = 5 13
΄ −5
2
3 ΅΄ 35 105 ΅ ΄ 5 20 ΅
−1 54 49 = 15 14
΄ −5
2
3 ΅΄ 93 84 ΅ ΄ 9 7 ΅
−1 50 86 = 8 20
΄ −5
2
3 ΅΄ 92 152 ΅ ΄ 26 26 ΅
The decoded message is 16, 8, 15, 14, 5, 13, 5, 20, 15, 14, 9, 7, 8, 20, 26, 26.
Replacing each number with its corresponding letter in the alphabet gives
PHONEMETONIGHTZZ, the original message with the two Zs as ﬁllers.

Matrix multiplication and inverse matrices are the basis for many computerized
encryption systems like those used for electronic transactions between banks and
income tax returns ﬁled over the Internet.

Transportation and communication networks can be represented using matrices,
called network matrices. Such matrices provide information on the number
of direct links between two vertices or points (such as people or places). The
advantage of depicting networks using matrices is that information on indirect
routes can be found by performing calculations with the network matrix.

To construct a network matrix, let each vertex (point) be represented as a row
and as a column in the matrix. Use 1 to represent a direct link and 0 to
represent no direct link. A vertex may be linked to another vertex in one
direction or in both directions. Assume that a vertex does not link with itself,
so each entry in the main diagonal is 0. Note that the network matrix provides
information only on direct links.


Example 5 Using Matrices to Model a Network

Matrixville Airlines offers ﬂights between London, England
eight cities as shown on the right.
a) Represent the network using a
Toronto New
matrix, A. Organize the matrix Vancouver Paris Delhi
so the cities are placed in
alphabetical order.
Kingston,
b) Calculate A2. What information Honolulu Jamaica
does it contain?
Buenos Aires
c) How many indirect routes with
exactly one change of planes are there from London to Buenos Aires?
d) Calculate A + A2. What information does it contain?
e) Explain what the entry from Vancouver to Paris in A + A2 represents.
f) Calculate A3. Compare this calculation with the one for A2.
g) Explain the signiﬁcance of any entry in matrix A3.

Solution

a) B H K L N P T V
0 0 0 0 0 1 1 0 B
0 0 0 0 0 0 1 0 H
0 0 0 1 0 0 1 0 K
A= 0 0 1 0 1 1 1 1 L
0 0 0 1 0 1 0 0 N
1 0 0 1 1 0 1 0 P
1 1 1 1 0 1 0 1 T
0 0 0 1 0 0 1 0 V

b) Since the dimensions of matrix A are 8 × 8, you may prefer to use a
calculator or software for this calculation.

2 1 1 2 1 1 1 1
1 1 1 1 0 1 0 1
1 1 2 1 1 2 1 2
A2 = 2 1 1 5 1 2 3 1
1 0 1 1 2 1 2 1
1 1 2 2 1 4 2 2
1 0 1 3 2 2 6 2
1 1 2 1 1 2 1 2
The entries in A2 show the number of indirect routes with exactly one
change of planes. A2 does not contain any information on direct routes.


c) There are two indirect routes with exactly one change of planes from
London to Buenos Aires.

London → Paris → Buenos Aires
London → Toronto → Buenos Aires

2 1 1 2 1 2 2 1
1 1 1 1 0 1 1 1
1 1 2 2 1 2 2 2
d) A + A2 = 2 1 2 5 2 3 4 2
1 0 1 2 2 2 2 1
2 1 2 3 2 4 3 2
2 1 2 4 2 3 6 2
1 1 2 2 1 2 2 2
Since A shows the number of direct routes and A2 shows the number of
routes with one change of planes, A + A2 shows the number of routes with
at most one change of planes.

e) The entry in row 8, column 6 of A + A2 shows that there are two routes
with a maximum of one change of planes from Vancouver to Paris.
Vancouver → Toronto → Paris
Vancouver → London → Paris

2 1 3 5 3 6 8 3
1 0 1 3 2 2 6 1
3 1 2 8 3 4 9 2
f) A3 = 5 3 8 8 7 11 12 8
3 2 3 7 2 6 5 3
6 2 4 11 6 6 12 4
8 6 9 12 5 12 8 9
3 1 2 8 3 4 9 2
The calculation of A3 = A2 × A is more laborious than that for A2 = A × A
since A2 has substantially fewer zero entries than A does. A calculator or
spreadsheet could be useful.

g) The entries in A3 tell you the number of indirect routes with exactly two
changes of planes between each pair of cities.


Key Concepts

• To multiply two matrices, their inner dimensions must be the same. The outer
dimensions give the dimensions of the resultant matrix: Am×n × Bn×p = Cm×p. To
find the entry with row i and column j of matrix AB, multiply the entries of
row i of matrix A with the corresponding entries of column j of matrix B, and
then add the resulting products together.

• The inverse of the 2 × 2 matrix A = a b is A−1 = ᎏᎏ d −b
΄ ΅ ΄ ΅
1
c d ad − bc −c a
provided that ad ≠ bc. Larger inverse matrices can be found using a graphing
calculator or a spreadsheet.

• To represent a network as a matrix, use a 1 to indicate a direct link and a 0 to
indicate no direct link. Calculations with the square of a network matrix and
its higher powers give information on the various direct and indirect routings
possible.


1. Explain how multiplying matrices is different from scalar multiplication of
matrices.

2. Describe the steps you would take to multiply
΄ 4 −2 ΅΄ 3
1 5 0
6
4
2 −1 .
5 7 ΅
3. Is it possible to find an inverse for a matrix that is not square? Why or why
not?

4. Explain why a network matrix must be square.
A B
5. Describe how you would represent the following network
as a matrix. How would you find the number of routes
C D
with up to three changeovers?

E

Practise Calculate, if possible,
a) BD b) DB c) B2 d) EA
A
e) AC f) CE g) DA
1. Let A =
΄
4 7
−3 −5 1΅
0 ,B= 2
−7 ΄ ΅
9 ,
0
2. Given A =
΄ 4 −1 ΅ and B = ΄−2 3 ΅, show
2 0

΄ ΅ ΄ ΅
1 5 8 1 0 0
C = 2 0 −4 , D = −3
−3 −2 8 2 ΄ 5΅
1 ,E= 2 .
−3 ΄
that A2 + 2B3 = 16 −30 . ΅
24 1


8. Application Calculators Galore has three
3. If A =
΄0 0 ΅
0 1 , show that A4 =
΄ 0
0 ΅
0 ,
0
stores in Matrixville. The downtown store
sold 12 business calculators, 40 scientific
the 2 × 2 zero matrix.
calculators, and 30 graphing calculators
during the past week. The northern store
4. Let A =
΄ 5 −1 ΅, B = ΄ −2 4 ΅, C = ΄ 1 −3΅.
2
0 3
0 0 7 sold 8 business calculators, 30 scientific
calculators, and 21 graphing calculators
Show that
during the same week, and the southern
a) A(B + C) = AB + AC store sold 10 business calculators,
(distributive property) 25 scientific calculators, and 23 graphing
b) (AB)C = A(BC ) calculators. What were the total weekly sales
(associative property) for each store if the average price of a
c) AB ≠ BA business calculator is $40, a scientific
(not commutative) calculator is $30, and a graphing calculator
is $150?
5. Find the inverse matrix, if it exists.
9. Application The manager at Sue’s Restaurant
4 −6
a)
΄ ΅ 0 −1
2 4
b)
΄ ΅
−2 3
c)
΄ 3
−6
0
1 ΅ prepares the following schedule for the next
week.
d)
΄ 5 3΅
4 2 ΄ 4 2΅
e) 10 5 Employee Mon. Tues.
Chris − 8
Wed. Thurs.
− 8
Fri. Sat. Sun.
8 − −
Wage Per Hour
$7.00
6. Use a graphing calculator or a spreadsheet Lee 4 4 − − 6.5 4 4 $6.75
to calculate the inverse matrix, if it exists. Jagjeet − 4 4 4 4 8 8 $7.75
Pierre − 3 3 3 3 8 − $6.75

΄ ΅
1 −3 1
Ming 8 8 8 8 − − − $11.00
a) A = −2 1 3
Bobby − − 3 5 5 8 − $8.00
0 −1 0
Nicole 3 3 3 3 3 − − $7.00

΄ ΅
−2 0 5 Louis 8 8 8 8 8 − − $12.00
b) B = 2 −1 −1 Glenda 8 − − 8 8 8 8 $13.00
3 4 0 Imran 3 4.5 4 3 5 − − $7.75
a) Create matrix A to represent the number

΄ ΅
2 −1 1 0 of hours worked per day for each
0 1 0 2
c) C = employee.
−2 −1 0 0
b) Create matrix B to represent the hourly
1 0 −1 0
wage earned by each employee.
Apply, Solve, Communicate c) Use a graphing calculator or spreadsheet
to calculate the earnings of each
B employee for the coming week.
7. For A =
΄−2 − 4 ΅ and B = ΄ 5 7΅, show that
2
5 3 4
d) What is the restaurant’s total payroll for
these employees?
a) (A−1)−1 = A
b) (AB)−1 = B −1A−1
c) (A t )−1 = (A−1) t


10. According to a 1998 general social survey c) What is the total cost of cloth and labour
conducted by Statistics Canada, the ten most for filling the order in part a)?
popular sports for people at least 15 years
old are as follows: 12. Use the coding matrix
each message.
΄ −2 −3 ΅ to encode
2
5
Sport Total (%) Male (%) Female (%)
Golf 7.4 11.1 3.9 a) BIRTHDAY PARTY FRIDAY
Ice Hockey 6.2 12.0 0.5 b) SEE YOU SATURDAY NIGHT
Baseball 5.5 8.0 3.1
Swimming 4.6 3.6 5.6 13. Application Use the decoding matrix

΄ −1 −3 ΅ to decode each message.
Basketball 3.2 4.6 1.9 2
Volleyball 3.1 3.3 2.8 2
Soccer 3.0 4.6 1.5 a) 64, 69, 38, 45, 54, 68, 31, 44, 5, 115, 3,
Tennis 2.7 3.6 1.8 70, 40, 83, 25, 49
Downhill/Alpine Skiing 2.7 2.9 2.6
b) 70, 47, 39, 31, 104, 45, 61, 25, 93, 68, 57,
Cycling 2.5 3.0 2.0
44, 55, 127, 28, 76
In 1998, about 11 937 000 males and
14. a) Create a secret message about 16 to 24
12 323 000 females in Canada were at least
letters long using the coding
15 years old. Determine how many males
and how many females declared each of the ΄
matrix 3 5 .
1 2 ΅
above sports as their favourite. Describe how
b) Trade messages with a classmate and
you used matrices to solve this problem.
decode each other’s messages.
11. Application A company manufacturing
15. Quality education at a school requires open
designer T-shirts produces five sizes: extra-
communication among many people.
small, small, medium, large, and extra-large.
Superintendent
The material and labour needed to produce
a box of 100 shirts depends on the size of
the shirts.
Cloth per Labour per Administration
Size shirt (m2) 100 shirts (h)
Extra-small 0.8 8
Small 0.9 8.5 Teachers Guidance
Medium 1.2 9
Large 1.5 10
Extra-large 2.0 11
Students Parents
a) How much cloth and labour are required
to fill an order for 1200 small, 1500 a) Represent this network as a matrix, A.
medium, 2500 large, and 2000 extra- b) Explain the meaning of any entry, ai j , of
large T-shirts? matrix A.
b) If the company pays $6.30 per square c) Describe what the sum of the entries in
metre for fabric and $10.70 per hour for the third column represents.
labour, find the cost per box for each size d) Calculate A2.
of T-shirt.


e) How many indirect links exist with
C
exactly one intermediary between the
18. Inquiry/Problem Solving Create your own
principal and parents? List these links.
network problem, then exchange problems
f) Calculate A + A2. Explain what with a classmate. Solve both problems and
information this matrix provides. compare your solutions with those of your
16. Network matrices provide another approach
classmate. Can you suggest any
to the Koenigsberg bridges example on improvements for either set of solutions?
page 44. 19. Show how you could use inverse matrices
Blacksmith to solve any system of equations in two
Bridge
Honey Wooden
variables whose matrix of coefﬁcients has
Bridge Bridge an inverse.
D
E G
20. Communication Research encryption
Merchants C
Bridge
techniques on the Internet. What is meant
by 128-bit encryption? How does the system
A F of private and public code keys work?
B High
Green Bridge 21. Inquiry/Problem Solving
Bridge Connecting
Bridge a) Suppose you receive a coded message
like the one in Example 4, but you do
Use network matrices to answer the not know the coding matrix or its
following questions. inverse. Describe how you could use a
a) How many ways can you get from computer to break the code and decipher
Honey Bridge to Connecting Bridge by the message.
crossing only one of the other bridges? b) Describe three methods you could use
List these routes. to make a matrix code harder to break.
b) How many ways can you get from
22. a) Show that, for any m × n matrix A and
Blacksmith Bridge to Connecting Bridge
any n × p matrix B, (AB)t = B tA t.
without crossing more than one of the
other bridges? b) Show that, if a square matrix C has an
inverse C –1, then C t also has an inverse,
c) Is it possible to travel from Wooden
and (C t )–1 = (C –1) t.
Bridge to Green Bridge without crossing
at least two other bridges?

17. Use network matrices to ﬁnd the number
pt
ha e of VIA Rail routes from
C

r

a) Toronto to Montréal with up to two
m
P

r
oble
change-overs
b) Kingston to London with up to three
change-overs


Review of Key Concepts

1.1 The Iterative Process 1.2 Data Management Software
Refer to the Key Concepts on page 10. Refer to the Key Concepts on page 21.

1. a) Draw a tree diagram showing your direct 5. List three types of software that can be used
ancestors going back four generations. for data management, giving an example of
b) How many direct ancestors do you have the data analysis you could do with each
in four generations? type.

2. a) Describe the algorithm used to build the 6. Evaluate each spreadsheet expression.
iteration shown. a) F2+G7–A12
b) Continue the iteration for eight more where F2=5, G7= –9, and A12=F2+G7
rows. b) PROD(D3,F9)

c) Describe the resulting iteration.
where D3=6 and F9=5
c) SQRT(B1)
MATH
MATHMATH where B1=144
MATH MATH
7. Describe how to reference cells A3 to A10
MATHMATHMATHMATH
MATH MATH in one sheet of a spreadsheet into cells B2
MATHMATH MATHMATH to B9 in another sheet.

3. a) Construct a Pythagoras fractal tree using 8. Use a spreadsheet to convert temperatures
the following algorithm. between −30° C and 30° C to the
Step 1: Construct a square. Fahrenheit scale, using the formula
Step 2: Construct an isosceles right Fahrenheit = 1.8 × Celsius + 32. Describe
triangle with the hypotenuse on how you would list temperatures at two-
one side of the square. degree intervals in the Celsius column.
Step 3: Construct a square on each of
the other sides of the triangle. 1.3 Databases
Repeat this process, with the newly Refer to the Key Concepts on page 31.
drawn squares to a total of four
9. Describe the characteristics of a well-
iterations.
organized database.
b) If the edges in the ﬁrst square are 4 cm,
determine the total area of all the squares 10. Outline a design for a database of a shoe
in the fourth iteration. store’s customer list.
c) Determine the total area of all the
11. a) Describe the types of data that are
squares in the diagram.
available from Statistics Canada’s
4. Design an iterative process using the percent E-STAT database.
reduction capabilities of a photocopier. b) What can you do with the data once
you have accessed them?


12. What phrase would you enter into a search 18. State whether each network is
engine to find i) connected
a) the top-selling cookbook in Canada? ii) traceable
b) the first winner of the Fields medal? iii) planar
c) a list of movies in which bagpipes are a) A C b) P Q
played?

1.4 Simulations U R
Refer to the Key Concepts on page 39. B D

13. List three commonly used simulations and T S
a reason why each is used. c) L

14. Write out the function to generate a random
J M
integer between 18 and 65 using K
a) a graphing calculator
N
b) a spreadsheet
19. For each network in question 18, verify that
15. A chocolate bar manufacturer prints one of
a repeating sequence of 50 brainteasers on V − E + R = 2, where V is the number of
the inside of the wrapper for each of its vertices, E is the number of edges, and R is
chocolate bars. Describe a manual the number of regions in a graph.
simulation you could use to estimate the 20.The following is a listing of viewing requests
chances of getting two chocolate bars with submitted by patrons of a classic film
the same brainteaser if you treat yourself to festival. Use graph theory to set up the
one of the bars every Friday for five weeks. shortest viewing schedule that has no
16. Outline how you would use technology to
conflicts for any of these patrons.
run a simulation 500 times for the scenario Person A: Gone With the Wind, Curse of The
in question 15. Mummy, Citizen Kane
Person B: Gone With the Wind, Jane Eyre
1.5 Graph Theory Person C: The Amazon Queen, West Side
Refer to the Key Concepts on page 48. Story, Citizen Kane
Person D: Jane Eyre, Gone With the Wind,
17. How many colours are needed to colour West Side Story
each of the following maps? Person E: The Amazon Queen, Ben Hur
a) b) A B
C C
D
A B D
E F
E
G

Review of Key Concepts • MHR 79

21. Below is a network showing the Calculate, if possible,
relationships among a group of children. a) A + C b) C − B
The vertices are adjacent if the children
are friends. c) A + B d) 3D
1
Sarah Mai e) −ᎏᎏ C f) 3(B + D)
2
g) A t + B h) B t + C t
Deqa Priya
25. The manager of a sporting goods store takes
inventory at the end of the month and ﬁnds
Tanya Afra
15 basketballs, 17 volleyballs, 4 footballs,
15 baseballs, 8 soccer balls, 12 packs of
a) Rewrite the network in table form. tennis balls, and 10 packs of golf balls. The
b) Are these children all friends with each manager orders and receives a shipment of
other? 10 basketballs, 3 volleyballs, 15 footballs,
c) Who has the most friends? 20 baseballs, 12 soccer balls, 5 packs of
d) Who has the fewest friends?
tennis balls, and 15 packs of golf balls.
During the next month, the store sells
1.6 Modelling With Matrices 17 basketballs, 13 volleyballs, 17 footballs,
Refer to the Key Concepts on page 59. 12 baseballs, 12 soccer balls, 16 packs of
tennis balls, and 23 packs of golf balls.

΄ ΅
2 −1 5
a) Represent the store’s stock using three
0 4 3
22. For the matrix A = , matrices, one each for the inventory, new
7 −8 −6
stock received, and items sold.
−2 9 1
b) How many of each item is in stock at the
a) state the dimensions end of the month?
b) state the value of entry c) At the beginning of the next month, the
i) a32 ii) a13 iii) a41 manager is asked to send 20% of the
c) list the entry with value store’s stock to a new branch that is
about to open. How many of each item
i) 3 ii) 9 iii) −1
will be left at the manager’s store?
23. Write a 4 × 3 matrix, A, with the property
26. Outline the procedure you would use to
that aij = i × j for all entries.
subtract one matrix from another

΄ ΅
8 −2 a) manually
2 −1 , B =
24. Given A =
΄ 3
−7 0 5 ΅ 3 4 ,
2 5
b) using a graphing calculator
c) using a spreadsheet

΄ ΅
4 3
1 −4 , and D = −1
C= ΄ −5
6
9 0 ΅ 6
7 .
2

80 MHR • Introduction to Probability

1.7 Problem Solving With Matrices 31. a) Write an equation to show the
Refer to the Key Concepts on page 74. relationship between a matrix and its
inverse.

΄ ΅
−1
27. Let A =
΄ −6 5΅, B = ΄ −5 7 ΅,
4
3
1 0
b) Show that 20
1.5 0
−1.5 −13 is the
−7.5 0.5 5

΄ ΅ ΄ ΅
3 6 −1 5

΄ ΅
C = 2 0 4 , and D = 4 . 4 2 6
−5 −2 8 −3 inverse of 10 0 2 .
Calculate, if possible, 5 3 9
a) AB b) BA c) A2
d) DC e) C 2
c) Find the inverse of
΄ 4 5 ΅.
2 3
28. a) Write the transpose of matrices 32. The following diagram illustrates the food

΄ ΅
A = 1 5 and B = 0 4 .
8 −2 ΄
6 −1 ΅ chains in a pond.

b) Show whether (AB) = B tA t.
t Plants

29. A small accounting firm charges $50 per Small Fish Large Fish
hour for preparing payrolls, $60 per hour
for corporate tax returns, and $75 per hour
for audited annual statements. The firm did Snails Bacteria
the following work for three of its clients:
XYZ Limited, payrolls 120 hours, tax a) Represent these food chains as a network
returns 10 hours, auditing 10 hours matrix, A.
YZX Limited, payrolls 60 hours, tax b) Calculate A2.
returns 8 hours, auditing 8 hours c) How many indirect links with exactly
ZXY Limited, payrolls 200 hours, tax one intermediate step are there from
returns 15 hours, auditing 20 hours plants to snails?
a) Use matrices to determine how much the d) Calculate A + A2. Explain the meaning
accounting firm should bill each client. of any entry in the resulting matrix.
b) How can you determine the total billed e) Calculate A3.
to the three clients? f) List all the links with two intermediate

30. Suppose you were to encode a message by
steps from plants to bacteria.
writing it in matrix form and multiplying by
a coding matrix. Would your message be
more secure if you then multiplied the
resulting matrices by another coding matrix
with the same dimensions as the first one?
Explain why or why not.


Chapter Test

ACHIEVEMENT CHART

Category Communication Application
Questions All 8, 9, 14 1, 2, 5, 6, 7, 8, 9, 14 9, 10, 13, 14

1. a) Describe an iterative 5. Suppose that, on January 10, you borrowed
process you could use $1000 at 6% per year compounded monthly
to draw the red path. (0.5% per month). You will be expected to
b) Complete the path. repay $88.88 a month for 1 year. However,
the final payment will be less than $88.88.
You set up a spreadsheet with the following
2. Find the first few terms of the recursion column headings: MONTH, BALANCE,
1 PAYMENT, INTEREST, PRINCIPAL,
formula tn = ᎏ , given t1 = 0. NEW BALANCE
tn − 1 + 2
Is there a pattern to these terms? If so, The first row of entries would be:
describe the pattern. MONTH: February
BALANCE: 1000.00
3. A “fan-out” calling system is frequently used PAYMENT: 88.88
to spread news quickly to a large number of INTEREST: 5.00
people such as volunteers for disaster relief. PRINCIPAL: 83.88
The first person calls three people. Each of NEW BALANCE: 916.12
those people calls an additional three people; Describe how you would
each of whom calls an additional three a) use the cell referencing formulas and
people, and so on. the Fill feature to complete the table
a) Use a tree diagram to illustrate a fan-out b) determine the size of the final payment
calling system with sufficient levels to on January 10 of the following year
call 50 people.
c) construct a line graph showing the
b) How many levels would be sufficient to declining balance
call 500 people?
6. Describe how you would design a database
4. Rewrite each of the following expressions as of the daily travel logs for a company’s
spreadsheet functions. salespersons.
a) C1+C2+C3+C4+C5+C6+C7+C8
7. Describe three different ways to generate
b) The smallest value between cells A5 random integers between 1 and 50.
and G5
8. a) Redraw this map as a
5 − ͙6ෆ
c) ᎏ network.
10 + 15
b) How many colours are
needed to colour the
map? Explain your reasoning.


9. A salesperson must visit each of the towns b) What is the value of entry a23?
on the following map. c) Identify the entry of matrix A with
Pinkford 55 Orangeton value −2.
67
Blacktown
d) Is it possible to calculate A2? Explain.
50 60
55

΄΅
46 2
Blueton
Brownhill Redville
35
86
12. Let A = 1 , B = [7 5 0], C = 4 8 ,
5 5 −3 ΄ ΅
53 38 40

΄ ΅
Whiteford
49 Greenside 8 −2
a) Is there a route that goes through each ΄
9 1 ΅
D = 2 −7 , and E = 5 0 .
−4 1
town only once? Explain.
b) Find the shortest route that begins and Calculate, if possible,
ends in Pinkford and goes through all the a) 2C + D b) A + B c) AD d) EC e) E t
towns. Show that it is the shortest route.
13. A local drama club staged a variety show for
10. The following map four evenings. The admission for adults was
shows the bridges $7.00, for students $4.00, and for children
of Uniontown, 13 years of age and under $2.00. On
situated on the Wednesday, 52 adult tickets, 127 student
banks of a river and on three islands. Use tickets, and 100 child tickets were sold; on
graph theory to determine if a continuous Thursday, 67 adult tickets, 139 student
path could traverse all the bridges once each. tickets, and 115 child tickets were sold; on
Friday, 46 adult tickets, 115 student tickets,

΄ ΅
4 −2 6
–8 5 9 and 102 child tickets were sold; and on
11. Let A = . Saturday, 40 adult tickets, 101 student
0 1 −1
3 −7 −3 tickets, and 89 child tickets were sold. Use
matrices to calculate how much money was
a) State the dimensions of matrix A.
collected from admissions.

ACHIEVEMENT CHECK

Knowledge/Understanding Thinking/Inquiry/Problem Solving Communication Application
14. The network diagram below gives the cost of ﬂights between Montréal
$579 $249
ﬁve Canadian cities.
$469
a) Construct a network matrix A for these routes. $199
Halifax
Vancouver $269
b) Calculate A2 and A3. Winnipeg
$438 $398
c) How many ways can a person travel from Halifax to $508
Vancouver by changing planes exactly twice? Describe Toronto
each route. Which route is most economical?

Chapter Test • MHR 83

To o l s f o r D a t a M a n a g e m e n t P r o j e c t

Wrap-Up
Implementing Your Action Plan 8. From your rankings, select the top five
1. With your whole class or a small group, universities or community colleges. Draw
brainstorm criteria for ranking universities a diagram of the distances from each
and community colleges. List the three university or college to the four others and
universities or colleges that you think will to your home. Then, use graph theory to
most likely be the best choices for you. determine the most efficient way to visit
each of the five universities or community
2. Have a class discussion on weighting colleges during a five-day period, such as a
systems. March break vacation.
3. Look up the Maclean’s university and 9. Based on your project, select your top three
community college rankings in a library or choices. Comment on how this selection
on the Internet. Note the criteria that compares with your original list of top
Maclean’s uses. choices.
4. Determine your own set of criteria. These
may include those that Maclean’s uses or
Suggested Resources
others, such as travelling distances,
• Maclean’s magazine rankings of universities
programs offered, size of the city or town
and community colleges
where you would be living, and
opportunities for part-time work. • Other publications ranking universities and
community colleges
5. Choose the ten criteria you consider most • University and community college calendars
important. Research any data you need to
• Guidance counsellors
rate universities and colleges with these
criteria. • Map of Ontario
• Spreadsheets
6. Assign a weighting factor to each of the ten
criteria. For example, living close to home Refer to section 9.3 for information on
may be worth a weighting of 5 and tuition implementing an action plan and Appendix C
cost may be worth a weighting of 7. for information on research techniques.

7. Use a spreadsheet and matrix methods to
determine an overall score for each
university or community college in
Ontario. Then, rank the universities or www.mcgrawhill.ca/links/MDM12
community colleges on the spreadsheet. For details of the Maclean’s rankings of universities
Compare your rankings with those in and colleges, visit the above web site and follow
Maclean’s magazine. Explain the similarities the links.
or differences.

84 MHR • Tools for Data Management Project

Evaluating Your Project Presentation
1. Reflect on your weighting formula and Prepare a written report on your findings.
whether you believe it fairly ranks the Include
universities and community colleges in • the raw data
Ontario.
• a rationale for your choice of criteria
2. Compare your rating system to that used • a rationale for your weightings
by one of your classmates. Can you suggest • a printout of your spreadsheet
improvements to either system?
• a diagram showing the distances between
3 What went well in this project? your five highest-ranked universities or
community colleges and the route you would
4. If you were to do the project over again, use to visit them
what would you change? Why?
• a summary of your findings
5. If you had more time, how would you
extend this project?

6. What factors could change between now
and when you make your final decision
about which university or college to attend?

Preparing for
the Culminating Project

Applying Project Skills Keeping on Track
Consider how the data management tools you Now is a good time to draw up a schedule
used on this project could be applied to the for your culminating project and to
culminating project in Chapter 9 to investigate methods for selecting a topic.
• access resources Refer to Chapter 9 for an overview of how to
• carry out research prepare a major project. Section 9.1 suggests
• carry out an action plan methods for choosing a topic. Also, consider
• evaluate your project how to find the information you will need in
• summarize your findings in a written report order to choose your topic.

Refine/Redefine

Define the Define Develop an Implement Evaluate Your Prepare Present Your Constructively
Problem Your Task Action Plan Your Action Investigation Written Investigation Critique the
Plan and Its Results Report and Its Results Presentations
of Others

Tools for Data Management Project: Wrap-Up • MHR 85

Career Connection
Cryptographer
In this digital era, information is sent with blinding
speed around the world. These transmissions need to be
both secure and accurate. Although best known for their
work on secret military codes, cryptographers also
design and test computerized encryption systems that
protect a huge range of sensitive data including
telephone conversations among world leaders, business
negotiations, data sent by credit-card readers in retail
stores, and ﬁnancial transactions on the Internet.
Encrypted passwords protect hackers from reading or
disrupting critical databases. Even many everyday
devices, such as garage-door openers and TV remote
controls, use codes.

Cryptographers also develop error-correcting codes.
Adding these special codes to a signal allows a computer
receiving it to detect and correct errors that have
occurred during transmission. Such codes have
numerous applications including CD players,
automotive computers, cable TV networks, and pictures
sent back to Earth by interplanetary spacecraft.

Modern cryptography is a marriage of mathematics and
computers. A cryptographer must have a background
in logic, matrices, combinatorics, and computer
programming as well as fractal, chaos, number, and
graph theory. Cryptographers work for a wide variety
of organizations including banks, government ofﬁces,
the military, software developers, and universities.


Visit the above web site and follow the links
for more information about a career as a
cryptographer and about other careers
related to mathematics.


Statistics Project

Life Expectancies
Background
Do women live longer than men? Do people live longer in warmer climates? Are
people living longer today than 50 years ago? Do factors such as education and
income affect life expectancy? In this project, you will answer such questions by
applying the statistical techniques described in the next two chapters.

Your Task
Research and analyse current data on life expectancies in Canada, and perhaps in
other countries. You will use statistical analysis to compare and contrast the data,
detect trends, predict future life expectancies, and identify factors that may affect
life expectancies.

Developing an Action Plan
You will need to ﬁnd sources of data on life expectancies and to choose the kinds
of comparisons you want to make. You will also have to decide on a method for
handling the data and appropriate techniques for analysing them.

<<Section Project: Introduction • MHR
Statistics number and title>> 87

2
2
PT ER
ER
Statistics of One Variable
CHA


Locate data to answer questions of signiﬁcance or personal interest, by 2.2
searching well-organized databases.

Use the Internet effectively as a source for databases. 2.2

Demonstrate an understanding of the purpose and the use of a variety 2.3, 2.4
of sampling techniques.

Describe different types of bias that may arise in surveys. 2.4

Illustrate sampling bias and variability by comparing the characteristics 2.4, 2.5, 2.6
of a known population with the characteristics of samples taken
repeatedly from that population, using different sampling techniques.

Organize and summarize data from secondary sources, using 2.1, 2.2, 2.5,
technology. 2.6

Compute, using technology, measures of one-variable statistics (i.e., 2.5, 2.6
the mean, median, mode, range, interquartile range, variance, and
standard deviation), and demonstrate an understanding of the
appropriate use of each measure.

Interpret one-variable statistics to describe characteristics of a data set. 2.5, 2.6

Describe the position of individual observations within a data set, using 2.6
z-scores and percentiles.

Explain examples of the use and misuse of statistics in the media. 2.4

Assess the validity of conclusions made on the basis of statistical studies, 2.5, 2.6
by analysing possible sources of bias in the studies and by calculating
and interpreting additional statistics, where possible.

Explain the meaning and the use in the media of indices based on 2.2
surveys.

In earlier times they had no
statistics, and so they had
to fall back on lies. Hence
the huge exaggerations of
primitive literature—giants
or miracles or wonders!
They did it with lies and
we do it with statistics; but
it is all the same.
—Stephen Leacock (1869–1944)

Facts are stubborn, but
statistics are more pliable.
—Mark Twain (1835–1910)

Chapter Problem
Contract Negotiations As these questions suggest, statistics could
François is a young NHL hockey player be used to argue both for and against
whose first major-league contract is up for a large salary increase for François.
renewal. His agent wants to bargain for a However, the statistics themselves are
better salary based on François’ strong not wrong or contradictory. François’
performance over his first five seasons with agent and the team’s manager will,
the team. Here are some of François’ understandably, each emphasize only the
statistics for the past five seasons. statistics that support their bargaining
positions. Such selective use of statistics
Season Games Goals Assists Points is one reason why they sometimes receive
negative comments such as the quotations
1 20 3 4 7 above. Also, even well-intentioned
2 45 7 11 18 researchers sometimes inadvertently use
3 76 19 25 44 biased methods and produce unreliable
4 80 19 37 56 results. This chapter explores such sources
5 82 28 36 64 of error and methods for avoiding them.
Total 303 76 113 189 Properly used, statistical analysis is a
1. How could François’ agent use these powerful tool for detecting trends and
statistics to argue for a substantial pay drawing conclusions, especially when you
increase for his client? have to deal with large sets of data.

2. Are there any trends in the data that the
team’s manager could use to justify a
more modest increase?



1. Fractions, percents, decimals The following 6. Graphing data Consider the following
amounts are the total cost for the items graph, which shows the average price
including the 7% goods and services tax of thingamajigs over time.
(GST) and an 8% provincial sales tax (PST).

Price of Thingamajigs ($)
Determine the price of each item. 1.90
1.80
a) watch $90.85
1.70
b) CD $19.54 1.60

c) bicycle $550.85 1.50
1.40
d) running shoes $74.39
0 1996 1997 1998 1999 2000 2001
2. Fractions, percents, decimals Year
a) How much will Josh make if he receives
an 8% increase on his pay of $12.50/h? a) What was the price of thingamajigs
in 1996?
b) What is the net increase in Josh’s take-
home pay if the payroll deductions total b) In what year did the price ﬁrst rise
17%? above $1.50?
c) Describe the overall trend over the
3. Fractions, percents, decimals What is the time period shown.
percent reduction on a sweater marked
d) Estimate the percent increase in the
down from $50 to $35?
price of thingamajigs from 1996 to 2001.
4. Fractions, percents, decimals Determine e) List the domain and range of these data.
the cost, including taxes, of a VCR sold at
a 25% discount from its original price of 7. Graphing data The table below gives the
$219. number of CDs sold at a music store on
each day of the week for one week.
5. Mean, median, mode Calculate the mean, Day Number of CDs Sold
median, and mode for each set of data.
Monday 48
a) 22, 26, 28, 27, 26 Tuesday 52
b) 11, 19, 14, 23, 16, 26, 30, 29 Wednesday 44
c) 10, 18, 30, 43, 18, 13, 10 Thursday 65
d) 70, 30, 25, 52, 12, 70 Friday 122

e) 370, 260, 155, 102, 126, 440 Saturday 152
Sunday 84
f) 24, 32, 37, 24, 32, 38, 32, 36, 35, 42
Display the data on a circle graph.

90 MHR • Statistics of One Variable

2.1 Data Analysis With Graphs

Statistics is the gathering, organization, analysis, and
presentation of numerical information. You can apply
statistical methods to almost any kind of data.
Researchers, advertisers, professors, and sports
announcers all make use of statistics. Often, researchers
gather large quantities of data since larger samples
usually give more accurate results. The first step in the
analysis of such data is to find ways to organize, analyse,
and present the information in an understandable form.

I N V E S T I G AT E & I N Q U I R E : U s i n g G r a p h s t o A n a l y s e D a t a

1. Work in groups or as a class to design a fast and efficient way to survey your
class about a simple numerical variable, such as the students’ heights or the
distances they travel to school.
2. Carry out your survey and record all the results in a table.
3. Consider how you could organize these results to look for any trends or
patterns. Would it make sense to change the order of the data or to divide
them into groups? Prepare an organized table and see if you can detect any
patterns in the data. Compare your table to those of your classmates. Which
methods work best? Can you suggest improvements to any of the tables?
4. Make a graph that shows how often each value or group of values occurs in
your data. Does your graph reveal any patterns in the data? Compare your
graph to those drawn by your classmates. Which graph shows the data most
clearly? Do any of the graphs have other advantages? Explain which graph
you think is the best overall.
5. Design a graph showing the total of the frequencies of all values of the
variable up to a given amount. Compare this cumulative-frequency graph
to those drawn by your classmates. Again, decide which design works best
and look for ways to improve your own graph and those of your classmates.

The unprocessed information collected for a study is called raw data. The quantity
being measured is the variable. A continuous variable can have any value within
a given range, while a discrete variable can have only certain separate values (often
integers). For example, the height of students in your school is a continuous
variable, but the number in each class is a discrete variable. Often, it is useful to
know how frequently the different values of a variable occur in a set of data.
Frequency tables and frequency diagrams can give a convenient overview of the
distribution of values of the variable and reveal trends in the data.

2.1 Data Analysis With Graphs • MHR 91

A histogram is a special form of bar graph in which the areas of the bars are
proportional to the frequencies of the values of the variable. The bars in a histogram
are connected and represent a continuous range of values. Histograms are used for
variables whose values can be arranged in numerical order, especially continuous
variables, such as weight, temperature, or travel time. Bar graphs can represent all
kinds of variables, including the frequencies of separate categories that have no set
order, such as hair colour or citizenship. A frequency polygon can illustrate the
same information as a histogram or bar graph. To form a frequency polygon, plot
frequencies versus variable values and then join the points with straight lines.

10 10 10
Frequency

Frequency

Frequency
5 5 5

0 0 Red 0
5 10 15 20 25 30 Blond Brown Black Purple Green 5 10 15 20 25
Travel Time to School (min) Hair Colour Travel Time to School (min)
Histogram Bar Graph Frequency Polygon

A cumulative-frequency graph or ogive shows the running
30
total of the frequencies from the lowest value up.
25
Cumulative Frequency
20

15
To learn more about histograms, visit the above
web site and follow the links. Write a short 10
description of how to construct a histogram.
5

0
5 10 15 20 25
Travel Time to School (min)

Example 1 Frequency Tables and Diagrams

Here are the sums of the two numbers from 50 rolls of a pair of standard dice.
11 4 4 10 8 7 6 6 5 10 7 9 8 8
4 7 9 11 12 10 3 7 6 9 5 8 6 8
2 6 7 5 11 2 5 5 6 6 5 2 10 9
6 5 5 5 3 9 8 2

a) Use a frequency table to organize these data.
b) Are any trends or patterns apparent in this table?
c) Use a graph to illustrate the information in the frequency table.


d) Create a cumulative-frequency table and graph for the data.
e) What proportion of the data has a value of 6 or less?

Solution

a) Go through the data and tally the frequency of each value of Sum Tally Frequency
the variable as shown in the table on the right. 2 |||| 4
3 || 2
b) The table does reveal a pattern that was not 4 ||| 3
obvious from the raw data. From the 5 |||| |||| 9
frequency column, notice that the middle 6 |||| ||| 8
values tend to be the most frequent while 7 |||| 5
the high and low values are much less 8 |||| | 6
frequent. 9 |||| 5
10 |||| 4
11 ||| 3
12 | 1

c) The bar graph or
8 8

Frequency
frequency polygon
Frequency

6 6
makes the pattern 4 4
in the data more 2 2
apparent. 0 0
2 3 4 5 6 7 8 9 10 11 12 2 3 4 5 6 7 8 9 10 11 12
Sum Sum

d) Add a column for cumulative frequencies to the table. Each value in this
column is the running total of the frequencies of each sum up to and
including the one listed in the corresponding row of the sum column.
Graph these cumulative frequencies against the values of the variable.
Sum Tally Frequency Cumulative Frequency
Cumulative Frequency

50
2 |||| 4 4 40
3 || 2 6 30
4 ||| 3 9 20
5 |||| |||| 9 18
10
6 |||| ||| 8 26
0
7 |||| 5 31 2 3 4 5 6 7 8 9 10 11 12
8 |||| | 6 37 Sum
9 |||| 5 42
10 |||| 4 46
11 ||| 3 49
12 | 1 50

e) From either the cumulative-frequency column or the diagram, you can
see that 26 of the 50 outcomes had a value of 6 or less.


When the number of measured values is large, data are usually grouped into
classes or intervals, which make tables and graphs easier to construct and
interpret. Generally, it is convenient to use from 5 to 20 equal intervals that
cover the entire range from the smallest to the largest value of the variable.
The interval width should be an even fraction or multiple of the measurement
unit for the variable. Technology is particularly helpful when you are working
with large sets of data.

Example 2 Working With Grouped Data

This table lists the daily high temperatures in July for a city in southern Ontario.
Day 1 2 3 4 5 6 7 8 9 10 11
Temperature (°C) 27 25 24 30 32 31 29 24 22 19 21

Day 12 13 14 15 16 17 18 19 20 21 22
Temperature (°C) 25 26 31 33 33 30 29 27 28 26 27

Day 23 24 25 26 27 28 29 30 31
Temperature (°C) 22 18 20 25 26 29 32 31 28

a) Group the data and construct a frequency table, a histogram or frequency See Appendix B for
polygon, and a cumulative-frequency graph. more detailed
b) On how many days was the maximum temperature 25°C or less? On how information about
many days did the temperature exceed 30°C? technology functions
and keystrokes.


a) The range of the data is 33°C − 18°C = 15°C. You could use ﬁve 3-degree
intervals, but then many of the recorded temperatures would fall on the
interval boundaries. You can avoid this problem by using eight 2-degree
intervals with the lower limit of the ﬁrst interval at 17.5°C. The upper limit
of the last interval will be 33.5°C.

Use the STAT EDIT menu to make sure that lists L1 to L4 are clear, and then
enter the temperature data into L1. Use STAT PLOT to turn on Plot1 and
select the histogram icon. Next, adjust the window settings. Set Xmin and
Xmax to the lower and upper limits for your intervals and set Xscl to the
interval width. Ymin should be 0. Press GRAPH to display the histogram,
then adjust Ymax and Yscl, if necessary.


You can now use the TRACE instruction and the arrow keys to determine the
tally for each of the intervals. Enter the midpoints of the intervals into L2
and the tallies into L3. Turn off Plot1 and set up Plot2 as an x-y line plot of
lists L2 and L3 to produce a frequency polygon.

Use the cumSum( function from the LIST OPS menu to find the running totals
of the frequencies in L3 and store the totals in L4. Now, an x-y line plot of L2
and L4 will produce a cumulative-frequency graph.

b) Since you know that all the temperatures were in whole degrees, you can see
from the cumulative frequencies in L4 that there were 11 days on which the
maximum temperature was no higher than 25°C. You can also get this
information from the cumulative-frequency graph.

You cannot determine the exact number of days with temperatures over
30°C from the grouped data because temperatures from 29.5°C to 31.5°C
are in the same interval. However, by interpolating the cumulative-
frequency graph, you can see that there were about 6 days on which the
maximum temperature was 31°C or higher.


a) Enter the temperature data into column A and the midpoints of the intervals
into column B. Use the COUNTIF function in column C to tally the
cumulative frequency for each interval. If you use absolute cell referencing,
you can copy the formula down the column and then change just the upper
limit in the counting condition. Next, find the frequency for each interval by
finding the difference between its cumulative frequency and the one for the
previous interval.

You can then use the Chart feature to produce a frequency polygon by
graphing columns B and D. Similarly, charting columns B and C will
produce a cumulative-frequency graph.


In Corel® Quattro® Pro, you can also use the Histogram tool in the
Tools/Numeric Tools/Analysis menu to automatically tally the frequencies
and cumulative frequencies.

b) As in the solution using a graphing calculator, you can see from the
cumulative frequencies that there were 11 days on which the maximum
temperature was no higher than 25°C. Also, you can estimate from the
cumulative-frequency graph that there were 6 days on which the maximum
temperature was 31°C or higher. Note that you could use the COUNTIF
function with the raw data to ﬁnd the exact number of days with
temperatures over 30°C.


A relative-frequency table or diagram shows the frequency of a data Project
group as a fraction or percent of the whole data set. Prep

You may ﬁnd
Example 3 Relative-Frequency Distribution
frequency-
Here are a class’ scores 78 81 55 60 65 86 44 90
distribution diagrams
obtained on a data- 77 71 62 39 80 72 70 64 useful for your
management examination. 88 73 61 70 75 96 51 73 statistics project.
59 68 65 81 78 67

a) Construct a frequency table that includes a column for relative frequency.
b) Construct a histogram and a frequency polygon.
c) Construct a relative-frequency histogram and a relative-frequency polygon.
d) What proportion of the students had marks between 70% and 79%?

Solution

a) The lowest and highest scores are Score (%) Midpoint Tally Frequency Relative Frequency
39% and 96%, which give a range 34.5−39.5 37 | 1 0.033
of 57%. An interval width of 5 is 39.5−44.5 42 | 1 0.033
convenient, so you could use 44.5−49.5 47 − 0 0
13 intervals as shown here. To 49.5−54.5 52 | 1 0.033
determine the relative frequencies, 54.5−59.5 57 || 2 0.067
divide the frequency by the total 59.5−64.5 62 |||| 4 0.133
number of scores. For example, the 64.5−69.5 67 |||| 4 0.133
relative frequency of the ﬁrst interval 69.5−74.5 72 |||| | 6 0.200
1 74.5−79.5 77 |||| 4 0.133
is ᎏᎏ, showing that approximately 79.5−84.5 82 ||| 3 0.100
30
3% of the class scored between 84.5−89.5 87 || 2 0.067
34.5% and 39.5%. 89.5−94.5 92 | 1 0.033
94.5−99.5 97 | 1 0.033

b) The frequency polygon can be superimposed onto the same grid
as the histogram.

6
Frequency

4

2

0 37 42 47 52 57 62 67 72 77 82 87 92 97
Score


c) Draw the relative-frequency histogram and 0.2

Relative Frequency
the relative-frequency polygon using the same
procedure as for a regular histogram and
frequency polygon. As you can see, the only 0.1

difference is the scale of the y-axis.

0 37 42 47 52 57 62 67 72 77 82 87 92 97
Score

d) To determine the proportion of students with marks in the 70s, add the relative
frequencies of the interval from 69.5 to 74.5 and the interval from 74.5 to 79.5:
0.200 + 0.133 = 0.333

Thus, 33% of the class had marks between 70% and 79%.

Categorical data are given labels rather than being measured numerically.
For example, surveys of blood types, citizenship, or favourite foods all produce
categorical data. Circle graphs (also known as pie charts) and pictographs
are often used instead of bar graphs to illustrate categorical data.

Example 4 Presenting Categorical Data

The table at the right shows Canadians’ primary use Primary Use Households (%)
of the Internet in 1999. E-mail 15.8
Electronic banking 4.2
Illustrate these data with Purchase of goods and services 3.6
a) a circle graph
Medical or health information 8.6
b) a pictograph Formal education/training 5.8
Government information 7.8
Other speciﬁc information 14.7
General browsing 14.2
Playing games 6.7
Chat groups 4.7
Other Internet services 5.8
Obtaining music 5.0
Listening to the radio 3.1


Solution

a) Home Internet Use
Listening to the Radio 3.1%
Obtaining Music 5.0%
E-mail 15.8%
Other Internet Services 5.8%

Chat Groups 4.7% Electronic Banking 4.2%
Purchase of Goods and Services 3.6%
Playing Games 6.7% Medical or Health Information 8.6%
Formal Education/Training 5.8%
General Browsing 14.2% Government Information 7.8%

Other Specific Information 14.7%

b) There are numerous ways to represent the data with a pictograph.
The one shown here has the advantages of being simple and visually
indicating that the data involve computers.
Home Internet Use
E-mail
Electronic Banking
Purchase of Goods and Services
Medical or Health Information
Formal Education/Training
Government Information
Other Specific Information
General Browsing
Playing Games
Chat Groups
Other Internet Services
Obtaining Music
Listening to the Radio

Each represents 2% of households.

You can see from the example above that circle graphs are good for showing the
sizes of categories relative to the whole and to each other. Pictographs can use a
wide variety of visual elements to clarify the data and make the graph more
interesting. However, with both circle graphs and pictographs, the relative
frequencies for the categories can be hard to read accurately. While a well-
designed pictograph can be a useful tool, you will sometimes see pictographs
with distorted or missing scales or confusing graphics.


Key Concepts

• Variables can be either continuous or discrete.

• Frequency-distribution tables and diagrams are useful methods of summarizing
large amounts of data.

• When the number of measured values is large, data are usually grouped into
classes or intervals. This technique is particularly helpful with continuous
variables.
• A frequency diagram shows the frequencies of values in each individual
interval, while a cumulative-frequency diagram shows the running total of
frequencies from the lowest interval up.

• A relative-frequency diagram shows the frequency of each interval as a
proportion of the whole data set.

• Categorical data can be presented in various forms, including bar graphs,
circle graphs (or pie charts), and pictographs.


1. a) What information does a histogram present?
b) Explain why you cannot use categorical data in a histogram.

2. a) What is the difference between a frequency diagram and a cumulative-
frequency diagram?
b) What are the advantages of each of these diagrams?

3. a) What is the difference between a frequency diagram and a relative-
frequency diagram?
b) What information can be easily read from a frequency diagram?
c) What information can be easily read from a relative-frequency diagram?

4. Describe the strengths and weaknesses of circle graphs and pictographs.


Practise b) Use the circle graph to determine what
percent of the people surveyed chose
A vegetarian dishes.
1. Explain the problem with the intervals in c) Sketch a pictograph for the data.
each of the following tables.
d) Use the pictograph to determine whether
a) Age (years) Frequency more than half of the respondents chose
28−32 6 red-meat dishes.
33−38 8
4. a) Estimate the number of hours you spent
38−42 11
each weekday on each of the following
42−48 9
activities: eating, sleeping, attending
48−52 4
class, homework, a job, household
b) Score (%) Frequency chores, recreation, other.
61−65 5 b) Present this information using a circle
66−70 11 graph.
71−75 7 c) Present the information using a
76−80 4 pictograph.
91−95 1
2. Would you choose a histogram or a bar 5. The examination scores for a biology class
graph with separated bars for the data listed are shown below.
below? Explain your choices.
68 77 91 66 52 58 79 94 81
a) the numbers from 100 rolls of a standard 60 73 57 44 58 71 78 80 54
die 87 43 61 90 41 76 55 75 49
b) the distances 40 athletes throw a shot-put
a) Determine the range for these data.
c) the ages of all players in a junior lacrosse
b) Determine a reasonable interval size
league
and number of intervals.
d) the heights of all players in a junior
c) Produce a frequency table for the
lacrosse league
grouped data.
3. A catering service conducted a survey asking d) Produce a histogram and frequency
respondents to choose from six different hot polygon for the grouped data.
meals. e) Produce a relative-frequency polygon
Meal Chosen Number for the data.
Chicken cordon bleu 16
f) Produce a cumulative-frequency polygon
New York steak 20 for the data.
Pasta primavera (vegetarian) 9
g) What do the frequency polygon, the
Lamb chop 12
relative-frequency polygon, and the
Grilled salmon 10 cumulative-frequency polygon each
Mushroom stir-fry with almonds (vegetarian) 5 illustrate best?
a) Create a circle graph to illustrate these
data.


b) Create a frequency table and diagram.
B
6. a) Sketch a bar graph to show the results c) Create a cumulative-frequency diagram.
you would expect if you were to roll a d) How might the store owner use this
standard die 30 times. information in planning sales
b) Perform the experiment or simulate it promotions?
with software or the random-number 9. The speeds of 24 motorists ticketed for
generator of a graphing calculator. exceeding a 60-km/h limit are listed below.
Record the results in a table.
75 72 66 80 75 70 71 82
c) Produce a bar graph for the data you
69 70 72 78 90 75 76 80
collected. 75 96 91 77 76 84 74 79
d) Compare the bar graphs from a) and c).
Account for any discrepancies you a) Construct a frequency-distribution table
observe. for these data.
b) Construct a histogram and frequency
7. Application In order to set a reasonable price polygon.
for a “bottomless” cup of coffee, a restaurant
c) Construct a cumulative-frequency
owner recorded the number of cups each
diagram.
customer ordered on a typical afternoon.
d) How many of the motorists exceeded
2 1 2 3 0 1 1 1 2 2 the speed limit by 15 km/h or less?
1 3 1 4 2 0 1 2 3 1
e) How many exceeded the speed limit by
a) Would you present these data in a over 20 km/h?
grouped or ungrouped format? Explain
your choice. 10. Communication This table summarizes the
pt
ha e salaries for François’ hockey team.
b) Create a frequency table and diagram.
C

r

Salary ($) Number of Players
m
P

r
c) Create a cumulative-frequency diagram. oble
300 000 2
d) How can the restaurant owner use this 500 000 3
information to set a price for a cup of 750 000 8
coffee? What additional information
900 000 6
would be helpful?
1 000 000 2
8. Application The list below shows the value 1 500 000 1
of purchases, in dollars, by 30 customers at 3 000 000 1
a clothing store. 4 000 000 1
55.40 48.26 28.31 14.12 88.90 34.45 a) Reorganize these data into appropriate
51.02 71.87 105.12 10.19 74.44 29.05 intervals and present them in a frequency
43.56 90.66 23.00 60.52 43.17 28.49 table.
67.03 16.18 76.05 45.68 22.76 36.73
39.92 112.48 81.21 56.73 47.19 34.45 b) Create a histogram for these data.
c) Identify and explain any unusual features
a) Would you present these data in a about this distribution.
grouped or ungrouped format? Explain
your choice.


11. Communication
a) What is the sum of all the relative
frequencies for any set of data?
b) Explain why this sum occurs.
b) Sketch a relative-frequency polygon to
12. The following relative-frequency polygon show the results you would expect if
was constructed for the examination scores these dice were rolled 100 times.
for a class of 25 students. Construct the c) Explain why your graph has the shape
frequency-distribution table for the students’ it does.
scores.
d) Use software or a graphing calculator
0.32
to simulate rolling the funny dice 100
0.28 times, and draw a relative-frequency
Relative Frequency

0.24 polygon for the results.
0.20 e) Account for any differences between
0.16 the diagrams in parts b) and d).
0.12
0.08 15. This cumulative-frequency diagram shows
0.04 the distribution of the examination scores
0 35 45 55 65 75 85 95
for a statistics class.
Score Cumulative Frequency
30
25
13. Inquiry/Problem Solving The manager of a
20
rock band suspects that MP3 web sites have 15
reduced sales of the band’s CDs. A survey of 10
fans last year showed that at least 50% had 5
purchased two or more of the band’s CDs.
0 34.5 44.5 54.5 64.5 74.5 84.5 94.5
A recent survey of 40 fans found they had
Score
purchased the following numbers of the
band’s CDs. a) What interval contains the greatest
2 1 2 1 3 1 4 1 0 1 number of scores? Explain how you can
0 2 4 1 0 5 2 3 4 1 tell.
2 1 1 1 3 1 0 5 4 2 b) How many scores fall within this interval?
3 1 1 0 2 2 0 0 1 3
16. Predict the shape of the relative-frequency
Does the new data support the manager’s
diagram for the examination scores of a
theory? Show the calculations you made to
ﬁrst-year university calculus class. Explain
reach your conclusion, and illustrate the
why you chose the shape you did. Assume
results with a diagram.
that students enrolled in a wide range of
C programs take this course. State any other
14. Inquiry/Problem Solving
assumptions that you need to make.
a) What are the possible outcomes for a roll
of two “funny dice” that have faces with
the numbers 1, 1, 3, 5, 6, and 7?


2.2 Indices

In the previous section, you used
tables and graphs of frequencies to
summarize data. Indices are another
way to summarize data and
recognize trends. An index relates
the value of a variable (or group of
variables) to a base level, which is
often the value on a particular date.
The base level is set so that the
index produces numbers that are
easy to understand and compare.
Indices are used to report on a wide
variety of variables, including prices
and wages, ultraviolet levels in
sunlight, and even the readability
of textbooks.

I N V E S T I G AT E & I N Q U I R E : C o n s u m e r P r i c e I n d e x

The graph below shows Statistics Canada’s
Unadjusted Consumer Price Index

118
consumer price index (CPI), which tracks 116
the cost of over 600 items that would be 114
(1992 = 100)

purchased by a typical family in Canada. 112
For this chart, the base is the cost of the 110
same items in 1992. 108
106
104
M J J J J J M
1996 1997 1998 1999 2000 2001

1. What trend do you see in this graph? Estimate the annual rate of increase.
2. Estimate the annual rate of increase for the period from 1992 to 1996.
Do you think the difference between this rate and the one from 1996 to
2001 is signiﬁcant? Why or why not?
3. What was the index value in February of 1998? What does this value tell
you about consumer prices at that time?


4. What would be the best way to estimate what the consumer price index
will be in May of 2003? Explain your reasoning.
5. Explain how the choice of the vertical scale in the graph emphasizes
changes in the index. Do you think this emphasis could be misleading?
Why or why not?

The best-known Canadian business index is the S&P/TSX Composite Index,
managed for the Toronto Stock Exchange by Standard & Poor’s Corporation.
Introduced in May, 2002, this index is a continuation of the TSE 300 Composite
Index®, which goes back to 1977. The S&P/TSX Composite Index is a measure
of the total market value of the shares of over 200 of the largest companies traded
on the Toronto Stock Exchange. The index is the current value of these stocks
divided by their total value in a base year and then multiplied by a scaling factor.
When there are signiﬁcant changes (such as takeovers or bankruptcies) in any of
the companies in the index, the scaling factor is adjusted so that the values of the
index remain directly comparable to earlier values. Note that the composite index
weights each company by the total value of its shares (its market capitalization)
rather than by the price of the individual shares. The S&P/TSX Composite Index
usually indicates trends for major Canadian corporations reasonably well, but it
does not always accurately reﬂect the overall Canadian stock market.

Time-series graphs are often used to show how indices change over time.
Such graphs plot variable values versus time and join the adjacent data points
with straight lines.

Example 1 Stock Market Index

The following table shows the TSE 300 Composite Index® from 1971 to 2001.
TSE 300 Index (1975 = 1000)

10 000

8000

6000

4000

2000

0
1971
1973
1975
1977
1979
1981
1983
1985
1987
1989
1991
1993
1995
1997
1999
2001

a) What does the notation “1975 = 1000” mean?
b) By what factor did the index grow over the period shown?
c) Estimate the rate of growth of the index during the 1980s.

2.2 Indices • MHR 105

Solution

a) The notation indicates that the index shows the stock prices relative to what
they were in 1975. This 1975 base has been set at 1000. An index value of
2000 would mean that overall the stocks of the 300 companies in the index
are selling for twice what they did in 1975.

b) From the graph, you can see that the index increased from about 1000 in 1971
to about 10 000 in 2001. Thus, the index increased by a factor of approximately
10 over this period.

c) To estimate the rate of growth of the index during the 1980s, approximate
the time-series graph with a straight line during that
10-year interval. Then, calculate the slope of
the line.
rise
m= ᎏ
run For more information on stock indices, visit the above
web site and follow the links. Write a brief
⋅ 3700 − 1700
= ᎏᎏ description of the rules for inclusion in
10
the various market indices.
= 200

The TSE 300 Composite Index® rose about 200 points a year during the 1980s.

Statistics Canada calculates a variety of carefully researched economic indices.
For example, there are price indices for new housing, raw materials, machinery
and equipment, industrial products, and farm products. Most of these indices
are available with breakdowns by province or region and by specific categories,
such as agriculture, forestry, or manufacturing. Statisticians, economists, and the
media make extensive use of these indices. (See section 1.3 for information on
how to access Statistics Canada data.)

The consumer price index (CPI) is the most widely reported of these
economic indices because it is an important measure of inflation. Inflation is Data in Action
a general increase in prices, which corresponds to a decrease in the value of Statistics Canada
money. To measure the average change in retail prices across Canada, usually publishes the
consumer price index
Statistics Canada monitors the retail prices of a set of over 600 goods and for each month in
services including food, shelter, clothing, transportation, household items, the third week of the
health and personal care, recreation and education, and alcohol and tobacco following month.
products. These items are representative of purchases by typical Canadians Over 60 000 price
and are weighted according to estimates of the total amount Canadians spend quotations are
collected for each
on each item. For example, milk has a weighting of 0.69% while tea has a
update.
weighting of only 0.06%.


Example 2 Consumer Price Index

The following graph shows the amount by which the consumer price index
changed since the same month of the previous year.
Percent Change in CPI

3

2

1

0
May J J J J J May
1996 1997 1998 1999 2000 2001

a) What does this graph tell you about changes in the CPI from 1996 to 2001?
b) Estimate the mean annual change in the CPI for this period.

Solution
a) Note that the graph above shows the annual changes in the CPI, unlike
the graph on page 104, which illustrates the value of the CPI for any
given month. From the above graph, you can see that the annual change
in the CPI varied between 0.5% and 4% from 1996 to 2001. Overall, Project
there is an upward trend in the annual change during this period. Prep

b) You can estimate the mean annual change by drawing a horizontal line If your statistics
such that the total area between the line and the parts of the curve project examines
above it is approximately equal to the total area between the line and how a variable
the parts of the curve below it. As shown above, this line meets the changes over time,
y-axis near 2%. a time-series graph
may be an effective
Thus, the mean annual increase in the CPI was roughly 2% from 1996
way to illustrate
to 2001.
your ﬁndings.

The consumer price index and the cost of living index are not
quite the same. The cost of living index measures the
cost of maintaining a constant standard of living. If
consumers like two similar products equally well, www.mcgrawhill.ca/links/MDM12
their standard of living does not change when they
switch from one to the other. For example, if you For more information about Statistics Canada
indices, visit the above web site and follow the
like both apples and pears, you might start buying
links to Statistics Canada.
more apples and fewer pears if the price of pears
went up while the price of apples was unchanged. Thus,
your cost of living index increases less than the consumer
price index does.


Indices are also used in many other ﬁelds, including science, sociology,
medicine, and engineering. There are even indices of the clarity of writing.

Example 3 Readability Index

The Gunning fog index is a measure of the readability of prose. This index
estimates the years of schooling required to read the material easily.

Gunning fog index = 0.4(average words per sentence + percent “hard” words)

where “hard” words are all words over two syllables long except proper nouns,
compounds of easy words, and verbs whose third syllable is ed or es.
a) Calculate the Gunning fog index for a book with an average sentence
length of 8 words and a 20% proportion of hard words.
b) What are the advantages and limitations of this index?

Solution
a) Gunning fog index = 0.4(8 + 20)
= 11.2
The Gunning fog index shows that the book is written at a level Project
appropriate for readers who have completed grade 11. Prep

b) The Gunning fog index is easy to use and understand. It generates a You may want to use
grade-level rating, which is often more useful than a readability rating an index to
on an arbitrary scale, such as 1 to 10 or 1 to 100. However, the index summarize and
assumes that bigger words and longer sentences always make prose compare sets of data
harder to read. A talented writer could use longer words and sentences in your statistics
and still be more readable than writers who cannot clearly express their project.
ideas. The Gunning fog index cannot, of course, evaluate literary merit.


Visit the above web site to ﬁnd a link to a
readability-index calculator. Determine the reading
level of a novel of your choice.


Key Concepts

• An index can summarize a set of data. Indices usually compare the values of a
variable or group of variables to a base value.

• Indices have a wide variety of applications in business, economics, science, and
other fields.

• A time-series graph is a line graph that shows how a variable changes over time.
• The consumer price index (CPI) tracks the overall price of a representative
basket of goods and services, making it a useful measure of inflation.


1. What are the key features of a time-series graph?

2. a) Name three groups who would be interested in the new housing price index.
b) How would this information be important for each group?

3. Explain why the consumer price index is not the same as the cost of living index.

Practise B
A 3. Refer to the graph of the TSE 300
Composite Index® on page 105.
1. Refer to the consumer price index graph on
page 104. a) When did this index first reach five times
its base value?
a) By how many index points did the CPI
increase from January, 1992 to January, b) Estimate the growth rate of the index
2000? from 1971 to 1977. What does this
growth rate suggest about the Canadian
b) Express this increase as a percent.
economy during this period?
c) Estimate what an item that cost
c) During what two-year period did the index
i) $7.50 in 1992 cost in April, 1998 grow most rapidly? Explain your answer.
ii) $55 in August, 1997 cost in May, 2000 d) Could a straight line be a useful
mathematical model for the TSE 300
Composite Index®? Explain why or why
2. a) Explain why there is a wide variety of not.
items in the CPI basket.
4. Communication
b) Is the percent increase for the price of
a) Define inflation.
each item in the CPI basket the same?
Explain. b) In what way do the consumer price index
and the new housing price index provide
a measure of inflation?


c) How would you expect these two indices b) Describe how the overall trend in energy
to be related? costs compares to that of the CPI for the
d) Why do you think that they would be period shown.
related in this way? c) What insight is gained by removing the
energy component of the CPI?
5. Application Consider the following time-
d) Estimate the overall increase in the
series graph for the consumer price index.
energy-adjusted CPI for the period shown.
Consumer Price Index

e) Discuss how your result in part d) compares
to the value found in part b) of Example 2.
(1992 = 100)

100

50 7. François’ agent wants to bargain for a better
pte
ha salary based on François’ statistics for his

C

r
0 first five seasons with the team.

m
P
1980 1984 1988 1992 1996 2000 r
oble
a) Produce a time-series graph for François’
a) Identify at least three features of this goals, assists, and points over the past
graph that are different from the CPI five years.
graph on page 104.
b) Calculate the mean number of goals,
b) Explain two advantages that the graph assists, and points per game played
shown here has over the one on page 104. during each of François’ five seasons.
c) Explain two disadvantages of the graph c) Generate a new time-series graph based
shown here compared to the one on on the data from part b).
page 104.
d) Which time-series graph will the agent
d) Estimate the year in which the CPI was likely use, and which will the team’s
at 50. manager likely use during the contract
e) Explain the significance of the result in negotiations? Explain.
part d) in terms of prices in 1992. e) Explain the method or technology that
you used to answer parts a) to d).
6. Application The following graph illustrates
the CPI both with and without energy price 8. Aerial surveys of wolves in Algonquin Park
changes. produced the following estimates of their
population density.
4.0
Year Wolves/100 km2
Percent Change in CPI

3.0
1988–89 4.91
All Items
1989–90 2.47
2.0 1990–91 2.80
1991–92 3.62
1.0
All Items 1992–93 2.53
Excluding Energy
1993–94 2.23
0
May J J J J May 1994–95 2.82
1997 1998 1999 2000 2001
1995–96 2.75
a) How is this graph different from the one 1996–97 2.33
on page 107? 1997–98 3.04
1998–99 1.59


a) Using 1988–89 as a base, construct an 12. Communication Use the Internet, a library,
index for these data. or other resources to research two indices
b) Comment on any trends that you not discussed in this section. Briefly describe
observe. what each index measures, recent trends in
the index, and any explanation or rationale
9. Use Statistics Canada web sites or other for these trends.
sources to find statistics for the following
and describe any trends you notice. 13. Inquiry/Problem Solving The pictograph
below shows total greenhouse-gas emissions
a) the population of Canada
for each province and territory in 1996.
b) the national unemployment rate
= 638
c) the gross domestic product = 50 200
= 99 800
10. Inquiry/Problem Solving = 149 500
a) Use data from E-STAT or other sources
= 199 100
to generate a time-series graph that
kilotonnes of
shows the annual number of crimes in CO2 equivalent
Canada for the period 1989−1999. If
using E-STAT, look in the Nation section
under Justice/Crimes and Offences.
b) Explain any patterns that you notice.
c) In what year did the number of crimes
peak?
d) Suggest possible reasons why the number a) Which two provinces have the highest
of crimes peaked in that year. What levels of greenhouse-gas emissions?
other statistics would you need to b) Are the diameters or areas of the circles
confirm whether these reasons are proportional to the numbers they
related to the peak in the number of represent? Justify your answer.
crimes? c) What are the advantages and
disadvantages of presenting these data
11. a) Use data from E-STAT or other sources
as a pictograph?
to generate a time-series graph that
shows the number of police officers in d) Which provinces have the highest levels
Canada for the period 1989−1999. If of greenhouse-gas emissions per
using E-STAT, look in the Nation section geographic area?
under Justice/Police services. e) Is your answer to part d) what you would
b) In what ways are the patterns in these have expected? How can you account for
data similar to the patterns in the data such relatively high levels in these areas?
in question 10? In what ways are the f) Research information from E-STAT
patterns different? or other sources to determine the
c) In what year did the number of police greenhouse-gas emissions per person
officers peak? for each province.
d) Explain how this information could affect
your answer to part d) of question 10.


ACHIEVEMENT CHECK a) Construct a Pareto chart for these data.
b) Describe the similarities and differences
Understanding Problem Solving between a Pareto chart and other
frequency diagrams.
14. The graph below shows the national
unemployment rate from January, 1997, Method Number of Respondents
to June, 2001. Automobile: alone 26
% Automobile: car pool 35
10.0
Bus/Streetcar 52
9.5
Train 40
Unemployment Rate
Seasonally Adjusted

9.0
8.5 Bicycle/Walking 13
8.0
7.5
7.0
6.5
6.0 www.mcgrawhill.ca/links/MDM12
J J J J J M
1997 1998 1999 2000 2001 For more information about Pareto charts, visit the above
web site and follow the links. Give two examples of
a) Describe the overall trend for the period
situations where you would use a Pareto chart.
shown. Explain your reasoning.
b) When did the unemployment rate reach
its lowest level?
c) Estimate the overall unemployment rate 16. Pick ﬁve careers of interest to you.
for the period shown. a) Use resources such as CANSIM II,
d) Explain what the term seasonally adjusted E-STAT, newspapers, or the Internet to
means. obtain information about entry-level
income levels for these professions.
e) Who is more likely to use this graph in
an election campaign, the governing b) Choose an effective method to present
party or an opposing party? Explain. your data.
f) How might an opposing party produce a c) Describe any signiﬁcant information
graph showing rising unemployment you discovered.
without changing the data? Why would
17. a) Research unemployment data for
they produce such a graph?
Ontario over the past 20 years.
b) Present the data in an appropriate form.
C c) Conduct additional research to account
15. A Pareto chart is a type of frequency diagram for any trends or unusual features of the
in which the frequencies for categorical data data.
are shown by connected bars arranged in d) Predict unemployment trends for both
descending order of frequency. In a random the short term and the long term.
survey, commuters listed their most Explain your predictions.
common method of travelling to the
downtown of a large city.


2.3 Sampling Techniques

Who will win the next federal election? Are
Canadians concerned about global warming? Should
a Canadian city bid to host the next Olympic Games?
Governments, political parties, advocacy groups, and
news agencies often want to know the public’s
opinions on such questions. Since it is not feasible to
ask every citizen directly, researchers often survey a
much smaller group and use the results to estimate
the opinions of the entire population.

I N V E S T I G AT E & I N Q U I R E : Extrapolating From a Sample

1. Work in groups or as a class to design a survey to determine the opinions of
students in your school on a subject such as favourite movies, extra-curricular
activities, or types of music.
2. Have everyone in your class answer the survey.
3. Decide how to categorize and record the results. Could you reﬁne the survey
questions to get results that are easier to work with? Explain the changes you
would make.
4. How could you organize and present the data to make it easier to recognize
any patterns? Can you draw any conclusions from the data?
5. a) Extrapolate your data to estimate the opinions of the entire school
population. Explain your method.
b) Describe any reasons why you think the estimates in part a) may be inaccurate.
c) How could you improve your survey methods to get more valid results?

In statistics, the term population refers to all individuals who belong to a
group being studied. In the investigation above, the population is all the
students in your school, and your class is a sample of that population. The
population for a statistical study depends on the kind of data being collected.

Example 1 Identifying a Population

Identify the population for each of the following questions.
a) Whom do you plan to vote for in the next Ontario election?
b) What is your favourite type of baseball glove?
c) Do women prefer to wear ordinary glasses or contact lenses?

2.3 Sampling Techniques • MHR 113

Solution

a) The population consists of those people in Ontario who will be eligible to
vote on election day.

b) The population would be just those people who play baseball. However,
you might want to narrow the population further. For example, you might
be interested only in answers from local or professional baseball players.

c) The population is all women who use corrective lenses.

Once you have identiﬁed the population, you need to decide how you will
obtain your data. If the population is small, it may be possible to survey the
entire group. For larger populations, you need to use an appropriate sampling
technique. If selected carefully, a relatively small sample can give quite accurate
results.

The group of individuals who actually have a chance of being selected is called
the sampling frame. The sampling frame varies depending on the sampling
technique used. Here are some of the most commonly used sampling
techniques.

Simple Random Sample
In a simple random sample, every member of the population has an equal
chance of being selected and the selection of any particular individual does not
affect the chances of any other individual being chosen. Choosing the sample
randomly reduces the risk that selected members will not be representative of
the whole population. You could select the sample by drawing names randomly
or by assigning each member of the population a unique number and then using
a random-number generator to determine which members to include.

Systematic Sample
For a systematic sample, you go through the population sequentially and select
members at regular intervals. The sample size and the population size determine
the sampling interval.
population size
interval = ᎏᎏ
sample size
For example, if you wanted the sample to be a tenth of the population, you
would select every tenth member of the population, starting with one chosen
randomly from among the ﬁrst ten in sequence.


Example 2 Designing a Systematic Sample

A telephone company is planning a marketing survey of its 760 000 customers.
For budget reasons, the company wants a sample size of about 250.
a) Suggest a method for selecting a systematic sample.
b) What expense is most likely to limit the sample size?

Solution

a) First, determine the sampling interval.
population size
interval = ᎏᎏ
sample size
760 000
= ᎏᎏ
250
= 3040
The company could randomly select one of the first 3040 names on its list
of customers and then choose every 3040th customer from that point on.
For simplicity, the company might choose to select every 3000th customer
instead.

b) The major cost is likely to be salaries for the staff to call and interview the
customers.

Stratified Sample
Sometimes a population includes groups of members who share common
characteristics, such as gender, age, or education level. Such groups are called
strata. A stratified sample has the same proportion of members from each
stratum as the population does.

Example 3 Designing a Stratified Sample

Before booking bands for the school dances, the students’ council at Statsville
High School wants to survey the music preferences of the student body. The
following table shows the enrolment at the school.
Grade Number of Students
9 255
10 232
11 209
12 184
Total 880

a) Design a stratified sample for a survey of 25% of the student body.
b) Suggest other ways to stratify this sample.


Solution

a) To obtain a stratified sample Grade Number of Students Relative Frequency Number Surveyed
with the correct proportions, 9 255 0.29 64
simply select 25% of the 10 232 0.26 58
students in each grade level 11 209 0.24 52
as shown on the right. 12 184 0.21 46
Total 880 1.00 220

b) The sample could be stratified according to gender or age instead of grade level.

Other Sampling Techniques
Cluster Sample: If certain groups are likely to be representative of the entire
population, you can use a random selection of such groups as a cluster sample. For
example, a fast-food chain could save time and money by surveying all its employees
at randomly selected locations instead of surveying randomly selected employees
throughout the chain.

Multi-Stage Sample: A multi-stage sample uses several levels of random
sampling. If, for example, your population consisted of all Ontario households, you
could first randomly sample from all cities and townships in Ontario, then randomly
sample from all subdivisions or blocks within the selected cities and townships, and
finally randomly sample from all houses within the selected subdivisions or blocks.

Voluntary-Response Sample: In a voluntary-response sample, the researcher
simply invites any member of the population to participate in the survey. The results
from the responses of such surveys can be skewed because the people who choose to
respond are often not representative of the population. Call-in shows and mail-in
surveys rely on voluntary-response samples.

Convenience Sample: Often, a sample is selected simply because it is easily
accessible. While obviously not as random as some of the other techniques, such
convenience samples can sometimes yield helpful information. The investigation
at the beginning of this section used your class as a convenience sample.

Key Concepts

• Α carefully selected sample can provide accurate information about a population.

• Selecting an appropriate sampling technique is important to ensure that the
sample reflects the characteristics of the population. Randomly selected samples
have a good chance of being representative of the population.

• The choice of sampling technique will depend on a number of factors, such as
the nature of the population, cost, convenience, and reliability.



1. What are the advantages and disadvantages of using a sample to estimate the
characteristics of a population?

2. Discuss whether a systematic sample is a random sample.

3. a) Explain the difference between stratified sampling and cluster sampling.
b) Suggest a situation in which it would be appropriate to use each of these
two sampling techniques.

Practise e) A statistician conducting a survey
randomly selects 20 cities from across
A Canada, then 5 neighbourhoods from
1. Identify the population for each of the each of the cities, and then 3 households
following questions. from each of the neighbourhoods.
a) Who should be the next president of f) The province randomly chooses 25
the students’ council? public schools to participate in a new
b) Who should be next year’s grade-10 fundraising initiative.
representative on the student council?
3. What type(s) of sample would be
c) What is the your favourite soft drink? appropriate for
d) Which Beatles song was the best? a) a survey of engineers, technicians, and
e) How effective is a new headache remedy? managers employed by a company?
b) determining the most popular pizza
2. Classify the sampling method used in each
topping?
of the following scenarios.
c) measuring customer satisfaction for a
a) A radio-show host invites listeners to call
department store?
in with their views on banning smoking
in restaurants. Apply, Solve, Communicate
b) The Heritage Ministry selects a sample
of recent immigrants such that the B
proportions from each country of origin 4. Natasha is organizing the annual family
are the same as for all immigrants last picnic and wants to arrange a menu that will
year. appeal to children, teens, and adults. She
c) A reporter stops people on a downtown estimates that she has enough time to survey
street to ask what they think of the city’s about a dozen people. How should Natasha
lakefront. design a stratified sample if she expects
13 children, 8 teens, and 16 adults to attend
d) A school guidance counsellor arranges
the picnic?
interviews with every fifth student on the
alphabetized attendance roster.


5. Communication Find out, or estimate, how 9. Application The host of a call-in program
many students attend your school. Describe invites listeners to comment on a recent
how you would design a systematic sample trade by the Toronto Maple Leafs. One
of these students. Assume that you can caller criticizes the host, stating that the
survey about 20 students. sampling technique is not random. The host
replies: “So what? It doesn’t matter!”
6. The newly elected Chancellor of the
a) What sampling technique is the call-in
Galactic Federation is interested in the
show using?
opinions of all citizens regarding economic
conditions in the galaxy. Unfortunately, she b) Is the caller’s statement correct? Explain.
does not have the resources to visit every c) Is the host’s response mathematically
populated planet or to send delegates to correct? Why or why not?
them. Describe how the Chancellor might
organize a multi-stage sample to carry out C
her survey. 10. Look in newspapers and periodicals or on
the Internet for an article about a study
7. Communication A community centre chooses involving a systematic, stratified, cluster,
15 of its members at random and asks them or multi-stage sample. Comment on the
to have each member of their families suitability of the sampling technique and
complete a short questionnaire. the validity of the study. Present your
a) What type of sample is the community answer in the form of a brief report. Include
centre using? any suggestions you have for improving the
b) Are the 15 community-centre members study.
a random sample of the community? 11. Inquiry/Problem Solving Design a data-
Explain. gathering method that uses a combination
c) To what extent are the family members of convenience and systematic sampling
randomly chosen? techniques.
8. Application A students’ council is conducting 12. Inquiry/Problem Solving Pick a professional
a poll of students as they enter the cafeteria. sport that has championship playoffs each
a) What sampling method is the student year.
council using? a) Design a multi-stage sample to gather
b) Discuss whether this method is your schoolmates’ opinions on which
appropriate for surveying students’ team is likely to win the next
opinions on championship.
i) the new mural in the cafeteria b) Describe how you would carry out your
ii) the location for the graduation prom study and illustrate your findings.
c) Would another sampling technique be c) Research the media to find what the
better for either of the surveys in part b)? professional commentators are
predicting. Do you think these opinions
would be more valid than the results of
your survey? Why or why not?


2.4 Bias in Surveys

The results of a survey can be accurate only if the sample is representative of the
population and the measurements are objective. The methods used for choosing
the sample and collecting the data must be free from bias. Statistical bias is
any factor that favours certain outcomes or responses and hence systematically
skews the survey results. Such bias is often unintentional. A researcher may
inadvertently use an unsuitable method or simply fail to recognize a factor
that prevents a sample from being fully random. Regrettably, some people
deliberately bias surveys in order to get the results they want. For this reason,
it is important to understand not only how to use statistics, but also how to
recognize the misuse of statistics.

I N V E S T I G AT E & I N Q U I R E : Bias in a Sur vey

1. What sampling technique is the pollster in this cartoon likely to be using?
2. What is wrong with his survey methods? How could he improve them?
3. Do you think the bias in this survey is intentional? Why or why not?
4. Will this bias seriously distort the results of the survey? Explain your
reasoning.
5. What point is the cartoonist making about survey methods?
6. Sketch your own cartoon or short comic strip about data management.

Sampling bias occurs when the sampling frame does not reﬂect the
characteristics of the population. Biased samples can result from problems
with either the sampling technique or the data-collection method.

2.4 Bias in Surveys • MHR 119

Example 1 Sampling Bias

Identify the bias in each of the following surveys and suggest how it could be
avoided.
a) A survey asked students at a high-school football game whether a fund for
extra-curricular activities should be used to buy new equipment for the
football team or instruments for the school band.
b) An aid agency in a developing country wants to know what proportion of
households have at least one personal computer. One of the agency’s staff
members conducts a survey by calling households randomly selected from
the telephone directory.

Solution

a) Since the sample includes only football fans, it is not representative of the
whole student body. A poor choice of sampling technique makes the results
of the survey invalid. A random sample selected from the entire student
body would give unbiased results.

b) There could be a signiﬁcant number of households without telephones.
Such households are unlikely to have computers. Since the telephone survey
excludes these households, it will overestimate the proportion of households
that have computers. By using a telephone survey as the data-collection
method, the researcher has inadvertently biased the sample. Visiting
randomly selected households would give a more accurate estimate of the
proportion that have computers. However, this method of data collection
would be more time-consuming and more costly than a telephone survey.

Non-response bias occurs when particular groups are under-represented in a
survey because they choose not to participate. Thus, non-response bias is a form
of sampling bias.

Example 2 Non-Response Bias

A science class asks every ﬁfth student entering the cafeteria to answer a survey
on environmental issues. Less than half agree to complete the questionnaire.
The completed questionnaires show that a high proportion of the respondents
are concerned about the environment and well-informed about environmental
issues. What bias could affect these results?

Solution
The students who chose not to participate in the survey are likely to be those
least interested in environmental issues. As a result, the sample may not be
representative of all the students at the school.


To avoid non-response bias, researchers must ensure that the sampling process is
truly random. For example, they could include questions that identify members of
particular groups to verify that they are properly represented in the sample.

Measurement bias occurs when the data-collection method consistently either
under- or overestimates a characteristic of the population. While random errors
tend to cancel out, a consistent measurement error will skew the results of a
survey. Often, measurement bias results from a data-collection process that affects
the variable it is measuring.

Example 3 Measurement Bias

Identify the bias in each of the following surveys and suggest how it could be avoided.
a) A highway engineer suggests that an economical way to survey traffic speeds on
an expressway would be to have the police officers who patrol the highway record
the speed of the traffic around them every half hour.
b) As part of a survey of the “Greatest Hits of All Time,” a radio station asks its
listeners: Which was the best song by the Beatles?
i) Help! ii) Nowhere Man
iii) In My Life iv) Other:
c) A poll by a tabloid newspaper includes the question: “Do you favour the proposed
bylaw in which the government will dictate whether you have the right to smoke
in a restaurant?”

Solution

a) Most drivers who are speeding will slow down when they see a police cruiser. A
survey by police cruisers would underestimate the average traffic speed. Here, the
data-collection method would systematically decrease the variable it is measuring.
A survey by unmarked cars or hidden speed sensors would give more accurate
results.

b) The question was intended to remind listeners of some of the Beatles’ early
recordings that might have been overshadowed by their later hits. However, some
people will choose one of the suggested songs as their answer even though they
would not have thought of these songs without prompting. Such leading questions
usually produce biased results. The survey would more accurately determine
listeners’ opinions if the question did not include any suggested answers.

c) This question distracts attention from the real issue, namely smoking in
restaurants, by suggesting that the government will infringe on the respondents’
rights. Such loaded questions contain wording or information intended to
influence the respondents’ answers. A question with straightforward neutral
language will produce more accurate data. For example, the question could read
simply: “Should smoking in restaurants be banned?”


Response bias occurs when participants in a survey deliberately give false Project
or misleading answers. The respondents might want to influence the results Prep
unduly, or they may simply be afraid or embarrassed to answer sensitive
questions honestly. When gathering
data for your
statistics project, you
Example 4 Response Bias will need to ensure
that the sampling
A teacher has just explained a particularly difficult concept to her class and process is free from
wants to check that all the students have grasped this concept. She realizes bias.
that if she asks those who did not understand to put up their hands, these
students may be too embarrassed to admit that they could not follow the
lesson. How could the teacher eliminate this response bias?

Solution

The teacher could say: “This material is very difficult. Does anyone want me
to go over it again?” This question is much less embarrassing for students to
answer honestly, since it suggests that it is normal to have difficulty with the
material. Better still, she could conduct a survey that lets the students answer
anonymously. The teacher could ask the students to rate their understanding
on a scale of 1 to 5 and mark the ratings on slips of paper, which they would
deposit in a box. The teacher can then use these ballots to decide whether to
review the challenging material at the next class.

As the last two examples illustrate, careful wording of survey questions is
essential for avoiding bias. Researchers can also use techniques such as follow-up
questions and guarantees of anonymity to eliminate response bias. For a study to
be valid, all aspects of the sampling process must be free from bias.

Key Concepts

• Sampling, measurement, response, and non-response bias can all invalidate
the results of a survey.

• Intentional bias can be used to manipulate statistics in favour of a certain
point of view.

• Unintentional bias can be introduced if the sampling and data-collection
methods are not chosen carefully.

• Leading and loaded questions contain language that can influence the
respondents’ answers.



1. Explain the difference between a measurement bias and a sampling bias.

2. Explain how a researcher could inadvertently bias a study.

3. Describe how each of the following might use intentional bias
a) the media
b) a marketing department
c) a lobby group

Practise 3. Communication Reword each of the following
questions to eliminate the measurement bias.
A a) In light of the current government’s weak
1. Classify the bias in each of the following policies, do you think that it is time for a
scenarios. refreshing change at the next federal
a) Members of a golf and country club are election?
polled regarding the construction of a b) Do you plan to support the current
highway interchange on part of their golf government at the next federal election,
course. in order that they can continue to
b) A group of city councillors are asked implement their effective policies?
whether they have ever taken part in c) Is ﬁrst-year calculus as brutal as they say?
an illegal protest.
d) Which of the following is your favourite
c) A random poll asks the following male movie star?
question: “The proposed casino will
i) Al Pacino ii) Keanu Reeves
produce a number of jobs and economic
activity in and around your city, and it iii) Robert DeNiro iv) Jack Nicholson
will also generate revenue for the v) Antonio Banderas vi) Other:
provincial government. Are you in favour e) Do you think that ﬁghting should be
of this forward-thinking initiative?” eliminated from professional hockey so
d) A survey uses a cluster sample of Toronto that skilled players can restore the high
residents to determine public opinion on standards of the game?
whether the provincial government should
increase funding for the public transit. B
4. Communication
Apply, Solve, Communicate a) Write your own example of a leading
question and a loaded question.
2. For each scenario in question 1, suggest how
the survey process could be changed to b) Write an unbiased version for each
eliminate bias. of these two questions.


ACHIEVEMENT CHECK 6. Application A talk-show host conducts an
on-air survey about re-instituting capital
Communication Application punishment in Canada. Six out of ten callers
voice their support for capital punishment.
5. A school principal wants to survey data-
The next day, the host claims that 60% of
management students to determine
Canadians are in favour of capital
whether having computer Internet access
punishment. Is this claim statistically valid?
at home improves their success in this
Explain your reasoning.
course.
a) What type of sample would you C
suggest? Why? Describe a technique 7. a) Locate an article from a newspaper,
for choosing the sample. periodical, or Internet site that involves
b) The following questions were drafted a study that contains bias.
for the survey questionnaire. Identify b) Brieﬂy describe the study and its
any bias in the questions and suggest a ﬁndings.
rewording to eliminate the bias. c) Describe the nature of the bias inherent
i) Can your family afford high-speed in the study.
Internet access? d) How has this bias affected the results of
ii) Answer the question that follows the study?
your mark in data management. e) Suggest how the study could have
Over 80%: How many hours per eliminated the bias.
week do you spend on the Internet
at home? 8. Inquiry/Problem Solving Do you think that
60−80%: Would home Internet the members of Parliament are a
access improve your mark in data representative sample of the population?
management ? Why or why not?
Below 60%: Would increased
Internet access at school improve
your mark in data management?
c) Suppose the goal is to convince the
school board that every data-
management student needs daily access
to computers and the Internet in the
classroom. How might you alter your
sampling technique to help achieve the
desired results in this survey? Would
these results still be statistically valid?


2.5 Measures of Central
Tendency
It is often convenient to use a central value
to summarize a set of data. People
frequently use a simple arithmetic average
for this purpose. However, there are several
different ways to ﬁnd values around which a
set of data tends to cluster. Such values are
known as measures of central tendency.

I N V E S T I G A T E & I N Q U I R E : N o t Yo u r A v e r a g e A v e r a g e

François is a NHL hockey player whose ﬁrst major-league contract is up
for renewal. His agent is bargaining with the team’s general manager.

Agent: Based on François’ strong performance, we can accept no less than
the team’s average salary.
Manager: Agreed, François deserves a substantial increase. The team is willing
to pay François the team’s average salary, which is $750 000 a season.
Agent: I’m certain that we calculated the average salary to be $1 000 000 per
season. You had better check your arithmetic.
Manager: There is no error, my friend. Half of the players earn $750 000 or
more, while half of the players receive $750 000 or less. $750 000 is a fair offer.

This table lists the current salaries for the team.
300 000 2
500 000 3
750 000 8
900 000 6
1 000 000 2
1 500 000 1
3 000 000 1
4 000 000 1

1. From looking at the table, do you think the agent or the manager is correct?
Explain why.

2.5 Measures of Central Tendency • MHR 125

2. Find the mean salary for the team. Describe how you calculated this
amount.
3. Find the median salary. What method did you use to ﬁnd it?
4. Were the statements by François’ agent and the team manager correct?
5. Explain the problem with the use of the term average in these
negotiations.

In statistics, the three most commonly used measures of central tendency are
the mean, median, and mode. Each of these measures has its particular
advantages and disadvantages for a given set of data.

A mean is deﬁned as the sum of the values of a variable divided by the number
of values. In statistics, it is important to distinguish between the mean of a
population and the mean of a sample of that population. The sample mean will
approximate the actual mean of the population, but the two means could have
different values. Different symbols are used to distinguish the two kinds of
means: The Greek letter mu, µ, represents a population mean, while −, read as
x
“x-bar,” represents a sample mean. Thus,

x1 + x2 + … + xN x1 + x2 + … + xn
− = ᎏᎏ
µ = ᎏᎏ and x
N n
∑x ∑x
=ᎏ =ᎏ
N n

where ∑x is the sum of all values of X in the population or sample, N is the
number of values in the entire population, and n is the number of values in a
sample. Note that ∑ , the capital Greek letter sigma, is used in mathematics
as a symbol for “the sum of.” If no limits are shown above or below the sigma,
the sum includes all of the data.

Usually, the mean is what people are referring to when they use the term
average in everyday conversation.

The median is the middle value of the data when they are ranked from highest
to lowest. When there is an even number of values, the median is the midpoint
between the two middle values.

The mode is the value that occurs most frequently in a distribution. Some
distributions do not have a mode, while others have several.

Some distributions have outliers, which are values distant from the majority of
the data. Outliers have a greater effect on means than on medians. For example,
the mean and median for the salaries of the hockey team in the investigation
have substantially different values because of the two very high salaries for the
team’s star players.


Example 1 Determining Mean, Median, and Mode

Two classes that wrote the same physics examination had the following results.
Class A 71 82 55 76 66 71 90 84 95 64 71 70 83 45 73 51 68
Class B 54 80 12 61 73 69 92 81 80 61 75 74 15 44 91 63 50 84

a) Determine the mean, median, and mode for each class.
b) Use the measures of central tendency to compare the performance of the
two classes.
c) What is the effect of any outliers on the mean and median?

Solution
a) For class A, the mean is
− ∑x
x=ᎏ For more information about means, medians, and
n modes, visit the above web site and follow the
71 + 82 + … + 68 links. For each measure, give an example of a
= ᎏᎏ situation where that measure is the best indicator
17
of the centre of the data.
1215
= ᎏᎏ
17
= 71.5
When the marks are ranked from highest to lowest, the middle value is 71.
Therefore, the median mark for class A is 71. The mode for class A is also 71
since this mark is the only one that occurs three times.
54 + 80 + … + 84
Similarly, the mean mark for class B is ᎏᎏ = 64.4. When the marks
18
are ranked from highest to lowest, the two middle values are 69 and 73, so the
69 + 73
median mark for class B is ᎏ = 71. There are two modes since the values 61
2
and 80 both occur twice. However, the sample is so small that all the values occur
only once or twice, so these modes may not be a reliable measure.

b) Although the mean score for class A is signiﬁcantly higher than that for class B, the
median marks for the two classes are the same. Notice that the measures of central
tendency for class A agree closely, but those for class B do not.

c) A closer examination of the raw data shows that, aside from the two extremely low
scores of 15 and 12 in class B, the distributions are not all that different. Without
these two outlying marks, the mean for class B would be 70.1, almost the same as
the mean for class A. Because of the relatively small size of class B, the effect of the
outliers on its mean is signiﬁcant. However, the values of these outliers have no
effect on the median for class B. Even if the two outlying marks were changed to
numbers in the 60s, the median mark would not change because it would still be
greater than the two marks.


The median is often a better measure of central tendency than the mean for small
data sets that contain outliers. For larger data sets, the effect of outliers on the
mean is less significant.

Example 2 Comparing Samples to a Population

Compare the measures of central tendency for each class in Example 1 to those
for all the students who wrote the physics examination.


Use the STAT EDIT menu to check that lists L1 and L2 are clear. Then, enter the data
for class A in L1 and the data for class B in L2. Next, use the augment( function
from the LIST OPS menu to combine L1 and L2, and store the result in L3.
You can use the mean( and median( functions from the LIST MATH menu
to find the mean and median for each of the three lists. You can also find these
measures by using the 1-Var Stats command from the STAT CALC menu. To find
the modes, sort the lists with the SortA( function from the LIST OPS menu, and
then scroll down through the lists to find the most frequent values. Alternatively,
you can use STAT PLOT to display a histogram for each list and read the x-values
for the tallest bars with the TRACE instruction.

Note that the mean for class A overestimates the population mean, while the
mean for class B underestimates it. The measures of central tendency for class
A are reasonably close to those for the whole population of students who wrote
the physics examination, but the two sets of measures are not identical. Because
both of the low-score outliers happen to be in class B, it is a less representative
sample of the population.

Enter the data for class A and class B in separate columns. The AVG and MEAN
functions in Corel® Quattro® Pro will calculate the mean for any range of cells
you specify, as will the AVERAGE function in Microsoft® Excel.
In both spreadsheets, you can use the MEDIAN, and MODE functions to find the
median and mode for each class and for the combined data for both classes. Note
that all these functions ignore any blank cells in a specified range. The MODE
function reports only one mode even if the data have two or more modes.


Solution 3 Using Fathom™

Drag the case table icon to the workspace and name the attribute for the first
column Marks. Enter the data for class A and change the name of the collection
from Collection1 to ClassA. Use the same method to enter the marks for class B
into a collection called ClassB. To create a collection with the combined data,
first open another case table and name the collection Both. Then, go back to the
class A case table and use the Edit menu to select all cases and then copy them.
Return to the Both case table and select Paste Cases from the Edit menu. Copy
the cases from the class B table in the same way.
Project
Now, right-click on the class A collection to open the inspector. Click the Prep
Measures tab, and create Mean, Median, and Mode measures. Use the Edit In your statistics
Formula menu to enter the formulas for these measures. Use the same project, you may find
procedure to find the mean, median, and mode for the other two measures of central
collections. Note from the screen below that Fathom™ uses a complicated tendency useful for
formula to find modes. See the Help menu or the Fathom™ section of describing your data.
Appendix B for details.


Chapter 8 discusses a method for calculating how representative of a population
a sample is likely to be.

Sometimes, certain data within a set are more significant than others. For
example, the mark on a final examination is often considered to be more
important than the mark on a term test for determining an overall grade for
a course. A weighted mean gives a measure of central tendency that reflects
the relative importance of the data:

− w1x1 + w2x2 + … + wnxn
x w = ᎏᎏᎏ
w1 + w2 + … + wn
∑wi xi
= ᎏi
∑wi
i

where ∑ wi xi is the sum of the weighted values and ∑ wi is the sum of the various
i i
weighting factors.

Weighted means are often used in calculations of indices.

Example 3 Calculating a Weighted Mean

The personnel manager for Statsville Marketing Limited considers five criteria
when interviewing a job applicant. The manager gives each applicant a score
between 1 and 5 in each category, with 5 as the highest score. Each category has
a weighting between 1 and 3. The following table lists a recent applicant’s scores
and the company’s weighting factors.
Criterion Score, xi Weighting Factor, wi
Education 4 2
Job experience 2 2
Interpersonal skills 5 3
Communication skills 5 3
References 4 1

a) Determine the weighted mean score for this job applicant.
b) How does this weighted mean differ from the unweighted mean?
c) What do the weighting factors indicate about the company’s hiring
priorities?


Solution

a) To compute the weighted mean, ﬁnd the sum of the products of each score
and its weighting factor.
∑wi xi
−
xw = ᎏi
∑ wi
i
2(4) + 2(2) + 3(5) + 3(5) + (1)4
= ᎏᎏᎏᎏ
2+2+3+3+1
46
= ᎏᎏ
11
= 4.2

Therefore, this applicant had a weighted-mean score of approximately 4.2.

b) The unweighted mean is simply the sum of unweighted scores divided by 5.
− ∑x
x=ᎏ
n
4+2+5+5+4
= ᎏᎏ
5
=4

Without the weighting factors, this applicant would have a mean score of
4 out of 5.

c) Judging by these weighting factors, the company places a high importance
on an applicant’s interpersonal and communication skills, moderate
importance on education and job experience, and some, but low, importance
on references.

When a set of data has been grouped into intervals, you can approximate
the mean using the formula

∑f m ∑f m
⋅ i i i
µ= ᎏ − ⋅ i i i
x= ᎏ
∑ fi ∑ fi
i i

where mi is the midpoint value of an interval and fi the frequency for that
interval.

You can estimate the median for grouped data by taking the midpoint of the
interval within which the median is found. This interval can be found by
analysing the cumulative frequencies.


Example 4 Calculating the Mean and Median for Grouped Data

A group of children were asked how many hours a day Number of Hours Number of Children, fi
they spend watching television. The table at the right 0−1 1
summarizes their responses.
1−2 4
a) Determine the mean and median number of hours 2−3 7
for this distribution. 3−4 3
b) Why are these values simply approximations? 4−5 2
5− 6 1

Solution
a) First, ﬁnd the midpoints and cumulative frequencies for the intervals. Then, use the
midpoints and the frequencies for the intervals to calculate an estimate for the mean.
Number of Midpoint, Number of Cumulative fixi
Hours xi Children, fi Frequency
0−1 0.5 1 1 0.5
1−2 1.5 4 5 6
2−3 2.5 7 12 17.5
3−4 3.5 3 15 10.5
4−5 4.5 2 17 9
5−6 5.5 1 18 5.5
∑ fi = 18 ∑ fi xi = 49
i i

∑f x
− ⋅ i i i
x= ᎏ
∑ fi
i
49
= ᎏᎏ
18
= 2.7

Therefore, the mean time the children spent watching television is approximately
2.7 h a day.

To determine the median, you must identify the interval in which the middle value
occurs. There are 18 data values, so the median is the mean of the ninth and tenth
values. According to the cumulative-frequency column, both of these occur within
the interval of 2−3 h. Therefore, an approximate value for the median is 2.5 h.

b) These values for the mean and median are approximate because you do not know
where the data lie within each interval. For example, the child whose viewing time
is listed in the ﬁrst interval could have watched anywhere from 0 to 60 min of
television a day. If the median value is close to one of the boundaries of the
interval, then taking the midpoint of the interval as the median could give an error
of almost 30 min.


Key Concepts

• The three principal measures of central tendency are the mean, median, and
mode. The measures for a sample can differ from those for the whole population.

• The mean is the sum of the values in a set of data divided by the number of
values in the set.

• The median is the middle value when the values are ranked in order. If there
are two middle values, then the median is the mean of these two middle values.

• The mode is the most frequently occurring value.

• Outliers can have a dramatic effect on the mean if the sample size is small.

• Α weighted mean can be a useful measure when all the data are not of equal
significance.

• For data grouped into intervals, the mean and median can be estimated using
the midpoints and frequencies of the intervals.


1. Describe a situation in which the most useful measure of central tendency is
a) the mean b) the median c) the mode

2. Explain why a weighted mean would be used to calculate an index such as the
consumer price index.
∑f m
− ⋅ i i i
3. Explain why the formula x = ᎏ gives only an approximate value for the
∑ fi
i
mean for grouped data.

Practise c) List a set of eight values that has two
modes.
A d) List a set of eight values that has a
1. For each set of data, calculate the mean, median that is one of the data values.
median, and mode.
a) 2.4 3.5 1.9 3.0 3.5 2.4 1.6 3.8 1.2 Apply, Solve, Communicate
2.4 3.1 2.7 1.7 2.2 3.3
3. Stacey got 87% on her term work in
b) 10 15 14 19 18 17 12 10 14 15 18 chemistry and 71% on the final examination.
20 9 14 11 18 What will her final grade be if the term
2. a) List a set of eight values that has no mark counts for 70% and the final
mode. examination counts for 30%?
b) List a set of eight values that has a
median that is not one of the data values.


4. Communication Determine which measure of 8. Application An academic award is to be
central tendency is most appropriate for granted to the student with the highest
each of the following sets of data. Justify overall score in four weighted categories.
your choice in each case. Here are the scores for the three finalists.
a) baseball cap sizes Criterion Weighting Paulo Janet Jamie
b) standardized test scores for 2000 students Academic
achievement 3 4 3 5
c) final grades for a class of 18 students
Extra-curricular
d) lifetimes of mass-produced items, such as activities 2 4 4 4
batteries or light bulbs Community
service 2 2 5 3
B Interview 1 5 5 4
5. An interviewer rates candidates out of 5 for
a) Calculate each student’s mean score
each of three criteria: experience, education,
without considering the weighting factors.
and interview performance. If the first two
criteria are each weighted twice as much as b) Calculate the weighted-mean score for
the interview, determine which of the each student.
following candidates should get the job. c) Who should win the award? Explain.
Criterion Nadia Enzo Stephan
9. Al, a shoe salesman, needs to restock his
Experience 4 5 5
best-selling sandal. Here is a list of the sizes
Education 4 4 3
of the pairs he sold last week. This sandal
Interview 4 3 4 does not come in half-sizes.

6. Determine the effect the two outliers have 10 7 6 8 7 10 5 10 7 9
on the mean mark for all the students in 11 4 6 7 10 10 7 8 10 7
9 7 10 4 7 7 10 11
Example 2. Explain why this effect is
different from the effect the outliers had on a) Determine the three measures of central
the mean mark for class B. tendency for these sandals.
7. Application The following table shows the b) Which measure has the greatest
grading system for Xabbu’s calculus course. significance for Al? Explain.
Term Mark Overall Mark c) What other value is also significant?
Knowledge and Term mark 70% d) Construct a histogram for the data.
understanding (K/U) 35% Final examination What might account for the shape of
Thinking, inquiry, problem 30% this histogram?
solving (TIPS) 25%
Communication (C) 15% 10. Communication Last year, the mean number
Application (A) 25% of goals scored by a player on Statsville’s
soccer team was 6.
a) Determine Xabbu’s term mark if he a) How many goals did the team score last
scored 82% in K/U, 71% in TIPS, 85% year if there were 15 players on the team?
in C, and 75% in A.
b) Explain how you arrived at the answer for
b) Determine Xabbu’s overall mark if he part a) and show why your method works.
scored 65% on the final examination.


11. Inquiry/Problem Solving The following table f) Determine a mean, median, and mode
shows the salary structure of Statsville Plush for the grouped data. Explain any
Toys, Inc. Assume that salaries exactly on an differences between these measures
interval boundary have been placed in the and the ones you calculated in part a).
higher interval.
13. The modal interval for grouped data is
Salary Range ($000) Number of Employees the interval that contains more data than
20−30 12 any other interval.
30−40 24 a) Determine the modal interval(s) for
40−50 32 your data in part d) of question 12.
50−60 19
b) Is the modal interval a useful measure
60−70 9 of central tendency for this particular
70−80 3 distribution? Why or why not?
80−90 0
90−100 1 14. a) Explain the effect outliers have on the
median of a distribution. Use examples
a) Determine the approximate mean salary
to support your explanation.
for an employee of this firm.
b) Explain the effect outliers have on the
b) Determine the approximate median
mode of a distribution. Consider
salary.
different cases and give examples of each.
c) How much does the outlier influence the
mean and median salaries? Use C
calculations to justify your answer. 15. The harmonic mean is defined as

΂ ΃
1 –1
∑ ᎏ , where n is the number of values
12. Inquiry/Problem Solving A group of friends i nx
i
and relatives get together every Sunday for in the set of data.
a little pick-up hockey. The ages of the 30
a) Use a harmonic mean to find the average
regulars are shown below.
price of gasoline for a driver who bought
22 28 32 45 48 19 20 52 50 21 $20 worth at 65¢/L last week and
30 46 21 38 45 49 18 25 23 46 another $20 worth at 70¢/L this week.
51 24 39 48 28 20 50 33 17 48
b) Describe the types of calculations for
a) Determine the mean, median, and mode which the harmonic mean is useful.
for this distribution.
16. The geometric mean is defined as
b) Which measure best describes these n

data? Explain your choice. x1 × x2 × … × xn
͙ෆෆෆ , where n is the number
of values in the set of data.
c) Group these data into six intervals and
produce a frequency table. a) Use the geometric mean to find the
average annual increase in a labour
d) Illustrate the grouped data with a
contract that gives a 4% raise the first
frequency diagram. Explain why the
year and a 2% raise for the next three
shape of this frequency diagram could be
years.
typical for such groups of hockey players.
b) Describe the types of calculations for
e) Produce a cumulative-frequency diagram.
which the geometric mean is useful.


2.6 Measures of Spread

The measures of central tendency indicate
the central values of a set of data. Often,
you will also want to know how closely the
data cluster around these centres.

I N V E S T I G AT E & I N Q U I R E : S p r e a d i n a S e t o f D a t a

For a game of basketball, a group of friends split into two randomly chosen
teams. The heights of the players are shown in the table below.
Falcons Ravens
Player Height (cm) Player Height (cm)
Laura 183 Sam 166
Jamie 165 Shannon 163
Deepa 148 Tracy 168
Colleen 146 Claudette 161
Ingrid 181 Maria 165
Justiss 178 Amy 166
Sheila 154 Selena 166

1. Judging by the raw data in this table, which team do you think has a
height advantage? Explain why.
2. Do the measures of central tendency conﬁrm that the teams are
mismatched? Why or why not?
3. Explain how the distributions of heights on the two teams might give
one of them an advantage. How could you use a diagram to illustrate
the key difference between the two teams?

The measures of spread or dispersion of a data set are quantities that indicate
how closely a set of data clusters around its centre. Just as there are several
measures of central tendency, there are also different measures of spread.


Standard Deviation and Variance
A deviation is the difference between an individual value in a set of data and
the mean for the data.

For a population, For a sample,
deviation = x − µ −
deviation = x − x

The larger the size of the deviations, the greater the spread in the data. Values less
than the mean have negative deviations. If you simply add up all the deviations for
a data set, they will cancel out. You could use the sum of the absolute values of the
deviations as a measure of spread. However, statisticians have shown that a root-
mean-square quantity is a more useful measure of spread. The standard deviation
is the square root of the mean of the squares of the deviations.
The lowercase Greek letter sigma, σ, is the symbol for the standard deviation
of a population, while the letter s stands for the standard deviation of a sample.
Population standard deviation Sample standard deviation
∑(x − µ)2 −
∑(x − x )2
σ= Ί๶ᎏᎏ
N
s= Ί๶
ᎏᎏ
n−1

where N is the number of data in the population and n is the number in the
sample.

Note that the formula for s has n − 1 in the denominator instead of n. This
denominator compensates for the fact that a sample taken from a population tends
to underestimate the deviations in the population. Remember that the sample
mean, −, is not necessarily equal to the population mean, µ. Since − is the central
x x
− than to µ. When n is large,
value of the sample, the sample data cluster closer to x
the formula for s approaches that for σ.

Also note that the standard deviation gives greater weight to the larger deviations
since it is based on the squares of the deviations.

The mean of the squares of the deviations is another useful measure. This quantity
is called the variance and is equal to the square of the standard deviation.
Population variance Sample variance
∑(x − µ)2 −
∑(x − x )2
σ 2 = ᎏᎏ s 2 = ᎏᎏ
N n−1

Example 1 Using a Formula to Calculate Standard Deviations

Use means and standard deviations to compare the distribution of heights for
the two basketball teams listed in the table on page 136.

2.6 Measures of Spread • MHR 137

Solution

Since you are considering the teams as two separate populations, use the
mean and standard deviation formulas for populations. First, calculate the
mean height for the Falcons.
∑x
µ= ᎏ
N
1155
= ᎏᎏ
7
= 165
Next, calculate all the deviations and their squares.
Falcons Height (cm) Deviation, x – µ (x – µ)2
Laura 183 18 324
Jamie 165 0 0
Deepa 148 −17 289
Colleen 146 −19 361
Ingrid 181 16 256
Justiss 178 13 169
Sheila 154 −11 121
Sum 1155 0 1520

Now, you can determine the standard deviation.

∑(x − µ)2
σ= Ί๶
ᎏᎏ
N

= Ί๶
1520
ᎏ
7
= 14.7

Therefore, the Falcons have a mean height of 165 cm with a standard deviation
of 14.7 cm.

Similarly, you can determine that the Ravens also have a mean height of 165 cm,
but their standard deviation is only 2.1 cm. Clearly, the Falcons have a much
greater spread in height than the Ravens. Since the two teams have the same
mean height, the difference in the standard deviations indicates that the Falcons
have some players who are taller than any of the Ravens, but also some
players who are shorter.

If you were to consider either of the basketball teams in the example above as a
sample of the whole group of players, you would use the formula for s to calculate
the team’s standard deviation. In this case, you would be using the sample to
estimate the characteristics of a larger population. However, the teams are very
small samples, so they could have signiﬁcant random variations, as the difference
in their standard deviations demonstrates.


For large samples the calculation of standard deviation can be quite tedious. See Appendix B for more
However, most business and scientiﬁc calculators have built-in functions for detailed information about
such calculations, as do spreadsheets and statistical software. technology functions and
keystrokes.

Example 2 Using Technology to Calculate Standard Deviations

A veterinarian has collected data on the life spans of a rare breed of cats.
Life Spans (in years)
16 18 19 12 11 15 20 21 18 15 16 13 16 22
18 19 17 14 9 14 15 19 20 15 15

Determine the mean, standard deviation, and the variance for these data.


Use the ClrList command to make sure list L1 is clear, then enter the data
into it. Use the 1-Var Stats command from the STAT CALC menu to calculate
a set of statistics including the mean and the standard deviation. Note that
the calculator displays both a sample standard deviation, Sx, and a
population standard deviation, σx. Use Sx since you are dealing with a
sample in this case. Find the variance by calculating the square of Sx.

The mean life span for this breed of cats is about 16.3 years with a standard
deviation of 3.2 years and a variance of 10.1. Note that variances are usually
stated without units. The units for this variance are years squared.


Enter the data into your spreadsheet
program. With Corel® Quattro® Pro, you
can use the AVG, STDS, and VARS functions
to calculate the mean, sample standard
deviation, and sample variance. In
Microsoft® Excel, the equivalent functions
are AVERAGE, STDEV, and VAR.



Drag a new case table onto the workspace, name the attribute for the ﬁrst
column Lifespan, and enter the data. Right-click to open the inspector, and click
the Measures tab. Create Mean, StdDev, and Variance measures and select the
formulas for the mean, standard deviation, and variance from the Edit Formula/
Functions/Statistical/One Attribute menu.

If you are working with grouped data, you can estimate the standard Project
deviation using the following formulas. Prep

For a population, For a sample, In your statistics
∑fi (mi − µ)2 −
∑fi(mi − x )2 project, you may
⋅
σ = ᎏᎏΊ๶ N
⋅
Ί๶
s = ᎏᎏ
n−1
wish to use an
appropriate measure
where fi is the frequency for a given interval and mi is the midpoint of the of spread to describe
interval. However, calculating standard deviations from raw, ungrouped the distribution of
data will give more accurate results. your data.

Quartiles and Interquartile Ranges
Quartiles divide a set of ordered data into four groups with equal numbers of
values, just as the median divides data into two equally sized groups. The three
“dividing points” are the ﬁrst quartile (Q1), the median (sometimes called the
second quartile or Q2), and the third quartile (Q3). Q1 and Q3 are the medians
of the lower and upper halves of the data.
Interquartile Range

Lowest Datum First Quartile Median Third Quartile Highest Datum
Q1 Q2 Q3


Recall that when there are an even number of data, you take the midpoint
between the two middle values as the median. If the number of data below the
median is even, Q1 is the midpoint between the two middle values in this half
of the data. Q3 is determined in a similar way.

The interquartile range is Q3 − Q1, which is the range of the middle half of the
data. The larger the interquartile range, the larger the spread of the central half
of the data. Thus, the interquartile range provides a measure of spread. The
semi-interquartile range is one half of the interquartile range. Both these
ranges indicate how closely the data are clustered around the median.

A box-and-whisker plot of the data illustrates these measures. The box shows
the first quartile, the median, and the third quartile. The ends of the “whiskers”
represent the lowest and highest values in the set of data. Thus, the length of the
box shows the interquartile range, while the left whisker shows the range of the
data below the first quartile, and the right whisker shows the range above the
third quartile.

Interquartile Range
Lowest Datum Highest Datum

Q1 Q3
Median
(Q2)

A modified box-and-whisker plot is often used when the data contain outliers.
By convention, any point that is at least 1.5 times the box length away from the
box is classified as an outlier. A modified box-and-whisker plot shows such
outliers as separate points instead of including them in the whiskers. This method
usually gives a clearer illustration of the distribution.

Interquartile Range
Lowest Datum Highest Datum

Q1 Q3 Outliers
Median
(Q2)


Example 3 Determining Quartiles and Interquartile Ranges

A random survey of people at a science-fiction convention asked them how
many times they had seen Star Wars. The results are shown below.
3 4 2 8 10 5 1 15 5 16 6 3 4 9 12 3 30 2 10 7

a) Determine the median, the first and third quartiles, and the interquartile and
semi-interquartile ranges. What information do these measures provide?
b) Prepare a suitable box plot of the data.
c) Compare the results in part a) to those from last year’s survey, which found
a median of 5.1 with an interquartile range of 8.0.


a) First, put the data into numerical order.

1 2 2 3 3 3 4 4 5 5 6 7 8 9 10 10 12 15 16 30

The median is either the middle datum or, as in this case, the mean of the
two middle data:
5+6
median = ᎏ
2
= 5.5

The median value of 5.5 indicates that half of the people surveyed had seen
Star Wars less than 5.5 times and the other half had seen it more than 5.5
times.

To determine Q1, find the median of the lower half of the data. Again, there
are two middle values, both of which are 3. Therefore, Q1 = 3.

Similarly, the two middle values of the upper half of the data are both 10, so
Q3 = 10.

Since Q1 and Q3 are the boundaries for the central half of the data, they show
that half of the people surveyed have seen Star Wars between 3 and 10 times.

Q3 − Q1 = 10 − 3
=7

Therefore, the interquartile range is 7. The semi-interquartile range is half
this value, or 3.5. These ranges indicate the spread of the central half of the
data.


b) The value of 30 at the end of the ordered data is clearly an outlier. Therefore,
a modiﬁed box-and-whisker plot will best illustrate this set of data.

0 5 10 15 20 25 30
Viewings of Star Wars

c) Comparing the two surveys shows that the median number of viewings is
higher this year and the data are somewhat less spread out.

a) Use the STAT EDIT menu to enter the data into a list. Use the 1-Var Stats
command from the CALC EDIT menu to calculate the statistics for your list.
Scroll down to see the values for the median, Q1, and Q3. Use the values for
Q1 and Q3 to calculate the interquartile and semi-interquartile ranges.

b) Use STAT PLOT to select a modiﬁed box plot of your list. Press GRAPH to
display the box-and-whisker plot and adjust the window settings, if necessary.


a) Drag a new case table onto the workspace, create an attribute called
StarWars, and enter your data. Open the inspector and create Median, Q1,
Q3, and IQR measures. Use the Edit Formula/Functions/Statistical/One
Attribute menu to enter the formulas for the median, quartiles, and
interquartile range.


b) Drag the graph icon onto the workspace, then drop the StarWars attribute
on the x-axis of the graph. Select Box Plot from the drop-down menu in the
upper right corner of the graph.

Although a quartile is, strictly speaking, a single value, people sometimes speak
of a datum being within a quartile. What they really mean is that the datum is in
the quarter whose upper boundary is the quartile. For example, if a value x1 is
“within the ﬁrst quartile,” then x1 ≤ Q1. Similarly, if x2 is “within the third
quartile,” then the median ≤ x2 ≤ Q3.

Example 4 Classifying Data by Quartiles

In a survey of low-risk mutual funds, the median annual yield was 7.2%, while Q1
was 5.9% and Q3 was 8.3%. Describe the following funds in terms of quartiles.
Mutual Fund Annual Yield (%)
XXY Value 7.5
YYZ Dividend 9.0
ZZZ Bond 7.2

Solution

The yield for the XXY Value fund was between the median and Q3. You might
see this fund described as being in the third quartile or having a third-quartile
yield.

YYZ Dividend’s yield was above Q3. This fund might be termed a fourth- or
top-quartile fund.

ZZZ Bond’s yield was equal to the median. This fund could be described as a
median fund or as having median performance.


Percentiles
Percentiles are similar to quartiles, except that percentiles divide the data into
100 intervals that have equal numbers of values. Thus, k percent of the data are
less than or equal to kth percentile, Pk , and (100 − k) percent are greater than or
equal to Pk. Standardized tests often use percentiles to convert raw scores to
scores on a scale from 1 to 100. As with quartiles, people sometimes use the
term percentile to refer to the intervals rather than their boundaries.

Example 5 Percentiles

An audio magazine tested 60 different models of speakers and gave each one an
overall rating based on sound quality, reliability, efﬁciency, and appearance. The
raw scores for the speakers are listed in ascending order below.
35 47 57 62 64 67 72 76 83 90
38 50 58 62 65 68 72 78 84 91
41 51 58 62 65 68 73 79 86 92
44 53 59 63 66 69 74 81 86 94
45 53 60 63 67 69 75 82 87 96
45 56 62 64 67 70 75 82 88 98

a) If the Audio Maximizer Ultra 3000 scored at the 50th percentile, what was
its raw score?
b) What is the 90th percentile for these data?
c) Does the SchmederVox’s score of 75 place it at the 75th percentile?

Solution

a) Half of the raw scores are less than or equal to the 50th percentile and half
are greater than or equal to it. From the table, you can see that 67 divides
the data in this way. Therefore, the Audio Maximizer Ultra 3000 had a raw
score of 67.

b) The 90th percentile is the boundary between the lower 90% of the scores
and the top 10%. In the table, you can see that the top 10% of the scores
are in the 10th column. Therefore, the 90th percentile is the midpoint
between values of 88 and 90, which is 89.

c) First, determine 75% of the number of raw scores.
60 × 75% = 45
There are 45 scores less than or equal to the 75th percentile. Therefore, the
75th percentile is the midpoint between the 45th and 46th scores. These
two scores are 79 and 81, so the 75th percentile is 80. The SchmederVox’s
score of 75 is below the 75th percentile.


Z-Scores
A z-score is the number of standard deviations that a datum is from the mean.
You calculate the z-score by dividing the deviation of a datum by the standard
deviation.
For a population, For a sample,
x−µ x−x −
z= ᎏ z= ᎏ
σ s

Variable values below the mean have negative z-scores, values above the mean
have positive z-scores, and values equal to the mean have a zero z-score.
Chapter 8 describes z-scores in more detail.

Example 6 Determining Z-Scores

Determine the z-scores for the Audio Maximizer Ultra 3000 and SchmederVox
speakers.

Solution

You can use a calculator, spreadsheet, or statistical software to determine that
the mean is 68.1 and the standard deviation is 15.2 for the speaker scores in
Example 4.

Now, use the mean and standard deviation to calculate the z-scores for the
two speakers.

For the Audio Maximizer Ultra 3000,
x−x −
z= ᎏ
s
67 − 68.1
= ᎏᎏ
15.2
= −0.072


For the SchmederVox,
x−x −
z= ᎏ
s
75 − 68.1
= ᎏᎏ
15.2
= 0.46

The Audio Maximizer Ultra 3000 has a z-score of −0.072, indicating that it is
approximately 7% of a standard deviation below the mean. The SchmederVox
speaker has a z-score of 0.46, indicating that it is approximately half a standard
deviation above the mean.

Key Concepts

• The variance and the standard deviation are measures of how closely a set of
data clusters around its mean. The variance and standard deviation of a sample
may differ from those of the population the sample is drawn from.

• Quartiles are values that divide a set of ordered data into four intervals with
equal numbers of data, while percentiles divide the data into 100 intervals.

• The interquartile range and semi-interquartile range are measures of how
closely a set of data clusters around its median.

• The z-score of a datum is a measure of how many standard deviations the
value is from the mean.


1. Explain how the term root-mean-square applies to the calculation of the
standard deviation.

2. Why does the semi-interquartile range give only an approximate measure of
how far the ﬁrst and third quartiles are from the median?

3. Describe the similarities and differences between the standard deviation and
the semi-interquartile range.

4. Are the median, the second quartile, and the 50th percentile always equal?
Explain why or why not.


A B
1. Determine the mean, standard deviation, 6. The board members of a provincial
and variance for the following samples. organization receive a car allowance for
a) Scores on a data management quiz travel to meetings. Here are the distances
(out of 10 with a bonus question): the board logged last year (in kilometres).

5 7 9 6 5 10 8 2 44 18 125 80 63 42 35 68 52
11 8 7 7 6 9 5 8 75 260 96 110 72 51

b) Costs for books purchased including a) Determine the mean, standard deviation,
taxes (in dollars): and variance for these data.
b) Determine the median, interquartile
12.55 15.31 21.98 45.35 19.81
33.89 29.53 30.19 38.20 range, and semi-interquartile range.
c) Illustrate these data using a box-and-
2. Determine the median, Q1, Q3, the whisker plot.
interquartile range, and semi-interquartile
d) Identify any outliers.
range for the following sets of data.
a) Number of home runs hit by players 7. The nurses’ union collects data on the hours
on the Statsville little league team: worked by operating-room nurses at the
Statsville General Hospital.
6 4 3 8 9 11 6 5 15
Hours Per Week Number of Employees
b) Final grades in a geography class: 12 1
88 56 72 67 59 48 81 62 32 5
90 75 75 43 71 64 78 84 35 7
38 8
3. For a recent standardized test, the median 42 5
was 88, Q1 was 67, and Q3 was 105. Describe
a) Determine the mean, variance, and
the following scores in terms of quartiles.
standard deviation for the nurses’ hours.
a) 8
b) Determine the median, interquartile
b) 81 range, and semi-interquartile range.
c) 103 c) Illustrate these data using a box-and-
whisker plot.
4. What percentile corresponds to
a) the ﬁrst quartile? 8. Application
b) the median? a) Predict the changes in the standard
c) the third quartile? deviation and the box-and-whisker plot
if the outlier were removed from the data
5. Convert these raw scores to z-scores. in question 7.
18 15 26 20 21 b) Remove the outlier and compare the
new results to your original results.
c) Account for any differences between your
prediction and your results in part b).


9. Application Here are the current salaries for semi-interquartile range? Give an example
pte
ha François’ team. or explain why one is not possible.
C

r

m
P

r
oble 14. Inquiry/Problem Solving A business-
300 000 2
travellers’ association rates hotels on a
500 000 3
variety of factors including price, cleanliness,
750 000 8 services, and amenities to produce an overall
900 000 6 score out of 100 for each hotel. Here are the
1 000 000 2 ratings for 50 hotels in a major city.
1 500 000 1
39 50 56 60 65 68 73 77 81 87
3 000 000 1 41 50 56 60 65 68 74 78 81 89
4 000 000 1 42 51 57 60 66 70 74 78 84 91
a) Determine the standard deviation, 44 53 58 62 67 71 75 79 85 94
48 55 59 63 68 73 76 80 86 96
variance, interquartile range, and
semi-interquartile range for these data. a) What score represents
b) Illustrate the data with a modiﬁed i) the 50th percentile?
box-and-whisker plot.
ii) the 95th percentile?
c) Determine the z-score of François’
b) What percentile corresponds to a rating
current salary of $300 000.
of 50?
d) What will the new z-score be if François’
c) The travellers’ association lists hotels
agent does get him a million-dollar
above the 90th percentile as “highly
contract?
recommended” and hotels between the
10. Communication Carol’s golf drives have a 75th and 90th percentiles as
mean of 185 m with a standard deviation “recommended.” What are the minimum
of 25 m, while her friend Chi-Yan shoots scores for the two levels of recommended
a mean distance of 170 m with a standard hotels?
deviation of 10 m. Explain which of the two
friends is likely to have a better score in a
round of golf. What assumptions do you ACHIEVEMENT CHECK

have to make for your answer? Knowledge/ Thinking/Inquiry/
11. Under what conditions will Q1 equal one of
15. a) A data-management teacher has two
the data points in a distribution?
classes whose midterm marks have
12. a) Construct a set of data in which Q1 = Q3 identical means. However, the standard
and describe a situation in which this deviations for each class are signiﬁcantly
equality might occur. different. Describe what these measures
tell you about the two classes.
b) Will such data sets always have a median
equal to Q1 and Q3? Explain your b) If two sets of data have the same mean,
reasoning. can one of them have a larger standard
deviation and a smaller interquartile
13. Is it possible for a set of data to have a range than the other? Give an example
standard deviation much smaller than its or explain why one is not possible.


a) Determine the midrange and
C
− interquartile range for these data.
16. Show that ∑(x − x ) = 0 for any distribution.
b) What are the similarities and differences
n(∑x ) − (∑x)
2 2 between these two measures of spread?
17. a) Show that s =
Ί๶๶
ᎏᎏ .
n(n − 1)
19. The mean absolute deviation of a set of
− −
∑|x − x |
(Hint: Use the fact that ∑ x = nx .) data is defined as ᎏ , where | x − x | is−
b) What are two advantages of using the n
formula in part a) for calculating the absolute value of the difference between
standard deviations? each data point and the mean.
a) Calculate the mean absolute deviation
18. Communication The midrange of a set of and the standard deviation for the data
data is defined as half of the sum of the in question 18.
highest value and the lowest value. The
b) What are the similarities and differences
incomes for the employees of Statsville
between these two measures of spread?
Lawn Ornaments Limited are listed below
(in thousands of dollars).
28 34 49 22 50 31 55 32 73 21
63 112 35 19 44 28 59 85 47 39

Career Connection
Statistician
Use of statistics today is so widespread that there are numerous career
opportunities for statisticians in a broad range of fields. Governments,
medical-research laboratories, sports agencies, financial groups, and
universities are just a few of the many organizations that employ statisticians.
Current trends suggest an ongoing need for statisticians in many areas.

A statistician is engaged in the collection, analysis, presentation, and
interpretation of data in a variety of forms. Statisticians provide insight
into which data are likely to be reliable and whether valid conclusions or
predictions can be drawn from them. A research statistician might develop
new statistical techniques or applications.

Because computers are essential for analysing large
amounts of data, a statistician should possess a
strong background in computers as well as
mathematics. Many positions call for a
minimum of a bachelor’s or master’s degree. For more information about a career as a
Research at a university or work for a statistician and other careers related to
consulting firm usually requires a doctorate. mathematics, visit the above web site
and follow the links.



2.1 Data Analysis With Graphs 2.2 Indices
Refer to the Key Concepts on page 100. Refer to the Key Concepts on page 109.

1. The following data show monthly sales of The following graph shows four categories
houses by a real-estate agency. from the basket of goods and services used
to calculate the consumer price index.
6 5 7 6 8 3 5 4 6
7 5 9 5 6 6 7 200
Fresh
Vegetables
a) Construct an ungrouped frequency

for Ontario (1992 = 100)
Consumer Price Index
150
table for this distribution. Coffee and
Tea
b) Create a frequency diagram. 100
Rent
c) Create a cumulative-frequency diagram.
50 Fuel Oil
and Other
2. A veterinary study recorded the masses in Fuel
grams of 25 kittens at birth. 0

1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
240 300 275 350 280 260 320
295 340 305 280 265 300 275
315 285 320 325 275 270 290 4. a) What is this type of graph called?
245 235 305 265 b) Which of the four categories had the
greatest increase during the period
a) Organize these data into groups. shown?
b) Create a frequency table and histogram. c) Why do all four graphs intersect
c) Create a frequency polygon. at 1992?
d) Create a relative-frequency diagram. d) Which category was
i) the most volatile?
3. A class of data-management students
listed their favourite board games. ii) the least volatile?

Game Frequency e) Suggest reasons for this difference
Pictionary® 10 in volatility.
Chess 5 5. a) If a tin of coffee cost $5.99 in 1992,
Trivial Pursuit® 8 what would you expect it to cost in
MONOPOLY® 3
i) 1995?
Balderdash® 6
ii) 1990?
Other 4
b) What rent would a typical tenant pay
a) What type of data does this table show? in 2000 for an apartment that had a rent
Explain your reasoning. of $550 per month in 1990?
b) Graph these data using an appropriate c) What might you expect to pay for
format. broccoli in 2000, if the average price
c) Explain why you chose the type of graph you paid in 1996 was $1.49 a bunch?
you did.


2.3 Sampling Techniques c) A budding musician plays a new song for
Refer to the Key Concepts on page 116. family members and friends to see if it is
good enough to record professionally.
6. a) Explain the difference between a
d) Every fourth person entering a public
stratiﬁed sample and a systematic sample.
library is asked: “Do you think Carol
b) Describe a situation where a convenience Shields should receive the Giller prize
sample would be an appropriate for her brilliant and critically acclaimed
technique. new novel?”
c) What are the advantages and
disadvantages of a voluntary-response 10. For each situation in question 9, suggest
sample? how the statistical process could be changed
to remove the bias.
7. Suppose you are conducting a survey that
you would like to be as representative as 2.5 Measures of Central Tendency
possible of the entire student body at your Refer to the Key Concepts on page 133.
school. However, you have time to visit
only six classes and to process data from a 11. a) Determine the mean, median, and mode
total of 30 students. for the data in question 1.
a) What sampling technique would you use? b) Which measure of central tendency best

b) Describe how you would select the
describes these data? Explain your
students for your sample. reasoning.

12. a) Use your grouped data from question 2
8. Drawing names from a hat and using a
random-number generator are two ways to to estimate the mean and median masses
obtain a simple random sample. Describe for the kittens.
two other ways of selecting a random sample. b) Determine the actual mean and median
masses from the raw data.
2.4 Bias in Surveys c) Explain any differences between your
Refer to the Key Concepts on page 122. answers to parts a) and b).

9. Identify the type of bias in each of the 13. a) For what type of “average” will the
following situations and state whether the following statement always be true?
bias is due to the sampling technique or “There are as many people with
the method of data collection. below-average ages as there are with
a) A survey asks a group of children above-average ages. ”
whether or not they should be allowed b) Is this statement likely to be true for
unlimited amounts of junk food. either of the other measures of central
b) A teachers asks students to raise their tendency discussed in this chapter?
hands if they have ever told a harmless lie. Why or why not?


14. Angela is applying to a university 17. a) Explain why you cannot calculate the
engineering program that weights an semi-interquartile range if you know
applicant’s eight best grade-12 marks as only the difference between either Q3
shown in the following table. and the median or median and Q1.
Subjects Weighting b) Explain how you could determine the
Calculus, chemistry, geometry semi-interquartile range if you did know
and discrete mathematics, physics 3 both of the differences in part a).
Computer science, data
management, English 2 18. a) For the data in question 2, determine
Other 1 i) the ﬁrst and third quartiles
Angela’s grade-12 ﬁnal marks are listed ii) the 10th, 25th, 75th, and 90th
below. percentiles
Subject Mark Subject Mark b) Would you expect any of the values in
Calculus 95 Computer science 84 part a) to be equal? Why or why not?
English 89 Chemistry 90
Geometry and 94 Mathematics of 87 19. The scores on a precision-driving test for
discrete mathematics data management prospective drivers at a transit company have
Physical education 80 Physics 92 a mean of 100 and a standard deviation of 15.
a) Calculate Angela’s weighted average. a) Determine the z-score for each of the

b) Calculate Angela’s unweighted average.
following raw scores.

c) Explain why the engineering program
i) 85 ii) 135 iii) 100 iv) 62
would use this weighting system. b) Determine the raw score corresponding
to each of the following z-scores.
15. Describe three situations where the mode
i) 1 ii) −2 iii) 1.5 iv) −1.2
would be the most appropriate measure of
central tendency. 20. Dr. Simba’s fourth-year class in animal
biology has only 12 students. Their scores on
2.6 Measures of Spread the midterm examination are shown below.
Refer to the Key Concepts on page 147. 50 71 65 54 84 69 82
67 52 52 86 85
16. a) Determine the standard deviation,
the interquartile range, and the semi- a) Calculate the mean and median for these
interquartile range for the data in data. Compare these two statistics.
question 1. b) Calculate the standard deviation and the
b) Create a box-and-whisker plot for these semi-interquartile range. Compare these
data. statistics and comment on what you notice.
c) Are there any outliers in the data? c) Which measure of spread is most
Justify your answer. suitable for describing this data set?
Explain why.


Chapter Test

ACHIEVEMENT CHART

Questions All 10, 11 4, 6, 7, 8, 9, 11 5, 6, 11

Use the following set of data-management ﬁnal 6. An interview committee graded three short-
examination scores to answer questions 1 listed candidates for a management position
through 5. as shown below. The scores are on a scale of
1 to 5, with 5 as the top score.
92 48 59 62 66 98 70 70 55 63
70 97 61 53 56 64 46 69 58 64 Criterion Weight Clarise Pina Steven
Education 2 3 3 4
1. a) Group these data into intervals and
Experience 2 4 5 3
create a frequency table.
Interpersonal skills 3 3 3 5
b) Produce a frequency diagram and
First interview 1 5 4 3
a frequency polygon.
Who should the committee hire based on
c) Produce a cumulative-frequency
these data? Justify your choice.
diagram.
7. Describe the type of sample used in each
2. Determine the
of the following scenarios.
a) three measures of central tendency
a) A proportionate number of boys and
b) standard deviation and variance girls are randomly selected from a class.
c) interquartile and semi-interquartile b) A software company randomly chooses
ranges a group of schools in a particular school
district to test a new timetable program.
3. a) Produce a modiﬁed box-and-whisker plot
for this distribution. c) A newspaper prints a questionnaire and
invites its readers to mail in their
b) Identify any outliers.
responses.
c) Identify and explain any other unusual
d) A telephone-survey company uses a
features of this graph.
random-number generator to select
4. Explain which of the three measures of which households to call.
central tendency is most appropriate to e) An interviewer polls people passing by
describe this distribution of marks and why on the street.
the other two measures are not appropriate.
8. A group of 8 children in a day-care centre
5. Students with scores above the 90th are to be interviewed about their favourite
percentile receive a book prize. games. Describe how you would select a
a) How many students will receive prizes? systematic sample if there are 52 children
b) What are these students’ scores?
at the centre.


9. a) Identify the bias in the following surveys iii) A random survey of corporate
and explain the effect it could have on executives asked: “Do you favour
their results. granting a cable-television licence
i) Parents of high-school students were for a new economics and business
asked: “Do you think that students channel?”
should be released from school a half b) Suggest how to eliminate the bias in
hour early on Friday, free to run each of the surveys in part a).
around and get into trouble?”
10. A mutual-fund company proudly advertises
ii) Audience members at an investment
that all of its funds have “ﬁrst-quartile
workshop were asked to raise their
performance.” What mathematical errors has
hands if they had been late with a bill
the company made in this advertisement?
payment within the last six months.

ACHIEVEMENT CHECK

11. The graph below shows the stock price for an Ontario technology
company over a one-month period in 2001.
30
28
Stock Price ($)

26
24
22
20
18
23 25 1 8 15 22

August 2001 September 2001

a) When did the stock reach its lowest value during the period shown?
Suggest a possible reason for this low point.
b) Compare the percent drop in stock price from September 1 to
September 8 to the drop during the following week.
c) Sketch a new graph and provide a commentary that the company
could use to encourage investors to buy the company’s stock.


3
PT ER
Statistics of Two Variables
CHA


Define the correlation coefficient as a measure of the fit of a scatter 3.1, 3.2, 3.3,
graph to a linear model. 3.5

Calculate the correlation coefficient for a set of data, using graphing 3.1, 3.2, 3.3,
calculators or statistical software. 3.5

Demonstrate an understanding of the distinction between cause-effect 3.1, 3.2, 3.3,
relationships and the mathematical correlation between variables. 3.4, 3.5

Describe possible misuses of regression. 3.2, 3.3, 3.5

Explain examples of the use and misuse of statistics in the media. 3.5

Assess the validity of conclusions made on the basis of statistical studies, 3.2, 3.3, 3.4,
by analysing possible sources of bias in the studies and by calculating 3.5
and interpreting additional statistics, where possible.

Demonstrate an understanding of the purpose and the use of a variety of 3.4, 3.5
sampling techniques.

Organize and summarize data from secondary sources, using 3.1, 3.2, 3.3,
technology. 3.4, 3.5

Locate data to answer questions of significance or personal interest, by 3.1, 3.2, 3.4,
searching well-organized databases. 3.5

Use the Internet effectively as a source for databases. 3.1, 3.2, 3.4,
3.5

Chapter Problem
Job Prospects 1. How could Gina graph this data to
Gina is in her second year of business estimate
studies at university and she is starting to a) her chances of finding a job in her
think about a job upon graduation. She has field when she graduates in two years?
two primary concerns—the job market and b) her starting salary?
expected income. Gina does some research
at the university’s placement centre and 2. What assumptions does Gina have to
finds employment statistics for graduates of make for her predictions? What other
her program and industry surveys of entry- factors could affect the accuracy of
level salaries. Gina’s estimates?
Number Mean Starting This chapter introduces statistical
Number of
Hired Upon Salary
Year Graduates techniques for measuring relationships
Graduation ($000)
1992 172 151 26 between two variables. As you will see,
1993 180 160 27 these techniques will enable Gina to make
1994 192 140 28 more precise estimates of her job prospects.
1995 170 147 27.5
1996 168 142 27
Two-variable statistics have an enormous
1997 176 155 26.5 range of applications including industrial
1998 180 160 27 processes, medical studies, and
1999 192 162 29 environmental issues—in fact, almost any
2000 200 172 31 field where you need to determine if a
2001 220 180 34 change in one variable affects another.



1. Scatter plots For each of the following sets of 5. Graphing exponential functions
data, create a scatter plot and describe any a) Identify the base and the numerical
patterns you see. coefficient for each of the following
a) x y b) x y functions.
3 18 4 6 i) y = 0.5(3)x ii) y = 2x iii) y = 100(0.5)x
5 15 7 2 b) Graph each of the functions in part a).
8 12 13 17
c) Explain what happens to the value of x as
9 10 14 5
the curves in part b) approach the x-axis.
12 8 23 19
15 4 24 11 6. Sigma notation Calculate each sum without
17 1 25 30 the use of technology.
8 5
33 21 a) Αi b) Αi 2

36 29 i=1 i=1
−
7. Sigma notation Given x = 2.5, calculate each
40 39
42 26 sum without the use of technology.
6 4
− −
46 32 a) Α (i − x )
i=1
b) Α (i − x )
i=1
2

2. Scatter plots For each plot in question 1, 8. Sigma notation
i) graph the line of best fit and calculate a) Repeat questions 6 and 7 using
its equation appropriate technology such as a
ii) estimate the x- and y-intercepts graphing calculator or a spreadsheet.
iii) estimate the value of y when x = 7 b) Explain the method that you chose.

3. Graphing linear equations Determine the 9. Sampling (Chapter 2) Briefly explain each
slope and y-intercept for the lines defined by of the following terms.
the following equations, and then graph the a) simple random sample
lines.
b) systematic sample
a) y = 3x − 4 b) y = −2x + 6
c) outlier
c) 12x − 6y = 7
10. Bias (Chapter 2)
4. Graphing quadratic functions Graph the
a) Explain the term measurement bias.
following functions and estimate any x- and
y-intercepts. b) Give an example of a survey method
containing unintentional measurement bias.
a) y = 2x2
c) Give an example of a survey method
b) y = x2 + 5x − 6
containing intentional measurement bias.
c) y = −3x2 + x + 2
d) Give an example of sampling bias.

158 MHR • Statistics of Two Variables

3.1 Scatter Plots and Linear Correlation

Does smoking cause lung cancer? Is job performance related to marks in high
school? Do pollution levels affect the ozone layer in the atmosphere? Often the
answers to such questions are not clear-cut, and inferences have to be made from
large sets of data. Two-variable statistics provide methods for detecting relationships
between variables and for developing mathematical models of these relationships.

The visual pattern in a graph or plot can often reveal the nature of the relationship
between two variables.

I N V E S T I G AT E & I N Q U I R E : V i s u a l i z i n g R e l a t i o n s h i p s B e t w e e n Va r i a b l e s

A study examines two new obedience-training
methods for dogs. The dogs were randomly selected
to receive from 5 to 16 h of training in one of
the two training programs. The dogs were assessed
using a performance test graded out of 20.
Rogers Method Laing System
Hours Score Hours Score
10 12 8 10
15 16 6 9
7 10 15 12
12 15 16 7
8 9 9 11
5 8 11 7
8 11 10 9
16 19 10 6
10 14 8 15
1. Could you determine which of the two training systems is more effective by
comparing the mean scores? Could you calculate another statistic that would
give a better comparison? Explain your reasoning.
2. Consider how you could plot the data for the Rogers Method. What do
you think would be the best method? Explain why.
3. Use this method to plot the data for the Rogers Method. Describe any
patterns you see in the plotted data.
4. Use the same method to plot the data for the Laing System and describe
any patterns you see.
5. Based on your data plots, which training method do you think is more
effective? Explain your answer.

3.1 Scatter Plots and Linear Correlation • MHR 159

6. Did your plotting method make it easy to compare the two sets of data?
Are there ways you could improve your method?
7. a) Suggest factors that could influence the test scores but have not been
taken into account.
b) How could these factors affect the validity of conclusions drawn from
the data provided?

In data analysis, you are often trying to discern whether one variable, the
dependent (or response) variable, is affected by another variable, the
independent (or explanatory) variable. Variables have a linear correlation
if changes in one variable tend to be proportional to changes in the other.
Variables X and Y have a perfect positive (or direct) linear correlation if
Y increases at a constant rate as X increases. Similarly, X and Y have a perfect
negative (or inverse) linear correlation if Y decreases at a constant rate as
X increases.

A scatter plot shows such relationships graphically, usually with the
independent variable as the horizontal axis and the dependent variable as
the vertical axis. The line of best fit is the straight line that passes as close
as possible to all of the points on a scatter plot. The stronger the correlation,
the more closely the data points cluster around the line of best fit.

Example 1 Classifying Linear Correlations

Classify the relationship between the variables X and Y for the data shown
in the following diagrams.
a) y b) y c) y

x x x

d) y e) y f) y

x x x


Solution

a) The data points are clustered around a line that rises to the right (positive
slope), indicating definitely that Y increases as X increases. Although the
points are not perfectly lined up, there is a strong positive linear correlation
between X and Y.

b) The data points are all exactly on a line that slopes down to the right, so
Y decreases as X increases. In fact, the changes in Y are exactly proportional
to the changes in X. There is a perfect negative linear correlation between X
and Y.

c) No discernible linear pattern exists. As X increases, Y appears to change
randomly. Therefore, there is zero linear correlation between X and Y.

d) A definite positive trend exists, but it is not as clear as the one in part a).
Here, X and Y have a moderate positive linear correlation.

e) A slight positive trend exists. X and Y have a weak positive linear correlation.

f) A definite negative trend exists, but it is hard to classify at a glance. Here,
X and Y have a moderate or strong negative linear correlation.

As Example 1 shows, a scatter plot often can give only a rough indication of the
correlation between two variables. Obviously, it would be useful to have a more
precise way to measure correlation. Karl Pearson (1857−1936) developed a
formula for estimating such a measure. Pearson, who also invented the term
standard deviation, was a key figure in the development of modern statistics.

The Correlation Coefficient
To develop a measure of correlation, mathematicians first defined the
covariance of two variables in a sample:
1
sXY = ᎏᎏ Α (x − x )( y − − )
− y
n−1
where n is the size of the sample, x represents individual values of the variable
X, y represents individual values of the variable Y, x is the mean of X, and − is
− y
the mean of Y.

Recall from Chapter 2 that the symbol Α means “the sum of.” Thus, the
covariance is the sum of the products of the deviations of x and y for all the data
points divided by n − 1. The covariance depends on how the deviations of the
two variables are related. For example, the covariance will have a large positive
value if both x − x and y − − tend to be large at the same time, and a negative
− y
value if one tends to be positive when the other is negative.


The correlation coefficient, r, is the covariance divided by the product of the
standard deviations for X and Y:
sXY
r = ᎏᎏ
sX × sY

where sX is the standard deviation of X and sY is the standard deviation of Y.
This coefficient gives a quantitative measure of the strength of a linear
correlation. In other words, the correlation coefficient indicates how closely the
data points cluster around the line of best fit. The correlation coefficient is also
called the Pearson product-moment coefficient of correlation (PPMC) or
Pearson’s r.

The correlation coefficient always has values in the range from −1 to 1. Consider
a perfect positive linear correlation first. For such correlations, changes in the
dependent variable Y are directly proportional to changes in the independent
variable X, so Y = aX + b, where a is a positive constant. It follows that

sXY = ᎏ Α (x − x )( y − − )
1 − ∑( y − − )2
Ί๶
y
y sY = ᎏᎏ
n−1 n−1
1 − − −
= ᎏ Α (x − x )[(ax + b) − (ax + b)] ∑[(ax + b) − (ax + b)]2
n−1 = Ί๶๶
ᎏᎏᎏ
n−1
1 −
= ᎏ Α (x − x )(ax − ax )− −
∑(ax − ax )2
n−1 = ᎏᎏΊ๶ n−1
1
= ᎏ Α a(x − x )2− −
a ∑(x − x )2
Ί๶
2
n−1 = ᎏᎏ
n−1
−
∑(x − x )2 −
∑(x − x )2
= a ᎏᎏ
n−1 Ί๶
= a ᎏᎏ
n−1
= as X
2
= asX

Substituting into the equation for the correlation coefficient gives
sXY
r= ᎏ
sX sY
as2
=ᎏ
X

sX (asX )
Y

=1 r =1

X


Similarly, r = −1 for a perfect negative linear correlation.

For two variables with no correlation, Y is r=0
equally likely to increase or decrease as X
increases. The terms in Α (x − x )( y − −) are
− y
randomly positive or negative and tend to
cancel each other. Therefore, the correlation

Y

Y
coefficient is close to zero if there is little or
no correlation between the variables. For
r = –0.5
moderate linear correlations, the summation
terms partially cancel out. X X

The following diagram illustrates how the correlation coefficient corresponds
to the strength of a linear correlation.
Negative Linear Correlation Positive Linear Correlation
Perfect Perfect
Strong Moderate Weak Weak Moderate Strong

–1 –0.67 – 0.33 0 0.33 0.67 1
Correlation Coefficient, r

−
Using algebraic manipulation and the fact that Α x = nx , Pearson showed that

n∑xy − (∑x)(∑y)
r = ᎏᎏᎏᎏ
͙[n∑x2 − (∑x)2][n∑y2 − (∑y)2]
ෆ ෆෆෆෆ

where n is the number of data points in the sample, x represents individual
values of the variable X, and y represents individual values of the variable Y.
(Note that Α x2 is the sum of the squares of all the individual values of X,
while ( Α x)2 is the square of the sum of all the individual values.)

Like the alternative formula for standard deviations (page 150), this formula for
r avoids having to calculate all the deviations individually. Many scientific and
statistical calculators have built-in functions for calculating the correlation
coefficient.

It is important to be aware that increasing the number of data points used in
determining a correlation improves the accuracy of the mathematical model.
Some of the examples and exercise questions have a fairly small set of data in
order to simplify the computations. Larger data sets can be found in the e-book
that accompanies this text.


Example 2 Applying the Correlation Coefﬁcient Formula

A farmer wants to determine whether there is a relationship between the mean
temperature during the growing season and the size of his wheat crop. He
assembles the following data for the last six crops.
Mean Temperature (°C) Yield (tonnes/hectare)
4 1.6
8 2.4
10 2.0
9 2.6
11 2.1
6 2.2
a) Does a scatter plot of these data indicate any linear correlation between
the two variables?
b) Compute the correlation coefﬁcient.
c) What can the farmer conclude about the relationship between the mean
temperatures during the growing season and the wheat yields on his farm?

Solution

a) The farmer wants to know whether the crop yield depends y
2.5
on temperature. Here, temperature is the independent

Yield (T/ha)
2
variable, X, and crop yield is the dependent variable, Y. The
1.5
scatter plot has a somewhat positive trend, so there appears 1
to be a moderate positive linear correlation. 0.5

0 2 4 6 8 10 12 14 x
Mean Temperature (ºC)

b) To compute r, set up a table to calculate the quantities required
by the formula.
Temperature, x Yield, y x2 y2 xy
4 1.6 16 2.56 6.4
8 2.4 64 5.76 19.2
10 2.0 100 4.00 20.0
9 2.6 81 6.76 23.4
11 2.1 121 4.41 23.1
6 2.2 36 4.84 13.2

Α x = 48 Α y = 12.9 Α x = 418
2
Αy 2
= 28.33 Α xy = 105.3


Now compute r, using the formula: Data in Action
n∑(xy) − (∑x)(∑y) From 1992 to 2001,
r = ᎏᎏᎏᎏ
͙ෆ− (∑x)2][n∑y2 − (∑yෆ
[n∑x2 ෆෆ ෆ)2] Canada produced an
average of 27 million
6(105.3) − (48)(12.9)
= ᎏᎏᎏᎏ tonnes of wheat a
[6(418) − (48)2][6(28.33) − (12.9)ෆ
͙ෆෆෆෆෆ2] year. About 70%
631.8 − 619.2 of this crop was
= ᎏᎏᎏᎏ exported.
(2508 − 2304)(169.98 − 166.41)
͙ෆෆෆෆෆ
12.6
=ᎏ
26.99
= 0.467

The correlation coefficient for crop yield versus mean temperature is
approximately 0.47, which confirms a moderate positive linear correlation.

c) It appears that the crop yield tends to increase somewhat as the mean temperature
for the growing season increases. However, the farmer cannot conclude that higher
temperatures cause greater crop yields. Other variables could account for the
correlation. For example, the lower temperatures could be associated with heavy
rains, which could lower yields by flooding fields or leaching nutrients from the soil.

The important principle that a correlation does not prove the existence of a cause-
and-effect relationship between two variables is discussed further in section 3.4.

Example 3 Using Technology to Determine Correlation Coefficients

Determine whether there is a linear correlation between horsepower and fuel
consumption for these five vehicles by creating a scatter plot and calculating the
correlation coefficient.
Vehicle Horsepower, x Fuel Consumption (L/100 km), y
Midsize sedan 105 6.7
Minivan 170 23.5
Small sports utility vehicle 124 5.9
Midsize motorcycle 17 3.4
Luxury sports car 296 8.4


Use the ClrList command to make sure lists L1 and L2 are clear, then enter See Appendix B for
more details on the
the horsepower data in L1 and the fuel consumption figures in L2.
graphing calculator
To display a scatter plot, first make sure that all functions in the Y= editor and software functions
are either clear or turned off. Then, use STAT PLOT to select PLOT1. used in this section.


Turn the plot on, select the scatter-plot icon, and enter L1 for XLIST and L2 for
YLIST. (Some of these settings may already be in place.) From the ZOOM menu,
select 9:ZoomStat. The calculator will automatically optimize the window
settings and display the scatter plot.

To calculate the correlation coefficient, from the CATALOG menu, select
DiagnosticOn, then select the LinReg(ax+b) instruction from the STAT CALC menu.
The calculator will perform a series of statistical calculations using the data in
lists L1 and L2. The last line on the screen shows that the correlation coefficient
is approximately 0.353.

Therefore, there is a moderate linear correlation
between horsepower and fuel consumption for
the five vehicles.


Set up three columns and enter the data from the table above. Highlight the
numerical data and use your spreadsheet’s Chart feature to display a scatter plot.
Both Corel® Quattro® Pro and Microsoft® Excel have a CORREL function that
allows you to calculate the correlation coefficient easily. The scatter plot and
correlation coefficient indicate a moderate correlation between horsepower and
fuel consumption.

Create a new collection by setting up a case table with three attributes: Vehicle,
Hp, and FuelUse. Enter the data for the five cases. To create a scatter plot, drag
the graph icon onto the work area and drop the Hp attribute on the x-axis and
the FuelUse attribute on the y-axis.


To calculate the correlation coefficient, right-click on the collection and select Inspect
Collection. Select the Measures tab and name a new measure PPMC. Right-click this
measure and select Edit Formula, then Functions/Statistical/Two Attributes/correlation.
When you enter the Hp and FuelUse attributes in the correlation function,
Fathom™ will calculate the correlation coefficient for these data.

Again, the scatter plot and correlation coefficient show a moderate linear
correlation.
Project
Prep

For your statistics
project, you may
be investigating
the linear
correlation
between two
variables. A
graphing
calculator or
computer software
may be a valuable
Notice that the scatter plots in Example 3 have an outlier at (170, 23.5). aid for this
Without this data point, you would have a strong positive linear correlation. analysis.
Section 3.2 examines the effect of outliers in more detail.

Key Concepts

• Statistical studies often find linear correlations between two variables.

• A scatter plot can often reveal the relationship between two variables. The
independent variable is usually plotted on the horizontal axis and the
dependent variable on the vertical axis.

• Two variables have a linear correlation if changes in one variable tend to be
proportional to changes in the other. Linear correlations can be positive or
negative and vary in strength from zero to perfect.

• The correlation coefficient, r, is a quantitative measure of the correlation
between two variables. Negative values indicate negative correlations while
positive values indicate positive correlations. The greater the absolute value
of r, the stronger the linear correlation, with zero indicating no correlation
at all and 1 indicating a perfect correlation.

• Manual calculations of correlation coefficients can be quite tedious, but a
variety of powerful technology tools are available for such calculations.



1. Describe the advantages and disadvantages of using a scatter plot or the
correlation coefficient to estimate the strength of a linear correlation.

2. a) What is the meaning of a correlation coefficient of
i) −1?
ii) 0?
iii) 0.5?
b) Can the correlation coefficient have a value greater than 1?
Why or why not?

3. A mathematics class finds a correlation coefficient of 0.25 for the students’
midterm marks and their driver’s test scores and a coefficient of −0.72 for
their weight-height ratios and times in a 1-km run. Which of these two
correlations is stronger? Explain your answer.

A B
1. Classify the type of linear correlation that 3. For a week prior to their final physics
you would expect with the following pairs examination, a group of friends collect
of variables. data to see whether time spent studying
a) hours of study, examination score or time spent watching TV had a stronger
correlation with their marks on the
b) speed in excess of the speed limit,
examination.
amount charged on a traffic fine
Hours Examination
c) hours of television watched per week,
Hours Studied Watching TV Score
final mark in calculus 10 8 72
d) a person’s height, sum of the digits in 11 7 67
the person’s telephone number 15 4 81
e) a person’s height, the person’s strength 14 3 93
8 9 54
2. Identify the independent variable and the
5 10 66
dependent variable in a correlational study
of a) Create a scatter plot of hours studied
a) heart disease and cholesterol level versus examination score. Classify the
b) hours of basketball practice and free- linear correlation.
throw success rate b) Create a similar scatter plot for the
c) amount of fertilizer used and height hours spent watching TV.
of plant c) Which independent variable has a
d) income and level of education stronger correlation with the
examination scores? Explain.
e) running speed and pulse rate


d) Calculate the correlation coefficient for c) Does the computed r-value agree with
hours studied versus examination score the classification you made in part a)?
and for hours watching TV versus Explain why or why not.
examination score. Do these answers d) Identify any outliers in the data.
support your answer to c)? Explain.
e) Suggest possible reasons for any outliers
4. Application Refer to the tables in the identified in part d).
investigation on page 159.
6. Application Six classmates compared their
a) Determine the correlation coefficient arm spans and their scores on a recent
and classify the linear correlation for mathematics test as shown in the following
the data for each training method. table. Span (m)
Arm Score
b) Suppose that you interchanged the 1.5 82
dependent and independent variables, 1.4 71
so that the test scores appear on the 1.7 75
horizontal axis of a scatter plot and the 1.6 66
hours of training appear on the vertical
1.6 90
axis. Predict the effect this change will
1.8 73
have on the scatter plot and the
correlation coefficient for each set of data.
a) Illustrate these data with a scatter plot.
c) Test your predictions by plotting the data
and calculating the correlation b) Determine the correlation coefficient
coefficients with the variables reversed. and classify the linear correlation.
Explain any differences between your c) What can the students conclude from
results and your predictions in part b). their data?

5. A company studied whether there was a 7. a) Use data in the table on page 157 to
pte
relationship between its employees’ years of ha create a scatter plot that compares the size
C

r

service and number of days absent. The data of graduating classes in Gina’s program to
m
P

r
oble
for eight randomly selected employees are the number of graduates who found jobs.
shown below. b) Classify the linear correlation.
Years of Days Absent c) Determine the linear correlation
Employee Service Last Year
coefficient.
Jim 5 2
Leah 2 6 8. a) Search sources such as E-STAT,
Efraim 7 3 CANSIM II, the Internet, newspapers,
Dawn 6 3 and magazines for pairs of variables that
Chris 4 4 exhibit
Cheyenne 8 0 i) a strong positive linear correlation
Karrie 1 2 ii) a strong negative linear correlation
Luke 10 1
iii) a weak or zero linear correlation
a) Create a scatter plot for these data and b) For each pair of variables in part a),
classify the linear correlation. identify the independent variable and
b) Calculate the correlation coefficient. the dependent variable.


9. Find a set of data for two variables known 13. a) Search sources such as newspapers,
to have a perfect positive linear correlation. magazines, and the Internet for a set of
Use these data to demonstrate that the two-variable data with
correlation coefficient for such variables is 1. i) a moderate positive linear correlation
Alternatively, find a set of data with a perfect
ii) a moderate negative correlation
negative correlation and show that the
correlation coefficient is −1. iii) a correlation in which |r| > 0.9
b) Outline any conclusions that you can
10. Communication make from each set of data. Are there
a) Would you expect to see a correlation any assumptions inherent in these
between the temperature at an outdoor conclusions? Explain.
track and the number of people using c) Pose at least two questions that could
the track? Why or why not? form the basis for further research.
b) Sketch a typical scatter plot of this type
of data. 14. a) Sketch scatter plots of three different
patterns of data that you think would
c) Explain the key features of your scatter
have zero linear correlation.
plot.
b) Explain why r would equal zero for each
11. Inquiry/Problem Solving Refer to data tables of these patterns.
in the investigation on page 159. c) Use Fathom™ or a spreadsheet to create
a) How could the Rogers Training a scatter plot that looks like one of your
Company graph the data so that their patterns and calculate the correlation
training method looks particularly good? coefficient. Adjust the data points to get
b) How could Laing Limited present the r as close to zero as you can.
same data in a way that favours their
training system?
c) How could a mathematically
knowledgeable consumer detect the
distortions in how the two companies
present the data?

C
a) Prove that interchanging the
independent and dependent variables
does not change the correlation
coefficient for any set of data.
b) Illustrate your proof with calculations
using a set of data selected from one of
the examples or exercise questions in
this section.


3.2 Linear Regression

Regression is an analytic technique for
determining the relationship between a dependent
variable and an independent variable. When the
two variables have a linear correlation, you can
develop a simple mathematical model of the
relationship between the two variables by finding
a line of best fit. You can then use the equation
for this line to make predictions by interpolation
(estimating between data points) and
extrapolation (estimating beyond the range of
the data).

I N V E S T I G AT E & I N Q U I R E : Modelling a Linear Relationship

A university would like to construct a mathematical model to predict
first-year marks for incoming students based on their achievement in grade 12.
A comparison of these marks for a random sample of first-year students is
shown below.
Grade 12 Average 85 90 76 78 88 84 76 96 86 85
First-Year Average 74 83 68 70 75 72 64 91 78 86

1. a) Construct a scatter plot for these data. Which variable should be
placed on the vertical axis? Explain.
b) Classify the linear correlation for this data, based on the scatter
plot.
2. a) Estimate and draw a line of best fit for the data.
b) Measure the slope and y-intercept for this line, and write an equation
for it in the form y = mx + b.
3. Use this linear model to predict
a) the first-year average for a student who had an 82 average in
grade 12
b) the grade-12 average for a student with a first-year average of 60
4. a) Use software or the linear regression instruction of a graphing
calculator to find the slope and y-intercept for the line of best fit.
(Note that most graphing calculators use a instead of m to represent
slope.)
b) Are this slope and y-intercept close to the ones you measured in
question 2? Why or why not?

3.2 Linear Regression • MHR 171

c) Estimate how much the new values for slope and y-intercept will change
your predictions in question 3. Check your estimate by recalculating your
predictions using the new values and explain any discrepancies.
5. List the factors that could affect the accuracy of these mathematical models.
Which factor do you think is most critical? How could you test how much
effect this factor could have?

It is fairly easy to “eyeball” a good estimate of the line of best fit on a scatter
plot when the linear correlation is strong. However, an analytic method using a
least-squares fit gives more accurate results, especially for weak correlations.

Consider the line of best fit in the following scatter plot. A dashed blue line
shows the residual or vertical deviation of each data point from the line of best
fit. The residual is the difference between the values of y at the data point and
at the point that lies on the line of best fit and has the same x-coordinate as the
data point. Notice that the residuals are positive for points above the line and
negative for points below the line. The boxes show the squares of the residuals.
y

x

For the line of best fit in the least-squares method,
• the sum of the residuals is zero (the positive and negative residuals cancel out)
• the sum of the squares of the residuals has the least possible value

Although the algebra is daunting, it can be shown that this line has the equation

n(∑xy) − (∑x)(∑y)
y = ax + b, where a = ᎏᎏ and b = − − ax y −
n(∑x2) − (∑x)2

Recall from Chapter 2 that x is the mean of x and − is the mean of y. Many
− y
statistics texts use an equation with the form y = a + bx, so you may sometimes
see the equations for a and b reversed.


Example 1 Applying the Least-Squares Formula

This table shows data for the full-time employees of a Age (years) Annual Income ($000)
small company. 33 33
a) Use a scatter plot to classify the correlation between 25 31
age and income. 19 18
b) Find the equation of the line of best ﬁt analytically. 44 52
c) Predict the income for a new employee who is 21 and 50 56
an employee retiring at age 65. 54 60
38 44
29 35

Solution

a) The scatter plot suggests a strong positive linear
65
correlation between age and income level.
55

Income
45
35
25
15

0 15 20 25 30 35 40 45 50 55
Age

b) To determine the equation of the line of best ﬁt, organize the data into
a table and compute the sums required for the formula.
Age, x Income, y x2 xy
33 33 1089 1089
25 31 625 775
19 18 361 342
44 52 1936 2288
50 56 2500 2800
54 60 2916 3240
38 44 1444 1672
29 35 841 1015

Α x = 292 Α y = 329 Αx 2
= 11 712 Α xy = 13 221
Substitute these totals into the formula for a.
n(∑xy) − (∑x)(∑y)
a = ᎏᎏ
n(∑x2) − (∑x)2
8(13 221) − (292)(329)
= ᎏᎏᎏ
8(11 712) − (292)2
9700
=ᎏ
8432
⋅ 1.15
=


To determine b, you also need the means of x and y.
− ∑x −= ᎏ∑y
x =ᎏ y b = − − ax
y −
n n
= 41.125 − 1.15(36.5)
292 329 = −0.85
=ᎏ =ᎏ
8 8
= 36.5 = 41.125
Now, substitute the values of a and b into the equation for the line of best fit.
y = ax + b
= 1.15x − 0.85
Therefore, the equation of the line of best fit is y = 1.15x − 0.85.

c) Use the equation of the line of best fit as a model.

For a 21-year-old employee, For a 65-year-old employee,
y = ax + b y = ax + b
= 1.15(21) − 0.85 = 1.15(65) − 0.85
= 23.3 = 73.9

Therefore, you would expect the new employee to have an income of about
$23 300 and the retiring employee to have an income of about $73 900. Note
that the second estimate is an extrapolation beyond the range of the data, so
it could be less accurate than the first estimate, which is interpolated between
two data points.

Note that the slope a indicates only how y varies with x on the line of best fit.
The slope does not tell you anything about the strength of the correlation
between the two variables. It is quite possible to have a weak correlation with
a large slope or a strong correlation with a small slope.

Example 2 Linear Regression Using Technology

Researchers monitoring the numbers of wolves and rabbits in a wildlife reserve
think that the wolf population depends on the rabbit population since wolves
prey on rabbits. Over the years, the researchers collected the following data.
Year 1994 1995 1996 1997 1998 1999 2000 2001
Rabbit Population 61 72 78 76 65 54 39 43
Wolf Population 26 33 42 49 37 30 24 19

a) Determine the line of best fit and the correlation coefficient for these data.
b) Graph the data and the line of best fit. Do these data support the
researchers’ theory?



a) You can use the calculator’s linear regression instruction to find both the line
of best fit and the correlation coefficient. Since the theory is that the wolf
population depends on the rabbit population, the rabbit population is the
independent variable and the wolf population is the dependent variable.

Use the STAT EDIT menu to enter the rabbit data into list L1 and the wolf
data into L2. Set DiagnosticOn, and then use the STAT CALC menu to select
LinReg(ax+b).

The equation of the line of best fit is y = 0.58x − 3.1 and the correlation
coefficient is 0.87.

b) Store the equation for the line of best fit as a function, Y1. Then, use the
STAT PLOT menu to set up the scatter plot. By displaying both Y1 and the
scatter plot, you can see how closely the data plots are distributed around
the line of best fit.

The correlation coefficient and the scatter plot show a strong positive linear
correlation between the variables. This correlation supports the researchers’
theory, but does not prove that changes in the rabbit population are the
cause of the changes in the wolf population.


Set up a table with the data for the rabbit and wolf populations. You can
calculate the correlation coefficient with the CORREL function. Use the Chart
feature to create a scatter plot.

In Corel® Quattro® Pro, you can find the equation of the line of best fit by
selecting Tools/Numeric Tools/Regression. Enter the cell ranges for the data,
and the program will display regression calculations including the constant (b),
the x-coefficient (or slope, a), and r 2.


In Microsoft® Excel, you can find the equation of the line of best fit by selecting
Chart/Add Trendline. Check that the default setting is Linear. Select the straight
line that appears on your chart, then click Format/Selected Trendline/Options.
Check the Display equation on chart box. You can also display r 2.

Project
Prep

When analysing
two-variable data
Solution 3 Using Fathom™ for your statistics
project, you may
Drag a new case table to the workspace, create attributes for Year, Rabbits, and wish to develop a
Wolves, and enter the data. Drag a new graph to the workspace, then drag the linear model,
Rabbits attribute to the x-axis and the Wolves attribute to the y-axis. From the particularly if a
Graph menu, select Least Squares Line. Fathom™ will display r 2 and the strong linear
equation for the line of best fit. To calculate the correlation coefficient directly, correlation is
select Inspect Collection, click the Measures tab, then create a new measure by evident.
selecting Functions/Statistical/Two Attributes/correlation and entering Rabbits
and Wolves as the attributes.


In Example 2, the sample size is small, so you should be cautious about
making generalizations from it. Small samples have a greater chance of not
being representative of the whole population. Also, outliers can seriously
affect the results of a regression on a small sample.

Example 3 The Effect of Outliers

To evaluate the performance of one of its instructors, a driving school
tabulates the number of hours of instruction and the driving-test scores
for the instructor’s students.
Instructional Hours 10 15 21 6 18 20 12
Student’s Score 78 85 96 75 84 45 82
a) What assumption is the management of the driving school making?
Is this assumption reasonable?
b) Analyse these data to determine whether they suggest that the instructor
is an effective teacher.
c) Comment on any data that seem unusual.
d) Determine the effect of any outliers on your analysis.

Solution
a) The management of the driving school is assuming that the correlation
between instructional hours and test scores is an indication of the
instructor’s teaching skills. Such a relationship could be difficult to prove
definitively. However, the assumption would be reasonable if the driving
school has found that some instructors have consistently strong
correlations between the time spent with their students and the students’
test scores while other instructors have consistently weaker correlations.

b) The number of hours of instruction is the independent variable. You
could analyse the data using any of the methods in the two previous
examples. For simplicity, a spreadsheet solution is shown here.

Except for an obvious outlier at (20, 45), the scatter plot below indicates
a strong positive linear correlation. At first glance, it appears that the
number of instructional hours is positively correlated to the students’ test
scores. However, the linear regression analysis yields a line of best fit with
the equation y = −0.13x + 80 and a correlation coefficient of −0.05.

These results indicate that there is virtually a zero linear correlation, and
the line of best fit even has a negative slope! The outlier has a dramatic
impact on the regression results because it is distant from the other data
points and the sample size is quite small. Although the scatter plot looked


favourable, the regression analysis suggests that the instructor’s lessons had
no positive effect on the students’ test results.

c) The fact that the outlier is substantially below all the other data points
suggests that some special circumstance may have caused an abnormal result.
For instance, there might have been an illness or emotional upset that
affected this one student’s performance on the driving test. In that case, it
would be reasonable to exclude this data point when evaluating the driving
instructor.

d) Remove the outlier from your data table and repeat your analysis.

Notice that the line of best ﬁt is now much closer to the data points and has
a positive slope. The correlation coefﬁcient, r, is 0.93, indicating a strong
positive linear correlation between the number of instructional hours and
the driver’s test scores. This result suggests that the instructor may be an
effective teacher after all. It is quite possible that the original analysis was
not a fair evaluation. However, to do a proper evaluation, you would need
a larger set of data, more information about the outlier, or, ideally, both.


As Example 3 demonstrates, outliers can skew a Project
regression analysis, but they could also simply Prep
indicate that the data really do have large variations.
A comprehensive analysis of a set of data should look If your statistics project involves a
for outliers, examine their possible causes and their linear relationship that contains
effect on the analysis, and discuss whether they outliers, you will need to consider
should be excluded from the calculations. As you carefully their impact on your results,
observed in Chapter 2, outliers have less effect on and how you will deal with them.
larger samples.


Visit the above web site and follow the links to
learn more about linear regression. Describe
an application of linear regression that
interests you.

Key Concepts

• Linear regression provides a means for analytically determining a line of best
fit. In the least-squares method, the line of best fit is the line which minimizes
the sum of the squares of the residuals while having the sum of the residuals
equal zero.

• You can use the equation of the line of best fit to predict the value of one of
the two variables given the value of the other variable.

• The correlation coefficient is a measure of how well a regression line fits a set
of data.

• Outliers and small sample sizes can reduce the accuracy of a linear model.


1. What does the correlation coefficient reveal about the line of best fit
generated by a linear regression?

2. Will the correlation coefficient always be negative when the slope of the
line of best fit is negative? Explain your reasoning.

3. Describe the problem that outliers present for a regression analysis and
outline what you could do to resolve this problem.


Practise a) Create a scatter plot and classify the
linear correlation.
A b) Apply the method of least squares to
1. Identify any outliers in the following sets of generate the equation of the line of
data and explain your choices. best fit.
a) X 25 34 43 55 92 105 16 c) Predict the mass of a trainee whose
Y 30 41 52 66 18 120 21 height is 165 cm.
X 5 7 6 6 4 8 d) Predict the height of a 79-kg trainee.
b)
Y 304 99 198 205 106 9 e) Explain any discrepancy between your
answer to part d) and the actual height of
2. a) Perform a linear regression analysis to the 79-kg trainee in the sample group.
generate the line of best fit for each set
of data in question 1. 6. A random survey of a small group of high-
school students collected information on the
b) Repeat the linear regressions in part a),
students’ ages and the number of books they
leaving out any outliers.
had read in the past year.
c) Compare the lines of best fit in parts a)
and b). Age (years) Books Read
16 5
Apply, Solve, Communicate 15 3
18 8
B 17 6
3. Use the formula for the method of least 16 4
squares to verify the slope and intercept
15 4
values you found for the data in the
14 5
investigation on page 171. Account for
17 15
any discrepancies.
a) Create a scatter plot for this data.
4. Use software or a graphing calculator to
Classify the linear correlation.
verify the regression results in Example 1.
b) Determine the correlation coefficient
5. Application The following table lists the and the equation of the line of best fit.
heights and masses for a group of fire- c) Identify the outlier.
department trainees.
d) Repeat part b) with the outlier excluded.
Height (cm) Mass (kg)
e) Does removing the outlier improve the
177 91
linear model? Explain.
185 88
f) Suggest other ways to improve the
173 82
model.
169 79
g) Do your results suggest that the number
188 87
of books a student reads depends on the
182 85
student’s age? Explain.
175 79


7. Application Market research has provided b) Determine the correlation coefficient
the following data on the monthly sales of and the equation of the line of best fit.
a licensed T-shirt for a popular rock band. c) Repeat the linear regression analysis with
Price ($) Monthly Sales any outliers removed.
10 2500 d) Repeat parts a) and b) using the data for
12 2200 the productions in 2002.
15 1600 e) Repeat parts a) and b) using the
18 1200 combined data for productions in both
20 800 2001 and 2002. Do there still appear to
24 250 be any outliers?
a) Create a scatter plot for these data. f) Which of the four linear equations
do you think is the best model for the
b) Use linear regression to model these
relationship between production costs
data.
and revenue? Explain your choice.
c) Predict the sales if the shirts are priced
g) Explain why the executive producer
at $19.
might choose to use the equation from
d) The vendor has 1500 shirts in stock and part d) to predict the income from
the band is going to finish its concert MDM’s 2003 productions.
tour in a month. What is the maximum
price the vendor can charge and still 9. At Gina’s university, there are 250 business
pt
avoid having shirts left over when the ha e students who expect to graduate in 2006.
C

r

band stops touring? a) Model the relationship between the total
m
P

r
oble
number of graduates and the number
8. Communication MDM Entertainment has
hired by performing a linear regression
produced a series of TV specials on the lives
on the data in the table on page 157.
of great mathematicians. The executive
Determine the equation of the line of
producer wants to know if there is a linear
best fit and the correlation coefficient.
correlation between production costs and
revenue from the sales of broadcast rights. b) Use this linear model to predict how
The costs and gross sales revenue for many graduates will be hired in 2006.
productions in 2001 and 2002 were as c) Identify any outliers in this scatter plot
follows (amounts in millions of dollars). and suggest possible reasons for an
2001 2002 outlier. Would any of these reasons
Cost ($M) Sales ($M) Cost ($M) Sales ($M) justify excluding the outlier from the
5.5 15.4 2.7 5.2 regression calculations?
4.1 12.1 1.9 1.0 d) Repeat part a) with the outlier removed.
1.8 6.9 3.4 3.4 e) Compare the results in parts a) and d).
3.2 9.4 2.1 1.9 What assumptions do you have to make?
4.2 1.5 1.4 1.5

a) Create a scatter plot using the data for
the productions in 2001. Do there
appear to be any outliers? Explain.


10. Communication Refer to Example 2, which ii) add a moveable line to the scatter plot
describes population data for wolves and and construct the geometric square
rabbits in a wildlife reserve. An alternate for the deviation of each data point
theory has it that the rabbit population from the moveable line
depends on the wolf population since the iii) generate a dynamic sum of the areas
wolves prey on the rabbits. of these squares
a) Create a scatter plot of rabbit population iv) manoeuvre the moveable line to the
versus wolf population and classify the position that minimizes the sum of
linear correlation. How are your data the areas of the squares.
points related to those in Example 2?
v) record the equation of this line
b) Determine the equation of the line of
and the equation of the line of best fit.
best fit for this set of data.
Graph this line on your scatter plot.
c) Compare the equations you found in
c) Is the equation of the line of best fit the
parts a) and b). Explain any differences
inverse of that found in Example 2?
or similarities.
Explain.
d) Plot both populations as a time series. 12. Application Use E-STAT or other sources
Can you recognize a pattern or to obtain the annual consumer price index
relationship between the two series? figures from 1914 to 2000.
Explain. a) Download this information into a
e) Does the time series suggest which spreadsheet or statistical software, or
population is the dependent variable? enter it into a graphing calculator. (If you
Explain. use a graphing calculator, enter the data
from every third year.) Find the line of
11. The following table lists the mathematics best fit and comment on whether a
of data management marks and grade 12 straight line appears to be a good model
averages for a small group of students. for the data.
Mathematics of Data Grade 12 b) What does the slope of the line of best
Management Mark Average fit tell you about the rate of inflation?
74 77
c) Find the slope of the line of best fit for
81 87
the data for just the last 20 years, and
66 68
then repeat the calculation using only
53 67 the data for the last 5 years.
92 85
d) What conclusions can you make by
45 55
comparing the three slopes? Explain
80 76 your reasoning.
a) Using FathomTM or The Geometer’s
Sketchpad,
i) create a scatter plot for these data


ACHIEVEMENT CHECK
C
14. Suppose that a set of data has a perfect linear
Understanding Problem Solving correlation except for two outliers, one above
13. The Worldwatch Institute has collected the line of best fit and the other an equal
the following data on concentrations of distance below it. The residuals of these two
carbon dioxide (CO2) in the atmosphere. outliers are equal in magnitude, but one is
positive and the other negative. Would you
Year CO2 Level (ppm)
agree that a perfect linear correlation exists
1975 331 because the effects of the two residuals
1976 332 cancel out? Support your opinion with
1977 333.7 mathematical reasoning and a diagram.
1978 335.3
1979 336.7 15. Inquiry/Problem Solving Recall the formulas
1980 338.5 for the line of best fit using the method of
1981 339.8 least squares that minimizes the squares of
1982 341 vertical deviations.
1983 342.6 a) Modify these formulas to produce a line
1984 344.3 of best fit that minimizes the squares of
1985 345.7 horizontal deviations.
1986 347
b) Do you think your modified formulas
1987 348.8
will produce the same equation as the
1988 351.4
regular least-squares formula?
1989 352.7
1990 354 c) Use your modified formula to calculate
1991 355.5 a line of best fit for one of the examples
1992 356.2 in this section. Does your line have the
1993 357 same equation as the line of best fit in
1994 358.8 the example? Is your equation the inverse
1995 360.7 of the equation in the example? Explain
why or why not.
a) Use technology to produce a scatter
plot of these data and describe any 16. a) Calculate the residuals for all of the data
correlation that exists. points in Example 3 on page 177.
Make a plot of these residuals versus the
b) Use a linear regression to find the line
independent variable, X, and comment
of best fit for the data. Discuss the
on any pattern you see.
reliability of this model.
b) Explain how you could use such residual
c) Use the regression equation to predict
plots to detect outliers.
the level of atmospheric CO2 that you
would expect today.
d) Research current CO2 levels. Are the
results close to the predicted level?
What factors could have affected the
trend?


3.3 Non-Linear Regression

Many relationships between two variables follow patterns that are not linear. For
example, square-law, exponential, and logarithmic relationships often appear in
the natural sciences. Non-linear regression is an analytical technique for finding
a curve of best fit for data from such relationships. The equation for this curve
can then be used to model the relationship between the two variables.

As you might expect, the calculations for curves are more complicated than those
for straight lines. Graphing calculators have built-in regression functions for a
variety of curves, as do some spreadsheets and statistical programs. Once you
enter the data and specify the type of curve, these technologies can automatically
find the best-fit curve of that type. They can also calculate the coefficient of
determination, r 2, which is a useful measure of how closely a curve fits the data.

I N V E S T I G AT E & I N Q U I R E : Bacterial Growth

A laboratory technician monitors the growth of a bacterial
culture by scanning it every hour and estimating the number
of bacteria. The initial population is unknown.
Time (h) 0 1 2 3 4 5 6 7
Population ? 10 21 43 82 168 320 475

1. a) Create a scatter plot and classify the linear correlation.
b) Determine the correlation coefficient and the line of
best fit.
c) Add the line of best fit to your scatter plot. Do you think
this line is a satisfactory model? Explain why or why not.
2. a) Use software or a graphing calculator to find a curve
of best fit with a
i) quadratic regression of the form y = ax2 + bx + c
ii) cubic regression of the form y = ax3 + bx2 + cx + d
b) Graph these curves onto a scatter plot of the data.
c) Record the equation and the coefficient of
determination, r 2, for the curves.
d) Use the equations to estimate the initial population See Appendix B for details
of the bacterial culture. Do these estimates seem on using technology for
reasonable? Why or why not? non-linear regressions.


3. a) Perform an exponential regression on the data. Graph the curve of best
fit and record its equation and coefficient of determination.
b) Use this model to estimate the initial population.
c) Do you think the exponential equation is a better model for the growth
of the bacterial culture than the quadratic or cubic equations? Explain
your reasoning.

Recall that Pearson’s correlation coefficient, r, is a measure of the linearity
of the data, so it can indicate only how closely a straight line fits the data.
However, the coefficient of determination, r 2, is defined such that it applies
to any type of regression curve.

variation in y explained by variation in x
r 2 = ᎏᎏᎏᎏᎏ
total variation in y
– )2
∑( yest − y
= ᎏᎏ
∑( y − – )2
y

where − is the mean y value, yest is the value estimated by the best-fit curve for
y
a given value of x, and y is the actual observed value for a given value of x.
Unexplained (x, y)
Total deviation
deviation
Explained (x,y est)
The total variation is the
deviation sum of the squares of the
Y

y deviations for all of the
individual data points.
Curve of best fit

X

The coefficient of determination can have values from 0 to 1. If the curve is a
perfect fit, then yest and y will be identical for each value of x. In this case, the
variation in x accounts for all of the variation in y, so r 2 = 1. Conversely, if the
curve is a poor fit, the total of ( yest − − )2 will be much smaller than the total of
y
( y − − )2, since the variation in x will account for only a small part of the total
y
variation in y. Therefore, r 2 will be close to 0. For any given type of regression,
the curve of best fit will be the one that has the highest value for r 2.

For graphing calculators and Microsoft® Excel, the procedures for non-linear
regression are almost identical to those for linear regression. At present, Corel®
Quattro® Pro and Fathom™ do not have built-in functions for non-linear
regression.

3.3 Non-Linear Regression • MHR 185

Exponential Regression
Exponential regressions produce equations with the form y = ab x or y = ae kx,
where e = 2.718 28…, an irrational number commonly used as the base for
exponents and logarithms. These two forms are equivalent, and it is
straightforward to convert from one to the other.

Example 1 Exponential Regression

Generate an exponential regression for the bacterial culture in the investigation
on page 184. Graph the curve of best fit and determine its equation and the
coefficient of determination.


Use the ClrList command from the STAT EDIT menu to clear lists L1 and L2,
and then enter the data. Set DiagnosticOn so that regression calculations will
display the coefficient of determination. From the STAT CALC menu, select the
non-linear regression function ExpReg. If you do not enter any list names, the
calculator will use L1 and L2 by default.

The equation for the curve of best fit is y = 5.70(1.93) x, and the coefficient of
determination is r 2 = 0.995. Store the equation as Y1. Use STAT PLOT to display
a scatter plot of the data along with Y1. From the ZOOM menu, select
9:ZoomStat to adjust the window settings automatically.


Enter the data into two columns. Next, highlight these columns and use the
Chart feature to create an x-y scatter plot.


Select Chart/Add Trendline and then choose Expontenial regression. Then, select
the curve that appears on your chart, and click Format/Selected Trendline/Options.
Check the option boxes to display the equation and r 2.

The equation of the best-fit curve is y = 5.7e0.66x and the coefficient of
determination is r 2 = 0.995. This equation appears different from the one found
⋅
with the graphing calculator. In fact, the two forms are equivalent, since e0.66 = 1.93.

Power and Polynomial Regression
In power regressions, the curve of best fit has an equation with the form y = ax b.

Example 2 Power Regression

For a physics project, a group of students videotape a ball dropped from the top
of a 4-m high ladder, which they have marked every 10 cm. During playback,
they stop the videotape every tenth of a second and compile the following table
for the distance the ball travelled.
Time (s) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Distance (m) 0.05 0.2 0.4 0.8 1.2 1.7 2.4 3.1 3.9 4.9

a) Does a linear model fit the data well?
b) Use a power regression to find a curve of best fit for the data. Does the
power-regression curve fit the data more closely than the linear model does?
c) Use the equation for the regression curve to predict
i) how long the ball would take to fall 10 m
ii) how far the ball would fall in 5 s



a) Although the linear correlation coefficient is 0.97, a scatter plot of the data
shows a definite curved pattern. Since b = −1.09, the linear model predicts
an initial position of about –1.1 m and clearly does not fit the first part of
the data well. Also, the pattern in the scatter plot suggests the linear model
could give inaccurate predictions for times beyond 1 s.

b) From the STAT CALC menu, select the non-linear regression function PwrReg
and then follow the same steps as in Example 1.

The equation for the curve of best fit is y = 4.83x2. The coefficient of
determination and a graph on the scatter plot show that the quadratic
curve is almost a perfect fit to the data.

c) Substitute the known values into the equation for the quadratic curve of
best fit:
i) 10 = 4.83x2 ii) y = 4.83(5)2
10
x2 = ᎏᎏ = 4.83(25)
4.83 = 121
Ί๶
10
x = ᎏᎏ
4.83
= 1.4
The quadratic model predicts that
i) the ball would take approximately 1.4 s to fall 10 m
ii) the ball would fall 121 m in 5 s


a) As in Solution 1, the scatter plot shows that a curve might be a
better model.


b) Use the Chart feature as in Example 1, but select Power when adding the
trend line.

The equation for the curve of best fit is y = 4.83x 2. The graph and the value
for r 2 show that the quadratic curve is almost a perfect fit to the data.

c) Use the equation for the curve of best fit to enter formulas for the two
values you want to predict, as shown in cells A13 and B14 in the screen
above.

Example 3 Polynomial Regression

Suppose that the laboratory technician takes further measurements of the
bacterial culture in Example 1.
Time (h) 8 9 10 11 12 13 14
Population 630 775 830 980 1105 1215 1410

a) Discuss the effectiveness of the exponential model from Example 1 for
the new data.
b) Find a new exponential curve of best fit.
c) Find a better curve of best fit. Comment on the effectiveness of the new
model.


Solution

a) If you add the new data to the scatter plot, you will see that the
exponential curve determined earlier, y = 5.7(1.9) x, is no longer a
good fit.

b) If you perform a new exponential regression on all 14 data points, you
obtain the equation y = 18(1.4) x with a coefficient of determination of
r 2 = 0.88. From the graph, you can see that this curve is not a
particularly good fit either.

Because of the wide range of non-linear regression options, you can
insist on a fairly high value of r 2 when searching for a curve of best fit
to model the data.

c) If you perform a quadratic regression, you get a much better fit with the
equation y = 4.0x2 + 55x − 122 and a coefficient of determination of
r 2 = 0.986.

This quadratic model will probably serve well for interpolating between
most of the data shown, but may not be accurate for times before 3 h
and after 14 h. At some point between 2 h and 3 h, the curve intersects
the x-axis, indicating a negative population prior to this time. Clearly
the quadratic model is not accurate in this range.

Similarly, if you zoom out, you will notice a problem beyond 14 h. The
rate of change of the quadratic curve continues to increase after 14 h,
but the trend of the data does not suggest such an increase. In fact,
from 7 h to 14 h the trend appears quite linear.

It is important to recognize the limitations of regression curves. One
interesting property of polynomial regressions is that for a set of n data
points, a polynomial function of degree n − 1 can be produced which
perfectly fits the data, that is, with r 2 = 1.

For example, you can determine the equation for a line (a first-degree
polynomial) with two points and the equation for a quadratic (a second- Project
degree polynomial) with three points. However, these polynomials are Prep
not always the best models for the data. Often, these curves can give
inaccurate predictions when extrapolated. Non-linear models may
be useful when you are
Sometimes, you can find that several different types of curves fit closely analysing two-variable
to a set of data. Extrapolating to an initial or final state may help data in your statistics
determine which model is the most suitable. Also, the mathematical project.
model should show a logical relationship between the variables.

Key Concepts

• Some relationships between two variables can be modelled using non-linear
regressions such as quadratic, cubic, power, polynomial, and exponential
curves.

• The coefficient of determination, r 2, is a measure of how well a regression
curve fits a set of data.

• Sometimes more than one type of regression curve can provide a good fit for
data. To be an effective model, however, the curve must be useful for
extrapolating beyond the data.


1. A data set for two variables has a linear correlation coefficient of 0.23. Does
this value preclude a strong correlation between the variables? Explain why
or why not.

2. A best-fit curve for a set of data has a coefficient of determination of r 2 = 0.76.
Describe some techniques you can use to improve the model.

Practise 2. For each set of data use software or a
graphing calculator to find the equation and
A coefficient of determination for a curve of
1. Match each of the following coefficients of best fit.
determination with one of the diagrams a) b) c)
below.
x y x y x y
a) 0 b) 0.5 c) 0.9 d) 1
−2.8 0.6 −2.7 1.6 1.1 2.5
i) ii)
−3.5 −5.8 −3.5 −3 3.5 11
−2 3 −2.2 3 2.8 8.6
−1 6 −0.5 −0.5 2.3 7
0.2 4 0 1.3 0 1
1 1 0.6 4.7 3.8 14
−1.5 5 −1.8 1.7 1.4 4.2
iii) iv) 1.4 −3.1 −3.8 −7 −4 0.2
0.7 3 −1.3 0.6 −1.3 0.6
−0.3 6.1 0.8 7 3 12
−3.3 −3.1 0.5 2.7 4.1 17
−4 −7 −1 1.5 2.2 5
2 −5.7 −3 −1.1 −2.7 0.4


Apply, Solve, Communicate Animal Mass (kg) BMR (kJ/day)
Frog 0.018 0.050
B Squirrel 0.90 1.0
3. The heights of a stand of pine trees were Cat 3.0 2.6
measured along with the area under the
Monkey 7.0 4.0
cone formed by their branches.
Baboon 30 14
Height (m) Area (m2)
Human 60 25
2.0 5.9
Dolphin 160 44
1.5 3.4
Camel 530 116
1.8 4.8
2.4 8.6 a) Create a scatter plot and explain why
2.2 7.3 Kleiber thought a power-regression
1.2 2.1 curve would fit the data.
1.8 4.9
3.1 14.4 b) Use a power regression to find the
equation of the curve of best fit. Can
a) Create a scatter plot of these data. you rewrite the equation so that it has
b) Determine the correlation coefficient exponents that are whole numbers?
and the equation of the line of best fit. Do so, if possible, or explain why not.
c) Use a power regression to calculate a c) Is this power equation a good
coefficient of determination and an mathematical model for the relationship
equation for a curve of best fit. between an animal’s mass and its basal
d) Which model do you think is more metabolic rate? Explain why or why not.
accurate? Explain why. d) Use the equation of the curve of best fit
e) Use the more accurate model to predict to predict the basal metabolic rate of
i) the area under a tree whose height is i) a 15-kg dog
2.7 m ii) a 2-tonne whale
ii) the height of a tree whose area is
5. Application As a sample of a radioactive
30 m2
element decays into more stable elements,
f) Suggest a reason why the height and the amount of radiation it gives off
circumference of a tree might be related in decreases. The level of radiation can be
the way that the model in part d) suggests. used to estimate how much of the original
element remains. Here are measurements
4. Application The biologist Max Kleiber
for a sample of radium-227.
(1893−1976) pioneered research on the
metabolisms of animals. In 1932, he Time (h) Radiation Level (%)
determined the relationship between an 0 100
animal’s mass and its energy requirements or 1 37
basal metabolic rate (BMR). Here are data 2 14
for eight animals. 3 5.0
4 1.8
5 0.7
6 0.3


a) Create a scatter plot for these data. b) Use the equation for this curve of best fit
b) Use an exponential regression to to estimate the power level at a distance of
find the equation for the curve of best i) 1.0 km from the transmitter
fit. ii) 4.0 km from the transmitter
c) Is this equation a good model for the iii) 50.0 km from the transmitter
radioactive decay of this element?
Explain why or why not. 8. Communication Logistic curves are often a
d) A half-life is the time it takes for half of good model for population growth. These
the sample to decay. Use the regression curves have equations with the form
c
equation to estimate the half-life of y = ᎏ , where a, b, and c are constants.
1 + ae−bx
radium-227.
Consider the following data for the bacterial
6. a) Create a time-series graph for the culture in Example 1:
pte
ha mean starting salary of the graduates Time (h) 0 1 2 3 4 5
C

r

who find jobs. Describe the pattern Population ? 10 21 43 82 168
m
P

r
oble
that you see.
Time (h) 6 7 8 9 10 11
b) Use non-linear regression to construct a
curve of best fit for the data. Record the Population 320 475 630 775 830 980
equation of the curve and the coefficient Time (h) 12 13 14 15 16 17
of determination.
Population 1105 1215 1410 1490 1550 1575
c) Comment on whether this equation is
a good model for the graduates’ starting Time (h) 18 19 20
salaries. Population 1590 1600 1600

a) Use software or a graphing calculator
7. An engineer testing the transmitter for a
new radio station measures the radiated to find the equation and coefficient of
power at various distances from the determination for the logistic curve
transmitter. The engineer’s readings are that best fits the data for the bacteria
in microwatts per square metre. population from 1 to 20 h.
b) Graph this curve on a scatter plot of
Distance (km) Power Level (µW/m2)
the data.
2.0 510
5.0 78 c) How well does this curve appear to

8.0 32
fit the entire data set? Describe the
shape of the curve.
10.0 19
12.0 14 d) Write a brief paragraph to explain

15.0 9
why you think a bacterial population
may exhibit this type of growth
20.0 5
pattern.
a) Find an equation for a curve of best fit
for these data that has a coefficient of
determination of at least 0.98.


9. Inquiry/Problem Solving The following 11. Inquiry/Problem Solving Use a software
table shows the estimated population of a program, such as Microsoft® Excel, to
crop-destroying insect. analyse these two sets of data:
Year Population (billions) Data Set A Data Set B
1995 100 x y x y
1996 130 2 5 2 6
1997 170 4 7 4 5
1998 220 6 2 7 –4
1999 285 8 5 9 1
2000 375 12 2
2001 490 a) For each set of data,
a) Determine an exponential curve of best i) determine the degree of polynomial
fit for the population data. regression that will generate a
b) Suppose that 100 million of an arachnid perfectly fit regression curve
that preys on the insect are imported ii) perform the polynomial regression
from overseas in 1995. Assuming the and record the value of r 2 and the
arachnid population doubles every year, equation of the regression curve
estimate when it would equal 10% of the b) Assess the effectiveness of the best-fit
insect population. polynomial curve as a model for the
c) What further information would trend of the set of data.
you need in order to estimate the c) For data set B,
population of the crop-destroying
i) explain why the best-fit polynomial
insect once the arachnids have been
curve is an unsatisfactory model
introduced?
ii) generate a better model and record
d) Write an expression for the size of this
the value of r 2 and the equation of
population.
your new best-fit curve
C iii) explain why this curve is a better
10. Use technology to calculate the coefficient model than the polynomial curve
of determination for two of the linear found in part a)
regression examples in section 3.2. Is there
any relationship between these coefficients
of determination and the linear correlation
coefficients for these examples?


3.4 Cause and Effect

Usually, the main reason for a correlational study is
to ﬁnd evidence of a cause-and-effect relationship.
A health researcher may wish to prove that even
mild exercise reduces the risk of heart disease. A
chemical company developing an oil additive
would like to demonstrate that it improves engine
performance. A school board may want to know
whether calculators help students learn
mathematics. In each of these cases, establishing
a strong correlation between the variables is just
the ﬁrst step in determining whether one affects
the other.

I N V E S T I G AT E & I N Q U I R E : C o r r e l a t i o n Ve r s u s C a u s e a n d E f f e c t

1. List the type of correlation that you would expect to observe between the
following pairs of variables. Also list whether you think the correlation is
due to a cause-and-effect relationship or some other factor.
a) hours spent practising at a golf driving range, golf drive distance
b) hours spent practising at a golf driving range, golf score
c) size of corn harvest, size of apple harvest
d) score on a geometry test, score on an algebra test
e) income, number of CDs purchased
2. Compare your list with those of your classmates and discuss any differences.
Would you change your list because of factors suggested by your classmates?
3. Suggest how you could verify whether there is a cause-and-effect relationship
between each pair of variables.

A strong correlation does not prove that the changes in one variable cause
changes in the other. There are various types and degrees of causal relationships
between variables.

Cause-and-Effect Relationship: A change in X produces a change in Y. Such
relationships are sometimes clearly evident, especially in physical processes.
For example, increasing the height from which you drop an object increases its
impact velocity. Similarly, increasing the speed of a production line increases
the number of items produced each day (and, perhaps, the rate of defects).

3.4 Cause and Effect • MHR 195

Common-Cause Factor: An external variable causes two variables to change
in the same way. For example, suppose that a town finds that its revenue from
parking fees at the public beach each summer correlates with the local tomato
harvest. It is extremely unlikely that cars parked at the beach have any effect on
the tomato crop. Instead good weather is a common-cause factor that increases
both the tomato crop and the number of people who park at the beach.

Reverse Cause-and-Effect Relationship: The dependent and independent
variables are reversed in the process of establishing causality. For example,
suppose that a researcher observes a positive linear correlation between the
amount of coffee consumed by a group of medical students and their levels of
anxiety. The researcher theorizes that drinking coffee causes nervousness, but
instead finds that nervous people are more likely to drink coffee.

Accidental Relationship: A correlation exists without any causal relationship
between variables. For example, the number of females enrolled in
undergraduate engineering programs and the number of “reality” shows on
television both increased for several years. These two variables have a positive
linear correlation, but it is likely entirely coincidental.

Presumed Relationship: A correlation does not seem to be accidental even
though no cause-and-effect relationship or common-cause factor is apparent.
For example, suppose you found a correlation between people’s level of fitness
and the number of adventure movies they watched. It seems logical that a
physically fit person might prefer adventure movies, but it would be difficult
to find a common cause or to prove that the one variable affects the other.

Example 1 Causal Relationships

Classify the relationships in the following situations.
a) The rate of a chemical reaction increases with temperature.
b) Leadership ability has a positive correlation with academic achievement.
c) The prices of butter and motorcycles have a strong positive correlation
over many years.
d) Sales of cellular telephones had a strong negative correlation with ozone
levels in the atmosphere over the last decade.
e) Traffic congestion has a strong correlation with the number of urban
expressways.


Solution

a) Cause-and-effect relationship: Higher temperatures cause faster reaction rates.

b) Presumed relationship: A positive correlation between leadership ability and
academic achievement seems logical, yet there is no apparent common-cause
factor or cause-and-effect relationship.

c) Common-cause factor: Inflation has caused parallel increases in the prices
of butter and motorcycles over the years.

d) Accidental relationship: The correlation between sales of cellular telephones
and ozone levels is largely coincidental. However, it is possible that the
chemicals used to manufacture cellular telephones cause a small portion
of the depletion of the ozone layer.

e) Cause-and-effect relationship and reverse cause-and-effect relationship:
Originally expressways were built to relieve traffic congestion, so traffic
congestion did lead to the construction of expressways in major cites
throughout North America. However, numerous studies over the last
20 years have shown that urban expressways cause traffic congestion by
encouraging more people to use cars.

As Example 1 demonstrates, several types of causal relationships can be involved
in the same situation. Determining the nature of causal relationships can be
further complicated by the presence of extraneous variables that affect either
the dependent or the independent variable. Here, extraneous means external
rather than irrelevant.
For example, you might expect to see a strong positive correlation between term
marks and final examination results for students in your class since both these
variables are affected by each student’s aptitude and study habits. However,
there are extraneous factors that could affect the examination results, including
the time each student had for studying before the examination, the individual
examination schedules, and varying abilities to work well under pressure.

In order to reduce the effect of extraneous variables, researchers often compare
an experimental group to a control group. These two groups should be as
similar as possible, so that extraneous variables will have about the same effect
on both groups. The researchers vary the independent variable for the
experimental group but not for the control group. Any difference in the
dependent variables for the two groups can then be attributed to the changes
in the independent variable.


Example 2 Using a Control Group

A medical researcher wants to test a new drug believed to help smokers
overcome the addictive effects of nicotine. Fifty people who want to quit
smoking volunteer for the study. The researcher carefully divides the volunteers
into two groups, each with an equal number of moderate and heavy smokers.
One group is given nicotine patches with the new drug, while the second group
uses ordinary nicotine patches. Fourteen people in the first group quit smoking
completely, as do nine people in the second group.
a) Identify the experimental group, the control group, the independent
variable, and the dependent variable.
b) Can the researcher conclude that the new drug is effective?
c) What further study should the researcher do?

Solution

a) The experimental group consists of the volunteers being given nicotine
patches with the new drug, while the control group consists of the
volunteers being given the ordinary patches. The independent variable is
the presence of the new drug, and the dependent variable is the number
of volunteers who quit smoking.

b) The results of the study are promising, but the researcher has not proven
that the new drug is effective. The sample size is relatively small, which is
prudent for an early trial of a new drug that could have unknown side-
effects. However, the sample is small enough that the results could be
affected by random statistical fluctuations or extraneous variables, such as
the volunteers’ work environments, previous attempts to quit, and the
influence of their families and friends.

c) Assuming that the new drug does not have any serious side-effects, the
researcher should conduct further studies with larger groups and try to
select the experimental and control groups to minimize the effect of all
extraneous variables. The researcher might also conduct a study with
several experimental groups that receive different dosages of the new drug.

When designing a study or interpreting a correlation, you often need
background knowledge and insight to recognize the causal relationships
present. Here are some techniques that can help determine whether a
correlation is the result of a cause-and-effect relationship.


• Use sampling methods that hold the extraneous variables constant.
Project
Prep
• Conduct similar investigations with different samples and check for
consistency in the results. In your statistics
• Remove, or account for, possible common-cause factors. project, you may
wish to consider
The later chapters in this book introduce probability theory and some cause-and-effect
statistical methods for a more quantitative approach to determining cause- relationships and
and-effect relationships. extraneous variables
that could affect
your study.

Key Concepts

• Correlation does not necessarily imply a cause-and-effect relationship.
Correlations can also result from common-cause factors, reverse cause-and-
effect relationships, accidental relationships, and presumed relationships.

• Extraneous variables can invalidate conclusions based on correlational
evidence.

• Comparison with a control group can help remove the effect of extraneous
variables in a study.


1. Why does a strong linear correlation not imply cause and effect?

2. What is the key characteristic of a reverse cause-and-effect relationship?

3. Explain the difference between a common-cause factor and an extraneous
variable.

4. Why are control groups used in statistical studies?

Practise b) score on physics examination, score on
calculus examination
A c) increase in pay, job performance
1. Identify the most likely type of causal
d) population of rabbits, consumer price
relationship between each of the following
index
pairs of variables. Assume that a strong
positive correlation has been observed with e) number of scholarships received, number
the ﬁrst variable as the independent variable. of job offers upon graduation
a) alcohol consumption, incidence of f) coffee consumption, insomnia
automobile accidents e) funding for athletic programs, number
of medals won at Olympic games


2. For each of the following common-cause 6. Application A random survey of students
relationships, identify the common-cause at Statsville High School found that their
factor. Assume a positive correlation interest in computer games is positively
between each pair of variables. correlated with their marks in mathematics.
a) number of push-ups performed in one a) How would you classify this causal
minute, number of sit-ups performed in relationship?
one minute b) Suppose that a follow-up study found
b) number of speeding tickets, number of that students who had increased the time
accidents they spent playing computer games
c) amount of money invested, amount of tended to improve their mathematics
money spent marks. Assuming that this study held all
extraneous variables constant, would you
Apply, Solve, Communicate change your assessment of the nature of
the causal relationship? Explain why or
3. A civil engineer examining traffic flow why not.
problems in a large city observes that the
number of traffic accidents is positively 7. a) The net assets of Custom Industrial
correlated with traffic density and concludes Renovations Inc., an industrial
that traffic density is likely to be a major construction contractor, has a strong
cause of accidents. What alternative negative linear correlation with those of
conclusion should the engineer consider? MuchMega-Fun, a toy distributor. How
would you classify the causal relationship
B between these two variables?
4. Communication An elementary school is
b) Suppose that the two companies are both
testing a new method for teaching grammar. subsidiaries of Diversified Holdings Ltd.,
Two similar classes are taught the same which often shifts investment capital
material, one with the established method between them. Explain how this additional
and the other with the new method. When information could change your
both classes take the same test, the class interpretation of the correlation in part a).
taught with the established method has
somewhat higher marks. 8. Communication Aunt Gisele simply cannot
a) What extraneous variables could sleep unless she has her evening herbal tea.
influence the results of this study? However, the package for the tea does not
list any ingredients known to induce sleep.
b) Explain whether the study gives the school
Outline how you would conduct a study to
enough evidence to reject the new method.
determine whether the tea really does help
c) What further studies would you
people sleep.
recommend for comparing the two
teaching methods? 9. Find out what a double-blind study is and
briefly explain the advantages of using this
5. Communication An investor observes a
technique in studies with a control group.
positive correlation between the stock price
of two competing computer companies. 10. a) The data on page 157 show a positive
pte
Explain what type of causal relationship is ha correlation between the size of the
C

r

likely to account for this correlation. graduating class and the number of
m
P

r
oble


graduates hired. Does this correlation 12. Search the E-STAT, CANSIM II, or other
mean that increasing the number of databases for a set of data on two variables
graduates causes a higher demand for with a positive linear correlation that you
them? Explain your answer. believe to be accidental. Explain your
b) A recession during the first half of the findings and reasoning.
1990s reduced the demand for business
C
graduates. Review the data on page 157
13. Use a library, the Internet, or other
and describe any trends that may be
caused by this recession. resources to find information on the
Hawthorne effect and the placebo effect.
Briefly explain what these effects are, how
ACHIEVEMENT CHECK they can affect a study, and how researchers
can avoid having their results skewed by
Communication Application these effects.

11. The table below lists numbers of divorces 14. Inquiry/Problem Solving In a behavioural
and personal bankruptcies in Canada for study of responses to violence, an
the years 1976 through 1985. experimental group was shown violent
Year Divorces Bankruptcies images, while a control group was shown
1976 54 207 10 049 neutral images. From the initial results, the
1977 55 370 12 772 researchers suspect that the gender of the
1978 57 155 15 938 people in the groups may be an extraneous
1979 59 474 17 876
variable. Suggest how the study could be
redesigned to
1980 62 019 21 025
1981 67 671 23 036 a) remove the extraneous variable
1982 70 436 30 643 b) determine whether gender is part of the
1983 68 567 26 822 cause-and-effect relationship
1984 65 172 22 022
15. Look for material in the media or on the
1985 61 976 19 752 Internet that incorrectly uses correlational
a) Create a scatter plot and classify the evidence to claim that a cause-and-effect
linear correlation between the number relationship exists between the two variables.
of divorces and the number of Briefly describe
bankruptcies. a) the nature of the correlational study
b) Perform a regression analysis. Record b) the cause and effect claimed or inferred
the equation of the line of best fit and c) the reasons why cause and effect was not
the correlation coefficient. properly proven, including any
c) Identify an external variable that could extraneous variables that were not
be a common-cause factor. accounted for
d) Describe what further investigation you d) how the study could be improved
could do to analyse the possible
relationship between divorces and
bankruptcies.


3.5 Critical Analysis

Newspapers and radio and television news programs often run stories involving
statistics. Indeed, the news media often commission election polls or surveys on
major issues. Although the networks and major newspapers are reasonably careful
about how they present statistics, their reporters and editors often face tight
deadlines and lack the time and mathematical knowledge to thoroughly critique
statistical material. You should be particularly careful about accepting statistical
evidence from sources that could be biased. Lobby groups and advertisers like to
use statistics because they appear scientific and objective. Unfortunately, statistics
from such sources
are sometimes flawed
by unintentional or,
occasionally, entirely
deliberate bias.
To judge the
conclusions of a
study properly, you
need information
about its sampling
and analytical
methods.

I N V E S T I G AT E & I N Q U I R E : Statistics in the Media

1. Find as many instances as you can of statistical claims made in the media or
on the Internet, including news stories, features, and advertisements. Collect
newspaper and magazine clippings, point-form notes of radio and television
stories, and printouts of web pages.
2. Compare the items you have collected with those found by your classmates.
What proportion of the items provide enough information to show that they
used valid statistical methods?
3. Select several of the items. For each one, discuss
a) the motivation for the statistical study
b) whether the statistical evidence
justifies the claim being made
The examples in this section illustrate how you learn more about how statistics can be misused.
can apply analytical tools to assess the results of Describe two examples of the misuse of
statistical studies. statistics.


Example 1 Sample Size and Technique Test Score Productivity
98 78
A manager wants to know if a new aptitude test accurately predicts
57 81
employee productivity. The manager has all 30 current employees 82 83
write the test and then compares their scores to their 76 44
productivities as measured in the most recent performance reviews. 65 62
72 89
The data is ordered alphabetically by employee surname. In order 91 85
to simplify the calculations, the manager selects a systematic 87 71
sample using every seventh employee. Based on this sample, the 81 76
39 71
manager concludes that the company should hire only applicants 50 66
who do well on the aptitude test. Determine whether the 75 90
manager’s analysis is valid. 71 48
89 80
82 83
Solution 95 72
56 72
A linear regression of the systematic sample produces a line of best 71 90
fit with the equation y = 0.55x + 33 and a correlation coefficient of 68 74
r = 0.98, showing a strong linear correlation between productivity 77 51
59 65
and scores on the aptitude test. Thus, these calculations seem to 83 47
support the manager’s conclusion. However, the manager has made 75 91
the questionable assumption that a systematic sample will be 66 77
48 63
representative of the population. The sample is so small that 61 58
statistical fluctuations could seriously affect the results. 78 55
70 73
y 68 75
84 64 69
Productivity

80
76
72
68
64

0 55 60 65 70 75 80 85 90 95 x
Test Score

Examine the raw data. A scatter plot with all 30 data points does not show
any clear correlation at all. A linear regression yields a line of best fit with the
equation y = 0.15x + 60 and a correlation coefficient of only 0.15.
y
90
Productivity

80
70
60
50
40

0 30 40 50 60 70 80 90 100 x
Test Score

3.5 Critical Analysis • MHR 203

Thus, the new aptitude test will probably be useless for predicting employee
productivity. Clearly, the sample was far from representative. The manager’s
choice of an inappropriate sampling technique has resulted in a sample size too
small to make any valid conclusions.

In Example 1, the manager should have done an analysis using all of the data
available. Even then the data set is still somewhat small to use as a basis for a
major decision such as changing the company’s hiring procedures. Remember
that small samples are also particularly vulnerable to the effects of outliers.

Example 2 Extraneous Variables and Sample Bias

An advertising blitz by SuperFast Computer Training Inc. features profiles of
some of its young graduates. The number of months of training that these
graduates took, their job titles, and their incomes appear prominently in the
advertisements.
Months of Income
Graduate
Training ($000)
Sarah, software developer 9 85
Zack, programmer 6 63
Eli, systems analyst 8 72
Yvette, computer technician 5 52
Kulwinder, web-site designer 6 66
Lynn, network administrator 4 60

a) Analyse the company’s data to determine the strength of the linear
correlation between the amount of training the graduates took and their
incomes. Classify the linear correlation and find the equation of the linear
model for the data.
b) Use this model to predict the income of a student who graduates from
the company’s two-year diploma program after 20 months of training.
Does this prediction seem reasonable?
c) Does the linear correlation show that SuperFast’s training accounts for
the graduates’ high incomes? Identify possible extraneous variables.
d) Discuss any problems with the sampling technique and the data.

Solution
a) The scatter plot for income versus months of training shows a definite
positive linear correlation. The regression line is y = 5.44x + 31.9, and
the correlation coefficient is 0.90. There appears to be a strong positive
correlation between the amount of training and income.


b) As shown in cell C9 in the screen above, substituting 20 months into the linear
regression equation predicts an income of approximately
y = 5.44(20) + 31.9
= 141
Therefore, the linear model predicts that a graduate who has taken 20 months
of training will make about $141 000 a year. This amount is extremely high
for a person with a two-year diploma and little or no job experience. The
prediction suggests that the linear model may not be accurate, especially when
applied to the company’s longer programs.
c) Although the correlation between SuperFast’s training and the graduates’
incomes appears to be quite strong, the correlation by itself does not prove
that the training causes the graduates’ high incomes. A number of extraneous
variables could contribute to the graduates’ success, including experience prior
to taking the training, aptitude for working with computers, access to a high-
end computer at home, family or social connections in the industry, and the
physical stamina to work very long hours.
d) The sample is small and could have intentional bias. There is no indication
that the individuals in the advertisements were randomly chosen from the
population of SuperFast’s students. Quite likely, the company carefully selected
the best success stories in order to give potential customers inﬂated
expectations of future earnings. Also, the company shows youthful graduates,
but does not actually state that the graduates earned their high incomes
immediately after graduation. It may well have taken the graduates years of
hard work to reach the income levels listed in the advertisements. Further, the
amounts given are incomes, not salaries. The income of a graduate working
for a small start-up company might include stock options that could turn out
to be worthless. In short, the advertisements do not give you enough
information to properly evaluate the data.


Example 2 had several fairly obvious extraneous variables. However, extraneous
variables are sometimes difﬁcult to recognize. Such hidden or lurking
variables can also invalidate conclusions drawn from statistical results.

Example 3 Detecting a Hidden Variable

An arts council is considering whether to fund the start-up of a local youth
orchestra. The council has a limited budget and knows that the number of
youth orchestras in the province has been increasing. The council needs to
know whether starting another youth orchestra will help the development of
young musicians. One measure of the success of such programs is the number
of youth-orchestra players who go on to professional orchestras. The council
has collected the following data.
Year Number of Youth Orchestras Number of Players Becoming Professionals
1991 10 16
1992 11 18
1993 12 20
1994 12 23
1995 14 26
1996 14 32
1997 16 13
1998 16 16
1999 18 20
2000 20 26
a) Does a linear regression allow you to determine whether the council
should fund a new youth orchestra? Can you draw any conclusions from
other analysis?
b) Suppose you discover that one of the country’s professional orchestras
went bankrupt in 1997. How does this information affect your analysis?

Solution

a) A scatter plot of the number of youth-orchestra members who go on to
play professionally versus the number of youth orchestras shows that there
may be a weak positive linear correlation. The correlation coefﬁcient is
0.16, indicating that the linear correlation is very weak. Therefore, you
might conclude that starting another youth orchestra will not help the
development of young musicians. However, notice that the data points seem
to form two clusters in the scatter plot, one on the left side and the other
on the right. This unusual pattern suggests the presence of a hidden
variable, which could affect your analysis. You will need more information
to determine the nature and effect of the possible hidden variable.


You have enough data to produce a time-series graph of the numbers of
young musicians who go on to professional orchestras. This graph also has
two clusters of data points. The numbers rise from 1991 to 1996, drop
substantially in 1997, and then rise again. This pattern suggests that
something unusual happened in 1997.

b) The collapse of a major orchestra means both that there is one less orchestra
hiring young musicians and that about a hundred experienced players are
suddenly available for work with the remaining professional orchestras.
The resulting drop in the number of young musicians hired by professional
orchestras could account for the clustering of data points you observed in
part a). Because of the change in the number of jobs available for young
musicians, it makes sense to analyse the clusters separately.


Observe that the two sets of data both exhibit a strong linear correlation.
The correlation coefﬁcients are 0.93 for the data prior to 1997 and 0.94 for
the data from 1997 on. The number of players who go on to professional
orchestras is strongly correlated to the number of youth orchestras. So,
funding the new orchestra may be a worthwhile project for the arts council.

The presence of a hidden variable, the collapse of a major orchestra,
distorted the data and masked the underlying pattern. However, splitting
the data into two sets results in smaller sample sizes, so you still have to
be cautious about drawing conclusions.

When evaluating claims based on statistical studies, you must assess the Project
methods used for collecting and analysing the data. Some critical questions Prep
are:
When collecting
• Is the sampling process free from intentional and unintentional bias? and analysing data
for your statistics
• Could any outliers or extraneous variables inﬂuence the results?
project, you can
• Are there any unusual patterns that suggest the presence of a hidden apply the concepts
variable? in this section to
ensure that your
• Has causality been inferred with only correlational evidence? conclusions are
valid.


Key Concepts

• Although the major media are usually responsible in how they present statistics,
you should be cautious about accepting any claim that does not include
information about the sampling technique and analytical methods used.

• Intentional or unintentional bias can invalidate statistical claims.

• Small sample sizes and inappropriate sampling techniques can distort the data
and lead to erroneous conclusions.

• Extraneous variables must be eliminated or accounted for.

• A hidden variable can skew statistical results and yet still be hard to detect.


1. Explain how a small sample size can lead to invalid conclusions.

2. A city councillor states that there are problems with the management of the
police department because the number of reported crimes in the city has risen
despite increased spending on law enforcement. Comment on the validity of
this argument.

3. Give an example of a hidden variable not mentioned in this section, and
explain why this variable would be hard to detect.

Apply, Solve, Communicate B
A 3. A student compares height and grade
average with four friends and collects
1. An educational researcher discovers that
the following data.
levels of mathematics anxiety are negatively
correlated with attendance in mathematics Height (cm) Grade Average (%)
class. The researcher theorizes that poor 171 73
attendance causes mathematics anxiety. 145 91
Suggest an alternate interpretation of the 162 70
evidence. 159 81
178 68
2. A survey ﬁnds a correlation between the
proportion of high school students who own From this table, the student concludes that
a car and the students’ ages. What hidden taller students tend to get lower marks.
variable could affect this study? a) Does a regression analysis support the
student’s conclusion?
b) Why are the results of this analysis invalid?
c) How can the student get more accurate
results?

4. Inquiry/Problem Solving A restaurant chain b) Is this prediction realistic? Explain.
randomly surveys its customers several times c) Explain why this model generated such
a year. Since the surveys show that the level an inaccurate prediction despite having
of customer satisfaction is rising over time, a high value for the coefficient of
the company concludes that its customer determination.
service is improving. Discuss the validity of
d) Suggest methods Gina could use to make
the surveys and the conclusion based on
a more accurate prediction.
these surveys.
7. Communication Find a newspaper or
5. Application A teacher offers the following data
magazine article, television commercial, or
to show that good attendance is important.
web page that misuses statistics of two
Days Absent Final Grade variables. Perform a critical analysis using
8 72 the techniques in this chapter. Present your
2 75 findings in a brief report.
0 82
8. Application A manufacturing company keeps
11 68
records of its overall annual production and
15 66
its number of employees. Data for a ten-year
20 30 period are shown below.
A student with a graphing calculator points Year Number of Employees Production (000)
out that the data indicate that anyone who
1992 158 75
misses 17 days or more is in danger of
1993 165 81
failing the course.
1994 172 84
a) Show how the student arrived at this
1995 148 68
conclusion.
1996 130 58
b) Identify and explain the problems that
1997 120 51
make this conclusion invalid.
1998 98 50
c) Outline statistical methods to avoid these 1999 105 57
problems. 2000 110 62
6. Using a graphing calculator, Gina found the 2001 120 70
apte
h cubic curve of best fit for the salary data in a) Create a scatter plot to see if there is a
C

r

the table on page 157. This curve has a linear correlation between annual
m
P

r
oble
coefficient of determination of 0.98, production and number of employees.
indicating an almost perfect fit to the data. Classify the correlation.
The equation of the cubic curve is b) At some point, the company began to lay
starting salary off workers. When did these layoffs begin?
= 0.0518y3 – 310y2 + 618 412y – 411 344 091 c) Does the scatter plot suggest the
where the salary is given in thousands of presence of a hidden variable? Could the
dollars and y is the year of graduation. layoffs account for the pattern you see?
a) What mean starting salary does this Explain why or why not.
model predict for Gina’s class when they d) The company’s productivity is its annual
graduate in 2005? production divided by the number of


employees. Create a time-series graph comment on any evidence of a hidden
for the company’s productivity. variable. Conduct further research to
e) Find the line of best fit for the graph in determine if there are any hidden variables.
part d). Write a brief report outlining your analysis
and conclusions.
f) The company has adopted a better
management system. When do you think 10. Inquiry/Problem Solving A study conducted
the new system was implemented? by Stanford University found that
Explain your reasoning. behavioural counselling for people who had
suffered a heart attack reduced the risk of a
C
further heart attack by 45%. Outline how
9. Search E-STAT, CANSIM II, or other
you would design such a study. List the
sources for time-series data for the price of independent and dependent variables you
a commodity such as gasoline, coffee, or would use and describe how you would
computer memory. Analyse the data and account for any extraneous variables.

Career Connection
Economist
Economists apply statistical methods to develop mathematical models of the
production and distribution of wealth. Governments, large businesses, and
consulting firms are employers of economists. Some of the functions
performed by an economist include
• recognizing and interpreting domestic and international market trends
• using supply and demand analysis to assess market potential and set prices
• identifying factors that affect economic growth, such as inflation and
unemployment
• advising governments on fiscal and monetary policies
• optimizing the economic activity of financial institutions and large
businesses

Typically, a bachelor’s degree in economics is necessary to enter this field.
However, many positions require a master’s or doctorate degree or
specialized training. Since economists often deal with large amounts of data,
a strong background in statistics and an ability to work with computers are
definite assets.

An economist can expect to earn a comfortable
living. Most employment opportunities for
economists are in large cities. The current
demand for economists is reasonably strong
and likely to remain so for the foreseeable Visit the above web site and follow the links to
future, as governments and large businesses will learn more about a career as an economist
continue to need the information and analysis and other related careers.
that economists provide.



3.1 Scatter Plots and Linear Correlation a) Create a scatter plot for these data.
Refer to the Key Concepts on page 167. Classify the linear correlation.
b) Determine the correlation coefficient.
1. a) Classify the linear correlation in each
scatter plot shown below. c) Can you make any conclusions about the
effect that watching television has on
y academic achievement? Explain.
14
12
10 3.2 Linear Regression
8 Refer to the Key Concepts on page 179.
6
3. Use the method of least squares to find the
4
2
equation for the line of best fit for the data
in question 2.
0
2 4 6 8 10 12 14 x
4. The scores for players’ first and second
y
10
games at a bowling tournament are shown
8
below.
6 First Game 169 150 202 230 187 177 164
4 Second Game 175 162 195 241 185 235 171
2
a) Create a scatter plot for these data.
0
2 4 6 8 10 12 14 x
y and the line of best fit.
10 c) Identify any outliers.
8
d) Repeat part b) with the outliers removed.
6
4 e) A player scores 250 in the first game.
2 Use both linear models to predict this
0 x player’s score for the second game.
2 4 6 8 10 12 14
How far apart are the two predictions?
b) Determine the correlation coefficient for
data points in the scatter plots in part a).
3.3 Non-Linear Regression
c) Do these correlation coefficients agree Refer to the Key Concepts on page 191.
with your answers in part a)?
5. An object is thrown straight up into the air.
2. A survey of a group of randomly selected The table below shows the height of the
students compared the number of hours of object as it ascends.
television they watched per week with their Time (s) 0 0.1 0.2 0.3 0.4 0.5 0.6
grade averages.
Height (m) 0 1 1.8 2.6 3.2 3.8 4.2
Hours Per Week 12 10 5 3 15 16 8
a) Create a scatter plot for these data.
Grade Average (%) 70 85 82 88 65 75 68


b) Perform a non-linear regression for these 8. a) Explain the relationship between
data. Record the equation of the curve of experimental and control groups.
best fit and the coefficient of b) Why is a control group needed in
determination. some statistical studies?
c) Use your model to predict the maximum
height of the object. 9. a) Explain the difference between an
accidental relationship and a presumed
d) Use your model to predict how long the
relationship.
object will be in the air.
b) Provide an example of each.
e) Do you think that your model is
accurate? Explain. 10. The price of eggs is positively correlated
with wages. Explain why you cannot
6. The table shows the Time (s) Distance (m)
conclude that raising the price of eggs
distance travelled by a 0 0 should produce a raise in pay.
car as a function of time. 2 6
a) Determine a curve 4 22 11. An educational researcher compiles data on
of best fit to model 6 50 Internet use and scholastic achievement for
the data. 8 90 a random selection of students, and observes
10 140
b) Do you think the a strong positive linear correlation. She
12 190
equation for this concludes that Internet use improves student
14 240
curve of best fit is 16 290
grades. Comment on the validity of this
a good model for 18 340 conclusion.
the situation? 20 380
Explain your 22 410 3.5 Critical Analysis
reasoning. 24 430 Refer to the Key Cconcepts on page 209.
c) Describe what the 26 440
28 440 12. A teacher is trying to determine whether a
driver did between
new spelling game enhances learning. In
0 and 28 s.
his gifted class, he finds a strong positive
correlation between use of the game and
3.4 Cause and Effect spelling-test scores. Should the teacher
Refer to the Key Concepts on page 199. recommend the use of the game in all
7. Define or explain the following terms and English classes at his school? Explain your
provide an example of each one. answer.
a) common-cause factor 13. a) Explain what is meant by the term hidden
b) reverse cause-and-effect relationship variable.
c) extraneous variable b) Explain how you might detect the presence
of a hidden variable in a set of data.


Chapter Test

ACHIEVEMENT CHART

Questions All 5, 7, 10 1, 5, 6, 8, 10 3, 4, 7, 10

1. Explain or define each of the following d) Use this model to predict the average
terms. word length in a book recommended
a) perfect negative linear correlation for 12-year olds.
b) experimental research Use the following information in order to
c) outlier answer questions 4–6.
d) extraneous variable
Jerome has kept track of the hours he spent
e) hidden variable studying and his marks on examinations.
2. Match the following. Subject Hours Studied Mark
Mathematics, grade 9 5 70
Correlation Type Coefficient, r
English, grade 9 3 65
a) strong negative linear 1
Science, grade 9 4 68
b) direct 0.6 Geography, grade 9 4 72
c) weak positive linear 0.3 French, grade 9 2 38
d) moderate positive linear −0.8 Mathematics, grade 10 7 74
e) perfect negative linear −1 English, grade 10 5 69
Science, grade 10 6 71
3. The following set of data relates mean word History, grade 10 5 75
length and recommended age level for a set Mathematics, grade 11 12 76
of children’s books.
English, grade 11 9 74
Recommended Age Mean Word Length
Physics, grade 11 14 78
4 3.5
6 5.5 4. a) Create a scatter plot for Jerome’s data
and classify the linear correlation.
5 4.6
6 5.0 b) Perform a regression analysis. Identify
7 5.2 the equation of the line of best fit as y1,
and record the correlation coefficient.
9 6.5
8 6.1 c) Identify any outliers.
5 4.9 d) Repeat part b) with the outlier removed.
a) Create a scatter plot and classify the Identify this line as y2.
linear correlation. 5. Which of the two linear models found in
b) Determine the correlation coefficient. question 4 gives a more optimistic
c) Determine the line of best fit. prediction for Jerome’s upcoming biology
examination? Explain.


6. a) Identify at least three extraneous a) Create a scatter plot for the data.
variables in Jerome’s study. b) Perform a quadratic regression. Record
b) Suggest some ways that Jerome might the equation of the curve of best fit and
improve the validity of his study. the coefficient of determination.
c) Repeat part b) for an exponential
7. A phosphorescent material can glow in the
regression.
dark by absorbing energy from light and
then gradually re-emitting it. The following d) Compare how well these two models fit
table shows the light levels for a the data.
phosphorescent plastic. e) According to each model, what will be
Time (h) Light Level (lumens) the light level after 10 h?
0 0.860 f) Which of these two models is superior
1 0.695 for extrapolating beyond 6 h? Explain.
2 0.562
8. Explain how you could minimize the effects
3 0.455
of extraneous variables in a correlation study.
4 0.367
5 0.305 9. Provide an example of a reverse cause-and-
6 0.247 effect relationship.

ACHIEVEMENT CHECK

10. The table shown on the right contains data Licensed Number of % of Drivers in Age
from the Ontario Road Safety Annual Report Age
Drivers Collisions Group in Collisions
for 1999. 16 85 050 1 725 2.0
a) Organize the data so that the age intervals 17 105 076 7 641 7.3
are consistent. Create a scatter plot of the 18 114 056 9 359 8.2
proportion of drivers involved in collisions 19 122 461 9 524 7.8
versus age. 20 123 677 9 320 7.5
b) Perform a regression analysis. Record the 21–24 519 131 36 024 6.9
equations of the curves of best fit for each 25–34 1 576 673 90 101 5.7
regression you try as well as the coefficient 35–44 1 895 323 90 813 4.8
of determination. 45–54 1 475 588 60 576 4.1
c) In Ontario, drivers over 80 must take vision 55–64 907 235 31 660 3.5
and knowledge tests every two years to 65–74 639 463 17 598 2.8
renew their licences. However, these drivers 75 and
no longer have to take road tests as part of older 354 581 9 732 2.7
the review. Advocacy groups for seniors had Total 7 918 314 374 073 4.7
lobbied the Ontario government for this
change. How could such groups have used
your data analysis to support their position?


Statistics Project

Wrap-Up
Implementing Your Action Plan Suggested Resources
1. Look up the most recent census data from • Statistics Canada web sites and publications
Statistics Canada. Pick a geographical • Embassies and consulates
region and study the data on age of all
• United Nations web sites and publications
respondents by gender. Conjecture a
such as UNICEF’s CyberSchoolbus and
relationship between age and the relative
World Health Organization reports
numbers of males and females. Use a table
and a graph to organize and present the • Statistical software (the Fathom™ sample
data. Does the set of data support your documents include census data for Beverly
conjecture? Hills, California)
• Spreadsheets
2. You may want to compare the data you
• Graphing calculators
analysed in step 1 to the corresponding
data for other regions of Canada or for
other countries. Identify any significant
similarities or differences between the data
sets. Suggest reasons for any differences www.mcgrawhill.ca/links/MDM12
you notice.
Visit the web site above to find links to various
3. Access data on life expectancies in Canada census databases.
for males and females from the 1920s to
the present. Do life expectancies appear to
be changing over time? Is there a
correlation between these two variables? Evaluating Your Project
If so, use regression analysis to predict To help assess your own project, consider the
future life expectancies for males and following questions.
females in Canada.
1. Are the data you selected appropriate?
4. Access census data on life expectancies in
2. Are your representations of the data
the various regions of Canada. Select
effective?
another attribute from the census data and
conjecture whether there is a correlation 3. Are the mathematical models that you used
between this variable and life expectancies. reliable?
Analyse data from different regions to see
if the data support your conjecture. 4. Who would be interested in your findings?
Is there a potential market for this
information?

216 MHR • Statistics Project

5. Are there questions that arose from your Presentation
research that warrant further investigation? Present the findings of your investigation in
How would you go about addressing these one or more of the following forms:
issues in a future project? • written report
6. If you were to do this project again, what • oral presentation
would you do differently? Why? • computer presentation (using software such
as Corel® Presentations™ or Microsoft®
Section 9.4 describes methods for evaluating
PowerPoint®)
your own work.
• web page
• display board
Remember to include a bibliography. See
section 9.5 and Appendix D for information on
how to prepare a presentation.

Preparing for
the Culminating Project

Applying Project Skills be the focus of your project and begun to
Throughout this statistics project, you have gather relevant data. Section 9.2 provides
developed skills in statistical research and suggestions to help you clearly define your
analysis that may be helpful in preparing task. Your next steps are to develop and
your culminating project: implement an action plan.

• making a conjecture or hypothesis Make sure there are enough data to support
• using technology to access, organize, and your work. Decide on the best way to
analyse data organize and present the data. Then,
determine what analysis you need to do. As
• applying a variety of statistical tools
you begin to work with the data, you may
• comparing two sets of data
find that they are not suitable or that further
• presenting your findings
research is necessary. Your analysis may lead
to a new approach or topic that you would
Keeping on Track
like to pursue. You may find it necessary to
At this point, you should have a good idea of refine or alter the focus of your project. Such
the basic nature of your culminating project. changes are a normal part of the development
You should have identified the issue that will and implementation process.

Refine/Redefine

Define the Define Develop an Implement Evaluate Your Prepare a Present Your Constructively
Problem Your Task Action Plan Your Action Investigation Written Investigation Critique the
Plan and Its Results Report and Its Results Presentations
of Others

Statistics Project: Wrap-Up • MHR 217

Cumulative Review: Chapters 1 to 3 4
Cumulative Review: Chapters 3 and

5. Classify the type of linear correlation that

΄ ΅
7 3
you would expect for each pair of variables.
1. Let A = 0 –2 , B = 8
–5 4 –5 ΄ 4 ΅
1 , and
a) air temperature, altitude
b) income, athletic ability

΄ ΅
–8 0
C= 5 6 . Calculate, if possible, c) people’s ages from 1 to 20 years, their
9 –3 masses
a) –2(A + C) b) AC d) people’s ages from 21 to 40 years, their
masses
c) (BA)t d) B2
e) C2 f) B –1 6. Identify the most likely causal relationship
between each of the following pairs of variables.
2. a) Describe the iterative process used to
a) grade point average, starting salary upon
generate the table below.
graduation
b) Continue the process until all the cells
b) grade in chemistry, grade in physics
are filled.
c) sales of symphony tickets, carrot harvest
17 16 15 14 13 d) monthly rainfall, monthly umbrella sales
18 5 4 3 12
6 1 2 11 7. a) Sketch a map that can be coloured using
7 8 9 10 only three colours.
b) Reconfigure your map as a network.

8. State whether each of the following
3. Which of the following would you consider networks is
to be databases? Explain your reasoning. i) connected ii) traceable iii) planar
a) a novel Provide evidence for your decisions.
b) school attendance records a) b) P
B T
c) the home page of a web site
d) an advertising flyer from a department A
store C
Q S
4. What sampling techniques are most likely
to be used for the following surveys? Explain D R
each of your choices. 9. Use a tree diagram to represent the
a) a radio call-in show administrative structure of a school that has
b) a political poll a principal, vice-principals, department
c) a scientific study heads, assistant heads, and teachers.

218 MHR • Cumulative Review: Chapters 1 to 3

10. A renowned jazz pianist living in Toronto b) Create a histogram and a cumulative-
often goes on tours in the United States. For frequency diagram for the data.
the tour shown below, which city has the c) What proportion of the families surveyed
most routes earn an annual income of $60 000 or less?
a) with exactly one stopover?
13. Classify the bias in each of the following
b) with no more than two stopovers?
situations. Explain your reasoning in each case.
Toronto
a) At a ﬁnancial planning seminar, the
Buffalo audience were asked to raise their hands
Detroit if they had ever considered declaring
New York
Cleveland
bankruptcy.
Chicago Philadelphia b) A supervisor asked an employee if he
Pittsburgh
would mind working late for a couple
Washington of hours on Friday evening.
c) A survey asked neighbourhood dog-
11. The following are responses to a survey that
owners if dogs should be allowed to run
asked: “On average, how many hours per free in the local park.
week do you read for pleasure?”
d) An irascible talk show host listed the
1 3 0 0 7 2 0 1 10 5 2 2 2 0 1 4 0 8 3 1 3 mayor’s blunders over the last year and
0 0 2 15 4 9 1 6 7 0 3 3 14 5 7 0 1 1 0 10 0 invited listeners to call in and express
Use a spreadsheet to their opinions on whether the mayor
should resign.
a) sort the data from smallest to largest value
b) determine the mean hours of pleasure 14. The scores in a recent bowling tournament
reading are shown in the following table.
c) organize the data into a frequency table 150 260 213 192 176 204 138 214 298 188
with appropriate intervals 168 195 225 170 260 254 195 177 149 224
d) make a histogram of the information in 260 222 167 182 207 221 185 163 112 189
part c)
a) Calculate the mean, median, and mode
12. The annual incomes of 40 families surveyed for this distribution. Which measure
at random are shown in the table. would be the most useful? Which would
be the least useful? Explain your choices.
Income ($000)
b) Determine the standard deviation, ﬁrst
28.5 38 61 109 42 56 19
quartile, third quartile, and interquartile
27 44.5 81 36 39 51 40.5
67 28 60 87 58 120 111 range.
73 65 34 54 16.5 135 70.5 c) Explain what each of the quantities in part
59 47 92 38 55 84.5 107 b) tells you about the distribution of scores.
71 59 26.5 76 50
d) What score is the 50th percentile for this
a) Group these data into 8 to 12 intervals distribution?
and create a frequency table.
Cumulative Review: Chapters 1 to 3 • MHR 219

e) Is the player who scored 222 above the a) Create a time-series graph for these data.
80th percentile? Explain why or why not. b) Based on this graph, what level of sales
would you predict for 2003?
15. The players on a school baseball team
compared their batting averages and the c) List three factors that could affect the
hours they spent at the batting practice. accuracy of your prediction.
Batting Average Practice Hours d) Compute an index value for the sales
0.220 20 each year using the 1997 sales as a base.
0.215 18 What information do the index values
0.185 15 provide?
0.170 14 e) Suppose that this salesperson is thinking
0.200 18 of changing jobs. Outline how she could
0.245 22 use the sales index to convince other
0.230 19
employers to hire her.
0.165 15
0.205 17 18. The following time-series graph shows the
a) Identify the independent variable and Consumer Price Index (CPI) for the period
dependent variable. Explain your choices. 1971 to 2001.
Consumer Price Index (CPI) 150
b) Produce a scatter plot for the data and
classify the linear correlation.
c) Determine the correlation coefficient and
(1992=100)

100
the equation of the line of best fit.
d) Use this linear model to predict the
batting average for players who had 50
batting practice for
i) 16 h ii) 13 h iii) 35 h
e) Discuss how accurate you think each of
1975

1980

1985

1990

1995

2000
these predictions will be.
Year
16. Describe a method you could use to detect
a) What is the base for this index? When
outliers in a sample. did the CPI equal half of this base value?
17. A bright, young car salesperson has made the b) Approximately how many times did the
following gross sales with her first employer. average price of goods double from 1971
Year Gross Sales ($ millions) to 1992?
1997 0.8 c) Which decade on this graph had the
1998 1.1 highest rate of inflation? Explain your
1999 1.6 answer.
2000 2.3
d) Estimate the overall rate of inflation for
2001 3.5
the period from 1971 to 2001.
2002 4.7

220 MHR • Cumulative Review: Chapters 1 to 3

Probability Project

Designing a Game
Background
Many games introduce elements of chance
with random processes. For example, card
games use shufﬂed cards, board games often
use dice, and bingo uses randomly selected
numbers.

Your Task
Design and then analyse a game for two or
more players, involving some form of
random process. One of the players may
assume the role of dealer or game master.

Developing an Action Plan
You will need to decide on one or more
instruments of chance, such as dice, cards,
coins, coloured balls, a random-number
generator, a spinner, or a nail maze.
Recommend a method of tracking progress
or keeping score, such as a game board or
tally sheet. Create the rules of the game.
Submit a proposal to your teacher outlining
the concept and purpose of your game.

Probability Project: Introduction • MHR
<<Section number and title>> 221

4
4
PT ER
ER
Permutations and Organized
CHA

Counting


Represent complex tasks or issues, using diagrams. 4.1

Solve introductory counting problems involving the additive and 4.1, 4.2, 4.3
multiplicative counting principles.

Express the answers to permutation and combination problems, using 4.2, 4.3
standard combinatorial symbols.

Evaluate expressions involving factorial notation, using appropriate 4.2, 4.3
methods.

Solve problems, using techniques for counting permutations where some 4.3
objects may be alike.

Identify patterns in Pascal’s triangle and relate the terms of Pascal’s 4.4, 4.5
n
triangle to values of ΂ r ΃, to the expansion of a binomial, and to the
solution of related problems.

Communicate clearly, coherently, and precisely the solutions to counting 4.1, 4.2, 4.3,
problems. 4.4, 4.5

Chapter Problem
Students’ Council Elections 1. In how many ways could the positions
Most high schools in Ontario have a of president and vice-president be filled
students’ council comprised of students by these ten students if all ten are
from each grade. These students are elected eligible for these positions? How many
representatives, and a part of their function ways are there if only the grades 11 and
is to act as a liaison between the staff and 12 students are eligible?
the students. Often, these students are
2. The grade representatives must
instrumental in fundraising and in
represent their current grade level.
coordinating events, such as school dances
In how many ways could the grade
and sports.
representative positions be filled?
A students’ council executive could consist of a
You could answer both of these questions
president, vice-president, secretary, treasurer,
by systematically listing all the possibilities
social convenor, fundraising chair, and four
and then counting them. In this chapter,
grade representatives. Suppose ten students
you will learn easier and more powerful
have been nominated to fill these positions.
techniques that can also be applied to much
Five of the nominees are from grade 12, three
more complex situations.
are from grade 11, and the other two are a
grade 9 and a grade 10 student.



1. Tree diagrams Draw a tree diagram to c) 3 by 5 grid? d) 4 by 5 grid?
illustrate the number of ways a quarter, a
dime, and a nickel can come up heads or
tails if you toss one after the other.

2. Tree diagrams
a) Draw a tree diagram to illustrate the
possible outcomes of tossing a coin and 5. Evaluating expressions Evaluate each
rolling a six-sided die. expression given x = 5, y = 4, and z = 3.
b) How many possible outcomes are there? 8y(x + 2)( y + 2)(z + 2)
a) ᎏᎏᎏ
3. Number patterns The manager of a grocery (x − 3)( y + 3)(z + 2)
store asks a stock clerk to arrange a display (x − 2)3( y + 2)2(z + 1)2
b) ᎏᎏᎏ
of canned vegetables in a triangular pyramid 2
y(x + 1)( y − 1)
like the one shown. Assume all cans are the (x + 4)( y − 2)(z + 3) (x − 1)2(z + 1)y
same size and shape. c) ᎏᎏᎏ + ᎏᎏ
( y − 1)(x − 3)z
4
(x − 3) ( y + 4)

6. Order of operations Evaluate.
a) 5(4) + (–1)3(3)2
(10 − 2)2(10 − 3)2
b) ᎏᎏᎏ
(10 − 2) − (10 − 3)
2 2

a) How many cans is the tallest complete 6(6 − 1)(6 − 2)(6 − 3)(6 − 4)(6 − 5)
pyramid that the clerk can make with c) ᎏᎏᎏᎏ
3(3 − 1)(3 − 2)
100 cans of vegetables?
50(50 − 1)(50 − 2)…(50 − 49)
b) How many cans make up the base level d) ᎏᎏᎏᎏ
48(48 − 1)(48 − 2)…(48 − 47)
of the pyramid in part a)?
12 × 11 × 10 × 9 10 × 9 × 8 × 7
c) How many cans are in the full pyramid e) ᎏᎏ + ᎏᎏ
2 4
6 2
in part a)?
8×7×6×5
d) What is the sequence of the numbers of − ᎏᎏ
42
cans in the levels of the pyramid?
7. Simplifying expressions Simplify.
4. Number patterns What is the greatest x2 − xy + 2x (4x + 8)2
possible number of rectangles that can a) ᎏᎏ b) ᎏ
2x 16
be drawn on a
14(3x2 + 6)
a) 1 by 5 grid? b) 2 by 5 grid? c) ᎏᎏ
7×6
x(x − 1)(x − 2)(x − 3)
d) ᎏᎏᎏ
2
x − 2x
2y + 1 16y + 4
e) ᎏ + ᎏ
x 4x
224 MHR • Permutations and Organized Counting

4.1 Organized Counting

The techniques and mathematical logic for counting possible arrangements or
outcomes are useful for a wide variety of applications. A computer programmer
writing software for a game or industrial process would use such techniques, as
would a coach planning a starting line-up, a conference manager arranging
a schedule of seminars, or a school board trying to make the most efﬁcient use
of its buses.

Combinatorics is the branch of mathematics dealing with ideas and methods
for counting, especially in complex situations. These techniques are also
valuable for probability calculations, as you will learn in Chapter 6.

I N V E S T I G AT E & I N Q U I R E : Licence Plates

Until 1997, most licence plates for passenger cars in Ontario had three
numbers followed by three letters. Suppose the provincial government
had wanted all the vehicles
registered in Ontario to have plates
with the letters O, N, and T.

1. Draw a diagram to illustrate all
the possibilities for arranging
these three letters assuming
that the letters can be repeated.
How many possibilities are
there?
2. How could you calculate the
number of possible three-letter groups
without listing them all?
3. Predict how many three-letter groups
the letters O, N, T, and G can
form.
4. How many three-letter groups
do you think there would be if
you had a choice of ﬁve letters?
5. Suggest a general strategy for
counting all the different
possibilities in situations like
those above.

4.1 Organized Counting • MHR 225

When you have to make a series of choices, you can usually determine the
total number of possibilities without actually counting each one individually.

Example 1 Travel Itineraries

Martin lives in Kingston and is planning a trip to Vienna, Austria. He checks
a web site offering inexpensive airfares and finds that if he travels through
London, England, the fare is much lower. There are three flights available
from Toronto to London and two flights from London to Vienna. If Martin
can take a bus, plane, or train from Kingston to Toronto, how many ways can
he travel from Kingston to Vienna?

Solution
Martin's Choices
You can use a tree diagram to illustrate and count Martin’s choices. Flight A
Flight 1
This diagram suggests another way to determine the number Flight 2
Flight 1
of options Martin has for his trip. Bus Flight B
Flight 2
Flight 1
Flight C
Choices for the first portion of trip: 3 Flight 2
Choices for the second portion of trip: 3 Flight A
Flight 1
Flight 2
Choices for the third portion of trip: 2
Flight 1
Total number of choices: 3 × 3 × 2 = 18 Train Flight B
Flight 2
Flight 1
Flight C
In all, Martin has 18 ways to travel from Kingston to Vienna. Flight 2
Flight 1
Flight A
Flight 2
Flight 1
Plane Flight B
Flight 2
Flight 1
Flight C
Flight 2
Example 2 Stereo Systems

Javon is looking at stereos in an electronics store. The store has five types of
receivers, four types of CD players, and five types of speakers. How many
different choices of stereo systems does this store offer?

Solution

For each choice of receiver, Javon could choose any one of the CD players.
Thus, there are 5 × 4 = 20 possible combinations of receivers and CD players.
For each of these combinations, Javon could then choose one of the five kinds
of speakers.

The store offers a total of 5 × 4 × 5 = 100 different stereo systems.


These types of counting problems illustrate the fundamental or multiplicative
counting principle:

If a task or process is made up of stages with separate choices, the total
number of choices is m × n × p × …, where m is the number of choices
for the first stage, n is the number of choices for the second stage, p is the
number of choices for the third stage, and so on.

Example 3 Applying the Fundamental Counting Principle Project
Prep
A school band often performs at benefits and other functions outside the
school, so its members are looking into buying band uniforms. The band You can use the
committee is considering four different white shirts, dress pants in grey, navy, fundamental or
or black, and black or grey vests with the school crest. How many different multiplicative
designs for the band uniform is the committee considering? counting principle
to help design
Solution the game for
your probability
First stage: choices for the white shirts, m = 4
project.
Second stage: choices for the dress pants, n = 3
Third stage: choices for the vests, p = 2
The total number of possibilities is
m×n×p=4×3×2
= 24
The band committee is considering 24 different possible uniforms.

In some situations, an indirect method makes a calculation easier.

Example 4 Indirect Method

Leora, a triathlete, has four pairs of running shoes loose in her gym bag.
In how many ways can she pull out two unmatched shoes one after the other?

Solution

You can find the number of ways of picking unmatched shoes by subtracting the
number of ways of picking matching ones from the total number of ways of
picking any two shoes.

There are eight possibilities when Leora pulls out the first shoe, but only seven
when she pulls out the second shoe. By the fundamental counting principle, the
number of ways Leora can pick any two shoes out of the bag is 8 × 7 = 56. She
could pick each of the matched pairs in two ways: left shoe then right shoe or right
shoe then left shoe. Thus, there are 4 × 2 = 8 ways of picking a matched pair.

Leora can pull out two unmatched shoes in 56 − 8 = 48 ways.


Sometimes you will have to count several subsets of possibilities separately.

Example 5 Signal Flags

Sailing ships used to send messages with signal flags flown from their masts.
How many different signals are possible with a set of four distinct flags if a
minimum of two flags is used for each signal?

Solution

A ship could fly two, three, or four signal flags.

Signals with two flags: 4 × 3 = 12
Signals with three flags: 4 × 3 × 2 = 24
Signals with four flags: 4 × 3 × 2 × 1 = 24
Total number of signals: 12 + 24 + 24 = 60

Thus, the total number of signals possible with these flags is 60.

In Example 5, you were counting actions that could not occur at the same time.
When counting such mutually exclusive actions, you can apply the additive
counting principle or rule of sum:
If one mutually exclusive action can occur in m ways, a second in n ways,
a third in p ways, and so on, then there are m + n + p … ways in which
one of these actions can occur.

Key Concepts

• Τree diagrams are a useful tool for organized counting.

• Ιf you can choose from m items of one type and n items of another, there are
m × n ways to choose one item of each type (fundamental or multiplicative
counting principle).

• If you can choose from either m items of one type or n items of another type,
then the total number of ways you can choose an item is m + n (additive
counting principle).

• Both the multiplicative and the additive counting principles also apply to
choices of three or more types of items.

• Sometimes an indirect method provides an easier way to solve a problem.



1. Explain the fundamental counting principle in your own words and give
an example of how you could apply it.
2. Are there situations where the fundamental counting principle does not
apply? If so, give one example.
3. Can you always use a tree diagram for organized counting? Explain your
reasoning.

A 6. Ten different books and four different pens
1. Construct a tree diagram to illustrate the are sitting on a table. One of each is
possible contents of a sandwich made from selected. Should you use the rule of sum or
white or brown bread, ham, chicken, or the product rule to count the number of
beef, and mustard or mayonnaise. How possible selections? Explain your reasoning.
many different sandwiches are possible?
B
2. In how many ways can you roll either a sum
7. Application A grade 9 student may build a
of 4 or a sum of 11 with a pair of dice?
timetable by selecting one course for each
3. In how many ways can you draw a 6 or a period, with no duplication of courses.
face card from a deck of 52 playing cards? Period 1 must be science, geography, or
physical education. Period 2 must be art,
4. How many ways are there to draw a 10 or
music, French, or business. Periods 3 and 4
a queen from the 24 cards in a euchre deck,
must each be mathematics or English.
which has four 10s and four queens?
a) Construct a tree diagram to illustrate the
5. Use tree diagrams to answer the following: choices for a student’s timetable.
a) How many different soccer uniforms are b) How many different timetables could a
possible if there is a choice of two types student choose?
of shirts, three types of shorts, and two
types of socks? 8. A standard die is rolled ﬁve times. How
many different outcomes are possible?
b) How many different three-scoop cones
can be made from vanilla, chocolate, and 9. A car manufacturer offers three kinds of
strawberry ice cream? upholstery material in ﬁve different colours
c) Suppose that a college program has six for this year’s model. How many upholstery
elective courses, three on English options would a buyer have? Explain your
literature and three on the other arts. If reasoning.
the college requires students to take one
of the English courses and one of the 10. Communication In how many ways can a
other arts courses, how many pairs of student answer a true-false test that has six
courses will satisfy these requirements? questions. Explain your reasoning.


11. The final score of a soccer game is 6 to 3. 17. Ten students have been nominated for
pte
How many different scores were possible ha a students’ council executive. Five of the

C

r
at half-time? nominees are from grade 12, three are

m
P
r
oble
from grade 11, and the other two are
12. A large room has a bank of five windows. from grades 9 and 10.
Each window is either open or closed. How
a) In how many ways could the nominees
many different arrangements of open and
fill the positions of president and vice-
closed windows are there?
president if all ten are eligible for these
13. Application A Canadian postal code uses six senior positions?
characters. The first, third, and fifth are b) How many ways are there to fill
letters, while the second, fourth, and sixth these positions if only grade 11 and
are digits. A U.S.A. zip code contains five grade 12 students are eligible?
characters, all digits.
18. Communication
a) How many codes are possible for each
country? a) How many different licence plates could
be made using three numbers followed
b) How many more possible codes does
by three letters?
the one country have than the other?
b) In 1997, Ontario began issuing licence
14. When three-digit area codes were plates with four letters followed by three
introduced in 1947, the first digit had to be numbers. How many different plates are
a number from 2 to 9 and the middle digit possible with this new system?
had to be either 1 or 0. How many area c) Research the licence plate formats used in
codes were possible under this system? the other provinces. Compare and contrast
15. Asha builds new homes and offers her
these formats briefly and suggest reasons
customers a choice of brick, aluminium for any differences between the formats.
siding, or wood for the exterior, cedar or 19. In how many ways can you arrange the
asphalt shingles for the roof, and radiators or letters of the word think so that the t and the
forced-air for the heating system. How many h are separated by at least one other letter?
different configurations is Asha offering?
20. Application Before the invention of the
16. a) In how many ways could you choose telephone, Samuel Morse (1791−1872)
two fives, one after the other, from a developed an efficient system for sending
deck of cards? messages as a series of dots and dashes
b) In how many ways could you choose a red (short or long pulses). International code, a
five and a spade, one after the other? modified version of Morse code, is still
c) In how many ways could you choose widely used.
a red five or a spade? a) How many different characters can the
d) In how many ways could you choose international code represent with one to
a red five or a heart? four pulses?
e) Explain which counting principles you b) How many pulses would be necessary
could apply in parts a) to d). to represent the 72 letters of the
Cambodian alphabet using a system
like Morse code?


ACHIEVEMENT CHECK 24. Inquiry/Problem Solving Your school is
purchasing a new type of combination lock
Communication Application for the student lockers. These locks have
40 positions on their dials and use a three-
21. Ten finalists are competing in a race at
number combination.
the Canada Games.
a) How many combinations are possible if
a) In how many different orders can the
consecutive numbers cannot be the
competitors finish the race? same?
b) How many ways could the gold, silver,
b) Are there any assumptions that you have
and bronze medals be awarded? made? Explain.
c) One of the finalists is a friend from
c) Assuming that the first number must be
your home town. How many of the dialled clockwise from 0, how many
possible finishes would include your different combinations are possible?
friend winning a medal?
d) Suppose the first number can also be
d) How many possible finishes would
dialled counterclockwise from 0. Explain
leave your friend out of the medal the effect this change has on the number
standings? of possible combinations.
e) Suppose one of the competitors is
e) If you need four numbers to open the
injured and cannot finish the race. lock, how many different combinations
How does that affect your previous are possible?
answers?
f) How would the competitor’s injury 25. Inquiry/Problem Solving In chess, a knight
affect your friend’s chances of winning can move either two squares horizontally
a medal? Explain your reasoning. plus one vertically or two squares vertically
What assumptions have you made? plus one horizontally.
a) If a knight starts from one corner of a
standard 8 × 8 chessboard, how many
C different squares could it reach after
22. A locksmith has ten types of blanks for i) one move?
keys. Each blank has five different cutting
ii) two moves?
positions and three different cutting depths
at each position, except the first position, iii) three moves?
which only has two depths. How many b) Could you use the fundamental counting
different keys are possible with these principle to calculate the answers for
blanks? part a)? Why or why not?

23. Communication How many 5-digit numbers
are there that include the digit 5 and exclude
the digit 8? Explain your solution.


4.2 Factorials and Permutations

In many situations, you need to determine the number of different orders
in which you can choose or place a set of items.

I N V E S T I G AT E & I N Q U I R E : N u m b e r s o f A r r a n g e m e n t s

Consider how many different ways a president and a vice-president could
be chosen from eight members of a students’ council.

1. a) Have one person in your class make two signs, writing President
on one and Vice-President on the other. Now, choose two people
to stand at the front of the class. Using the signs to indicate which
person holds each position, decide in how many ways you can
choose a president and a vice-president from the two people at
the front of the class.
b) Choose three students to
be at the front of the class.
Again using the signs to
indicate who holds each
position, determine how
many ways you can choose
a president and a vice-
president from the three
people at the front of the
class.
c) Repeat the process with
four students. Do you see
a pattern in the number of
ways a president and a
vice-president can be
chosen from the different
sizes of groups? If so, what
is the pattern? If not,
continue the process with ﬁve students and then with six students.
d) When you see a pattern, predict how many ways a president and
a vice-president can be chosen from the eight members of the
students’ council.
e) Suggest other ways of simulating the selection of a president and
a vice-president for the students’ council.


2. Suppose that each of the eight members of the students’ council has to
give a brief speech at an assembly. Consider how you could determine
the number of different orders in which they could speak.
a) Choose two students from your class and list all the possible orders
in which they could speak.
b) Choose three students and list all the possible orders in which they
could speak.
c) Repeat this process with four students.
d) Is there an easy method to organize the list so that you could
include all the possibilities?
e) Is this method related to your results in question 1? Explain.
f) Can you use your method to predict the number of different
orders in which eight students could give speeches?

Many counting and probability calculations involve the product of a series of
consecutive integers. You can use factorial notation to write such expressions
more easily. For any natural number n,

n! = n × (n − 1) × (n − 2) × (n − 3) × … × 3 × 2 × 1

This expression is read as n factorial.

Example 1 Evaluating Factorials

Calculate each factorial.
a) 2! b) 4! c) 8!

Solution
a) 2! = 2 × 1
=2

b) 4! = 4 × 3 × 2 × 1
= 24

c) 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
= 40 320

As you can see from Example 1, n! increases dramatically as n becomes larger.
However, calculators and computer software provide an easy means of
calculating the larger factorials. Most scientiﬁc and graphing calculators have
a factorial key or function.

4.2 Factorials and Permutations • MHR 233

Example 2 Using Technology to Evaluate Factorials

Calculate.
a) 21! b) 53! c) 70!


Enter the number on the home screen and then use the ! function on the
MATH PRB menu to calculate the factorial.
a) 21! = 21 × 20 × 19 × 18 × … × 2 × 1
= 5.1091 × 1019

b) 53! = 53 × 52 × 51 × … × 3 × 2 × 1
= 4.2749 × 1069

c) Entering 70! on a graphing calculator gives an ERR:OVERFLOW message since
70! > 10100 which is the largest number the calculator can handle. In fact, 69! is
the largest factorial you can calculate directly on TI-83 series calculators.


Both Corel® Quattro® Pro and Microsoft® Excel have a built-in factorial
function with the syntax FACT(n).


Example 3 Evaluating Factorial Expressions

Evaluate.
10! 83!
a) ᎏ b) ᎏ
5! 79!

Solution

In both these expressions, you can divide out the common terms in the
numerator and denominator.
10! 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
a) ᎏ = ᎏᎏᎏᎏ
5! 5×4×3×2×1
= 10 × 9 × 8 × 7 × 6
= 30 240

83! 83 × 82 × 81 × 80 × 79 × 78 × … × 2 × 1
b) ᎏ = ᎏᎏᎏᎏᎏ
79! 79 × 78 × … × 2 × 1
= 83 × 82 × 81 × 80
= 44 102 880

Note that by dividing out the common terms, you can use a calculator to evaluate
this expression even though the factorials are too large for the calculator.

Example 4 Counting Possibilities
The senior choir has rehearsed five songs for an upcoming assembly.
In how many different orders can the choir perform the songs?

Solution
There are five ways to choose the first song, four ways to choose the second,
three ways to choose the third, two ways to choose the fourth, and only one way
to choose the final song. Using the fundamental counting principle, the total
number of different ways is
5 × 4 × 3 × 2 × 1 = 5!
= 120
The choir can sing the five songs in 120 different orders.

Example 5 Indirect Method
In how many ways could ten questions on a test be arranged, if the easiest
question and the most difficult question
a) are side-by-side?
b) are not side-by-side?


Solution

a) Treat the easiest question and the most difficult question as a unit making
nine items that are to be arranged. The two questions can be arranged in 2!
ways within their unit.
9! × 2! = 725 760
The questions can be arranged in 725 760 ways if the easiest question
and the most difficult question are side-by-side.

b) Use the indirect method. The number of arrangements with the easiest and
most difficult questions separated is equal to the total number of possible
arrangements less the number with the two questions side-by-side:
10! − 9! × 2! = 3 628 800 − 725 760
= 2 903 040
The questions can be arranged in 2 903 040 ways if the easiest question
and the most difficult question are not side-by-side.

A permutation of n distinct items is an arrangement of all the items in a definite
order. The total number of such permutations is denoted by nPn or P(n, n).

There are n possible ways of choosing the first item, n − 1 ways of choosing the
second, n − 2 ways of choosing the third, and so on. Applying the fundamental
counting principle as in Example 5 gives
P = n × (n − 1) × (n − 2) × (n − 3) × … × 3 × 2 × 1
n n
= n!

Example 6 Applying the Permutation Formula
In how many different orders can eight nominees for the students’ council give
their speeches at an assembly?

Solution
P = 8!
8 8
=8×7×6×5×4×3×2×1
= 40 320
There are 40 320 different orders in which the eight nominees can give their speeches.

Example 7 Student Government
In how many ways could a president and a vice-president be chosen from a group
of eight nominees?

Solution
Using the fundamental counting principle, there are 8 × 7, or 56, ways to choose
a president and a vice-president.


A permutation of n distinct items taken r at a time is an arrangement of r
of the n items in a deﬁnite order. Such permutations are sometimes called
r-arrangements of n items. The total number of possible arrangements of
r items out of a set of n is denoted by nPr or P(n, r).

There are n ways of choosing the ﬁrst item, n − 1 ways of choosing the second
item, and so on down to n − r + 1 ways of choosing the rth item. Using the
fundamental counting principle,

P = n(n − 1)(n − 2)…(n − r + 1)
n r
Project
Prep
It is often more convenient to rewrite this expression in terms of factorials.
The permutations
formula could be
n!
P = ᎏᎏ a useful tool for
n r
(n − r)!
your probability
project.
The denominator divides out completely, as in Example 3, so these two ways
of writing nPr are equivalent.

Example 8 Applying the Permutation Formula

In a card game, each player is dealt a face down “reserve” of 13 cards that
can be turned up and used one by one during the game. How many different
sequences of reserve cards could a player have?


Here, you are taking 13 cards from a deck of 52.
52!
P = ᎏᎏ
52 13
(52 − 13)!
52!
=ᎏ
39!
= 52 × 51 × 50 × … × 41 × 40
= 3.9542 × 1021
There are approximately 3.95 × 1021 different sequences of reserve cards a
player could have.

Use the nPr function on the MATH PRB menu.

There are approximately 3.95 × 1021 different
sequences of reserve cards a player could turn
up during one game.



Both Corel® Quattro® Pro and Microsoft® Excel have a permutations function
with the syntax PERMUT(n,r).

There are approximately 3.95 × 1021 different sequences of reserve cards a
player could turn up during one game.

Key Concepts

• A factorial indicates the multiplication of consecutive natural numbers.
n! = n(n − 1)(n − 2) × … × 1.
• The number of permutations of n distinct items chosen n at a time in a
deﬁnite order is nPn = n!

• The number of permutations of r items taken from n distinct items is
n!
P = ᎏ.
n r
(n − r)!


1. Explain why it is convenient to write the expression for the number of
possible permutations in terms of factorials.

2. a) Is (−3)! possible? Explain your answer.
b) In how many ways can you order an empty list, or zero items? What does
this tell you about the value of 0!? Check your answer using a calculator.


Practise B
A 7. Simplify each of the following in factorial
form. Do not evaluate.
1. Express in factorial notation.
a) 12 × 11 × 10 × 9!
a) 6 × 5 × 4 × 3 × 2 × 1
b) 72 × 7!
b) 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
c) (n + 4)(n + 5)(n +3)!
c) 3 × 2 × 1
d) 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 8. Communication Explain how a factorial is an
iterative process.
2. Evaluate.
7! 11! 9. Seven children are to line up for a photograph.
a) ᎏ b) ᎏ
4! 9! a) How many different arrangements are
8! 15! possible?
c) ᎏ d) ᎏ
5! 2! 3! 8! b) How many arrangements are possible if
85! 14! Brenda is in the middle?
e) ᎏ f) ᎏ
82! 4! 5! c) How many arrangements are possible if
Ahmed is on the far left and Yen is on
3. Express in the form nPr. the far right?
a) 6 × 5 × 4 d) How many arrangements are possible if
b) 9 × 8 × 7 × 6 Hanh and Brian must be together?
c) 20 × 19 × 18 × 17
10. A 12-volume encyclopedia is to be placed on
d) 101 × 100 × 99 × 98 × 97 a shelf. How many incorrect arrangements
e) 76 × 75 × 74 × 73 × 72 × 71 × 70 are there?
4. Evaluate without using technology. 11. In how many ways can the 12 members of
a) P(10, 4) b) P(16, 4) c) 5P2 a volleyball team line up, if the captain and
d) 9P4 e) 7!
assistant captain must remain together?

12. Ten people are to be seated at a rectangular
5. Use either a spreadsheet or a graphing or
scientiﬁc calculator to verify your answers table for dinner. Tanya will sit at the head of
to question 4. the table. Henry must not sit beside either
Wilson or Nancy. In how many ways can the
Apply, Solve, Communicate people be seated for dinner?

6. a) How many ways can you arrange the 13. Application Joanne prefers classical and
letters in the word factor? pop music. If her friend Charlene has ﬁve
classical CDs, four country and western
b) How many ways can Ismail arrange
CDs, and seven pop CDs, in how many
four different textbooks on the shelf in
orders can Joanne and Charlene play the
his locker?
CDs Joanne likes?
c) How many ways can Laura colour
4 adjacent regions on a map if she has 14. In how many ways can the valedictorian,
a set of 12 coloured pencils? class poet, and presenter of the class gift
be chosen from a class of 20 students?


15. Application If you have a standard deck of ACHIEVEMENT CHECK
52 cards, in how many different ways can Knowledge/ Thinking/Inquiry/
you deal out Understanding Problem Solving

a) 5 cards? b) 10 cards? 20. Wayne has a briefcase with a three-digit
c) 5 red cards? d) 4 queens? combination lock. He can set the
combination himself, and his favourite
16. Inquiry/Problem Solving Suppose you are digits are 3, 4, 5, 6, and 7. Each digit can
designing a coding system for data relayed be used at most once.
by a satellite. To make transmissions errors
a) How many permutations of three of
easier to detect, each code must have no
these five digits are there?
repeated digits.
b) If you think of each permutation as a
a) If you need 60 000 different codes, how
three-digit number, how many of these
many digits long should each code be?
numbers would be odd numbers?
b) How many ten-digit codes can you
c) How many of the three-digit numbers
create if the first three digits must be 1,
are even numbers and begin with a 4?
3, or 6?
d) How many of the three-digit numbers are
17. Arnold Schoenberg (1874 −1951) pioneered even numbers and do not begin with a 4?
serialism, a technique for composing music e) Is there a connection among the four
based on a tone row, a sequence in which answers above? If so, state what it is and
each of the 12 tones in an octave is played why it occurs.
only once. How many tone rows are possible?

18. Consider the students’ council described on C
pte
ha page 223 at the beginning of this chapter.
C

r

a) In how many ways can the secretary, 21. TI-83 series calculators use the definition
m
P

r

΂ − ᎏ2ᎏ΃! = ͙␲. Research the origin of this
oble 1
treasurer, social convenor, and ෆ
fundraising chair be elected if all ten
nominees are eligible for any of these definition and explain why it is useful for
positions? mathematical calculations.
b) In how many ways can the council be 22. Communication How many different ways
chosen if the president and vice- can six people be seated at a round table?
president must be grade 12 students and Explain your reasoning.
the grade representatives must represent
their current grade level? 23. What is the highest power of 2 that divides
evenly into 100! ?
19. Inquiry/Problem Solving A student has
volunteered to photograph the school’s 24. A committee of three teachers are to select
championship basketball team for the the winner from among ten students
yearbook. In order to get the perfect nominated for special award. The teachers
picture, the student plans to photograph the each make a list of their top three choices in
ten players and their coach lined up in every order. The lists have only one name in
possible order. Determine whether this plan common, and that name has a different rank
is practical. on each list. In how many ways could the
teachers have made their lists?


4.3 Permutations With Some Identical Items

Often, you will deal with permutations in which some items are identical.

I N V E S T I G AT E & I N Q U I R E : W h a t I s i n a N a m e ?

1. In their mathematics class,
John and Jenn calculate the
number of permutations of
all the letters of their first
names.
a) How many permutations
do you think John finds?
b) List all the permutations
of John’s name.
c) How many permutations
do you think Jenn finds?
d) List all the permutations
of Jenn’s name.
e) Why do you think there
are different numbers of
permutations for the two
names?
2. a) List all the permutations of the letters in your first name. Is the
number of permutations different from what you would calculate
using the nPn = n! formula? If so, explain why.
b) List and count all the permutations of a word that has two identical
pairs of letters. Compare your results with those your classmates
found with other words. What effect do the identical letters have
on the number of different permutations?
c) Predict how many permutations you could make with the letters in the
word googol. Work with several classmates to verify your prediction by
writing out and counting all of the possible permutations.
3. Suggest a general formula for the number of permutations of a word that
has two or more identical letters.

As the investigation above suggests, you can develop a general formula for
permutations in which some items are identical.

4.3 Permutations With Some Identical Items • MHR 241

Example 1 Permutations With Some Identical Elements

Compare the different permutations for the words DOLE, DOLL, and LOLL.

Solution

The following are all the permutations of DOLE :
DOLE DOEL DLOE DLEO DEOL DELO
ODLE ODEL OLDE OLED OEDL OELD
LODE LOED LDOE LDEO LEOD LEDO
EOLD EODL ELOD ELDO EDOL EDLO

There are 24 permutations of the four letters in DOLE. This number matches
what you would calculate using 4P4 = 4!

To keep track of the permutations of the letters in the word DOLL, use a
subscript to distinguish the one L from the other.
DOLL1 DOL1L DLO L1 DL L1O DL1OL DL1LO
ODLL1 ODL1L OLDL1 OLL1D OL1DL OL1LD
LODL1 LOL1D LDOL1 LDL1O LL1OD LL1DO
L1OLD L1ODL L1LOD L1LDO L1DOL L1DLO

Of the 24 arrangements listed here, only 12 are actually different from each
other. Since the two Ls are in fact identical, each of the permutations shown in
black is duplicated by one of the permutations shown in red. If the two Ls in a
permutation trade places, the resulting permutation is the same as the original
one. The two Ls can trade places in 2P2 = 2! ways.

Thus, the number of different arrangements is
4! 24
ᎏ=ᎏ
2! 2
= 12

In other words, to ﬁnd the number of permutations, you divide the total number
of arrangements by the number of ways in which you can arrange the identical
letters. For the letters in DOLL, there are four ways to choose the ﬁrst letter,
three ways to choose the second, two ways to choose the third, and one way to
choose the fourth. You then divide by the 2! or 2 ways that you can arrange the
two Ls.

Similarly, you can use subscripts to distinguish the three Ls in LOLL, and then
highlight the duplicate arrangements.
L2OLL1 L2OL1L L2LOL1 L2LL1O L2L1OL L2L1LO
OL2LL1 OL2L1L OLL2L1 OLL1L2 OL1L2L OL1LL2
LOL2L1 LOL1L2 LL2OL1 LL2L1O LL1OL2 LL1L2O
L1OLL2 L1OL2L L1LOL2 L1LL2O L1L2OL L1L2LO


The arrangements shown in black are the only different ones. As with the other
two words, there are 24 possible arrangements if you distinguish between the
identical Ls. Here, the three identical Ls can trade places in 3P3 = 3! ways.
4!
Thus, the number of permutations is ᎏ = 4.
3!

You can generalize the argument in Example 1 to show that the number of
n!
permutations of a set of n items of which a are identical is ᎏ .
a!

Example 2 Tile Patterns

Tanisha is laying out tiles for the edge of a mosaic. How many patterns can she
make if she uses four yellow tiles and one each of blue, green, red, and grey
tiles?

Solution

Here, n = 8 and a = 4.
8!
ᎏ =8×7×6×5
4!
= 1680
Tanisha can make 1680 different patterns with the eight tiles.

Example 3 Permutation With Several Sets of Identical Elements

The word bookkeeper is unusual in that it has three consecutive double letters.
How many permutations are there of the letters in bookkeeper?

Solution
If each letter were different, there would be 10! permutations, but there are two
os, two ks, and three es. You must divide by 2! twice to allow for the duplication
of the os and ks, and then divide by 3! to allow for the three es:
10! 10 × 9 × 8 × 7 × 6 × 5 × 4
ᎏ = ᎏᎏᎏ
2!2!3! 2×2
= 151 200
There are 151 200 permutations of the letters in bookkeeper.

The number of permutations of a set of n objects containing a identical
objects of one kind, b identical objects of a second kind, c identical objects
n!
of a third kind, and so on is ᎏᎏ.
a!b!c!…


Example 4 Applying the Formula for Several Sets of Identical Elements

Barbara is hanging a display of clothing imprinted with the school’s crest on a
line on a wall in the cafeteria. She has ﬁve sweatshirts, three T-shirts, and four
pairs of sweatpants. In how many ways can Barbara arrange the display?

Solution
Project
Here, a = 5, b = 3, c = 4, and the total number of items is 12. Prep

So, The game you
n! 12! design for your
ᎏ=ᎏ
a!b!c! 5!3!4! probability project
= 27 720 could involve
Barbara can arrange the display in 27 720 different ways. permutations of
identical objects.

Key Concepts

• When dealing with permutations of n items that include a identical items
of one type, b identical items of another type, and so on, you can use the
n!
formula ᎏ .
a!b!c!…


1. Explain why there are fewer permutations of a given number of items if some
of the items are identical.

2. a) Explain why the formula for the numbers of permutations when some items
are identical has the denominator a!b!c!… instead of a × b × c… .
b) Will there ever be cases where this denominator is larger than the
numerator? Explain.
c) Will there ever be a case where the formula does not give a whole number
answer? What can you conclude about the denominator and the numerator?


Practise 8. a) Calculate the number of permutations for
each of the jumbled words in this puzzle.
A b) Estimate how long it would take to solve
1. Identify the indistinguishable items in each this puzzle by systematically writing out
situation. the permutations.
a) The letters of the word mathematics are
arranged.
b) Dina has six notebooks, two green and
four white.
c) The cafeteria prepares 50 chicken
sandwiches, 100 hamburgers, and
70 plates of French fries.
d) Thomas and Richard, identical twins,
are sitting with Marianna and Megan.

2. How many permutations are there of all
the letters in each name?
a) Inverary b) Beamsville
c) Mattawa d) Penetanguishene

3. How many different five-digit numbers © Tribune Media Services, Inc. All Rights Reserved. Reprinted with Permission.

can be formed using three 2s and two 5s?

4. How many different six-digit numbers are
possible using the following numbers? www.mcgrawhill.ca/links/MDM12

a) 1, 2, 3, 4, 5, 6 b) 1, 1, 1, 2, 3, 4 For more word jumbles and other puzzles, visit
c) 1, 3, 3, 4, 4, 5 d) 6, 6, 6, 6, 7, 8 the above web site and follow the links. Find
or generate two puzzles for a classmate
Apply, Solve, Communicate to solve.

B 9. Application Roberta is a pilot for a small
5. Communication A coin is tossed eight times. airline. If she flies to Sudbury three times,
In how many different orders could five Timmins twice, and Thunder Bay five times
heads and three tails occur? Explain your before returning home, how many different
reasoning. itineraries could she follow? Explain your
reasoning.
6. Inquiry/Problem Solving How many 7-digit
even numbers less than 3 000 000 can be 10. After their training run, six members of a
formed using all the digits 1, 2, 2, 3, 5, 5, 6? track team split a bag of assorted doughnuts.
How many ways can the team share the
7. Kathryn’s soccer team played a good season,
doughnuts if the bag contains
finishing with 16 wins, 3 losses, and 1 tie. In
a) six different doughnuts?
how many orders could these results have
happened? Explain your reasoning. b) three each of two varieties?
c) two each of three varieties?


11. As a project for the photography class, 15. Ten students have been nominated for the
pte
Haseeb wants to create a linear collage ha positions of secretary, treasurer, social

C

r
of photos of his friends. He creates a convenor, and fundraising chair. In how

m
P
r
oble
template with 20 spaces in a row. If many ways can these positions be filled if
Haseeb has 5 identical photos of each the Norman twins are running and plan to
of 4 friends, in how many ways can he switch positions on occasion for fun since
make his collage? no one can tell them apart?

12. Communication A used car lot has four 16. Inquiry/Problem Solving In how many ways
green flags, three red flags, and two blue can all the letters of the word CANADA be
flags in a bin. In how many ways can the arranged if the consonants must always be
owner arrange these flags on a wire in the order in which they occur in the word
stretched across the lot? Explain your itself?
reasoning.
C
13. Application Malik wants to skateboard over 17. Glen works part time stocking shelves in a
to visit his friend Gord who lives six blocks grocery store. The manager asks him to
away. Gord’s house is two blocks west and make a pyramid display using 72 cans of
four blocks north of Malik’s house. Each corn, 36 cans of peas, and 57 cans of carrots.
time Malik goes over, he likes to take a Assume all the cans are the same size and
different route. How many different routes shape. On his break, Glen tries to work out
are there for Malik if he only travels west or how many different ways he could arrange
north? the cans into a pyramid shape with a
triangular base.
a) Write a formula for the number of
ACHIEVEMENT CHECK
different ways Glen could stack the
Knowledge/ Thinking/Inquiry/ cans in the pyramid.
b) Estimate how long it will take Glen to
14. Fran is working on a word puzzle and is
calculate this number of permutations
looking for four-letter “scrambles” from
by hand.
the clue word calculate.
c) Use computer software or a calculator
a) How many of the possible four-letter
to complete the calculation.
scrambles contain four different letters?
b) How many contain two as and one 18. How many different ways are there of
other pair of identical letters? arranging seven green and eight brown
c) How many scrambles consist of any
bottles in a row, so that exactly one pair
two pairs of identical letters? of green bottles is side-by-side?
d) What possibilities have you not yet 19. In how many ways could a class of
taken into account? Find the number 18 students divide into groups of
of scrambles for each of these cases. 3 students each?
e) What is the total number of four-letter
scrambles taking all cases into account?


4.4 Pascal’s Triangle

The array of numbers shown below is called Pascal’s
triangle in honour of French mathematician, Blaise
Pascal (1623−1662). Although it is believed that the
14th century Chinese mathematician Chu Shi-kie
knew of this array and some of its applications, Pascal
discovered it independently at age 13. Pascal found
many mathematical uses for the array, especially in
probability theory.

Pascal’s method for building his triangle is a simple
iterative process similar to those described in,
section 1.1. In Pascal’s triangle, each term is equal
to the sum of the two terms immediately above it.
The ﬁrst and last terms in each row are both equal
to 1 since the only term immediately above them is
also always a 1.

If tn,r represents the term in row n, position r, then
tn,r = tn-1,r-1 + tn-1,r .

For example, t6,2 = t5,1 + t5,2. Note that both the row
and position labelling begin with 0.
Chu Shi-kie’s triangle
1 Row 0 t0,0
1 1 Row 1 t1,0 t1,1
1 2 1 Row 2 t2,0 t2,1 t2,2
1 3 3 1 Row 3 t3,0 t3,1 t3,2 t3,3
1 4 6 4 1 Row 4 t4,0 t4,1 t4,2 t4,3 t4,4
1 5 10 10 5 1 Row 5 t5,0 t5,1 t5,2 t5,3 t5,4 t5,5
1 6 15 20 15 6 1 Row 6 t6,0 t6,1 t6,2 t6,3 t6,4 t6,5 t6,6


learn more about Pascal’s triangle. Write a brief
report about an application or an aspect of
Pascal’s triangle that interests you.

4.4 Pascal’s Triangle • MHR 247

I N V E S T I G AT E & I N Q U I R E : R o w S u m s

1. Find the sums of the numbers in each of the first six rows of Pascal’s
triangle and list these sums in a table.
2. Predict the sum of the entries in
a) row 7 b) row 8 c) row 9
3. Verify your predictions by calculating the sums of the numbers in rows
7, 8, and 9.
4. Predict the sum of the entries in row n of Pascal’s triangle.
5. List any other patterns you find in Pascal’s triangle. Compare your list
with those of your classmates. Do their lists suggest further patterns you
could look for?

In his book Mathematical Carnival, Martin Gardner describes Pascal’s triangle
as “so simple that a 10-year old can write it down, yet it contains such
inexhaustible riches and links with so many seemingly unrelated aspects of
mathematics, that it is surely one of the most elegant of number arrays.”

Example 1 Pascal’s Method

a) The first six terms in row 25 of Pascal’s triangle are 1, 25, 300, 2300,
12 650, and 53 130. Determine the first six terms in row 26.
b) Use Pascal’s method to write a formula for each of the following terms:
i) t12,5
ii) t40,32
iii) tn+1,r+1

Solution

a) t26,0 = 1 t26,1 = 1 + 25 t26,2 = 25 + 300
= 26 = 325
t26,3 = 300 + 2300 t26,4 = 2300 + 12 650 t26,5 = 12 650 + 53 130
= 2600 = 14 950 = 65 780

b) i) t12,5 = t11,4 + t11,5
ii) t40,32 = t39,31 + t39,32
iii) tn+1,r+1 = tn,r + tn,r+1


Example 2 Row Sums

Which row in Pascal’s triangle has the sum of its terms equal to 32 768?

Solution

From the investigation on page 248, you know that the sum of the
terms in any row n is 2n. Dividing 32 768 by 2 repeatedly, you ﬁnd that
32 768 = 215. Thus, it is row 15 of Pascal’s triangle that has terms totalling 32 768.

Example 3 Divisibility

Determine whether tn,2 is divisible by tn,1 in each row of Pascal’s triangle.

Solution
tn,2
Row ᎏ Divisible?
tn,1
0 and 1 n/a n/a
2 0.5 no
3 1 yes
4 1.5 no
5 2 yes
6 2.5 no
7 3 yes

It appears that tn,2 is divisible by tn,1 only in odd-numbered rows.
However, 2tn,2 is divisible by tn,1 in all rows that have three or more terms.

Example 4 Triangular Numbers

Coins can be arranged in the shape of an equilateral triangle as shown.

a) Continue the pattern to determine the numbers of coins in triangles
with four, ﬁve, and six rows.
b) Locate these numbers in Pascal’s triangle.
c) Relate Pascal’s triangle to the number of coins in a triangle with n rows.
d) How many coins are in a triangle with 12 rows?


Solution

a) The numbers of coins in the triangles follow the pattern 1 + 2 + 3 + … as
shown in the table below.

b) The numbers of coins in the triangles match the entries on the third
diagonal of Pascal’s triangle.
Number of Rows Number of Coins Term in Pascal’s Triangle 1
1 1 t2,2 1 1
2 3 t3,2 1 2 1
3 6 t4,2 1 3 3 1
1 41 6 4
4 10 t5,2
1 5 10 10 5 1
5 15 t6,2 1 6 15 20 15 6 1
6 21 t7,2 1 7 21 35 35 21 7 1

c) Compare the entries in the first and third columns of the table. The row
number of the term from Pascal’s triangle is always one greater than the
number of rows in the equilateral triangle. The position of the term in the
row, r, is always 2. Thus, the number of coins in a triangle with n rows is
equal to the term tn+1,2 in Pascal’s triangle.

d) t12+1,2 = t13,2
= 78
A triangle with 12 rows contains 78 coins.

Numbers that correspond to the number of items stacked in a triangular array
are known as triangular numbers. Notice that the nth triangular number is
also the sum of the first n positive integers.

Example 5 Perfect Squares

Can you find a relationship between perfect squares and the sums of pairs of
entries in Pascal’s triangle?

Solution
Again, look at the third diagonal in n n2 Entries in Pascal’s Triangle Terms in Pascal’s Triangle
Pascal’s triangle. 1 1 1 t2,2
2 4 1+3 t2,2 + t3,2
3 9 3+6 t3,2 + t4,2
4 16 6 + 10 t4,2 + t5,2

Each perfect square greater than 1 is equal to the sum of a pair of adjacent
terms on the third diagonal of Pascal’s triangle: n2 = tn,2 + tn+1,2 for n > 1.


Key Concepts

• Each term in Pascal’s triangle is equal to the sum of the two adjacent terms in
the row immediately above: tn,r = tn-1,r-1 + tn-1,r where tn,r represents the r th term
in row n.

• The sum of the terms in row n of Pascal’s triangle is 2n.

• Τhe terms in the third diagonal of Pascal’s triangle are triangular numbers.
Many other number patterns occur in Pascal’s triangle.


1. Describe the symmetry in Pascal’s triangle.

2. Explain why the triangular numbers in Example 4 occur in Pascal’s triangle.

A B
1. For future use, make a diagram of the ﬁrst 5. Inquiry/Problem Solving
12 rows of Pascal’s triangle. a) Alternately add and subtract the terms
in each of the ﬁrst seven rows of Pascal’s
2. Express as a single term from Pascal’s
triangle and list the results in a table
triangle.
similar to the one below.
a) t7,2 + t7,3
Row Sum/Difference Result
b) t51,40 + t51,41 0 1 1
c) t18,12 − t17,12 1 1−1 0
d) tn,r − tn-1,r 2 1−2+1 0
3 1 − 3 + 3 −1 0
3. Determine the sum of the terms in each of
Ӈ
these rows in Pascal’s triangle.
a) row 12 b) Predict the result of alternately adding
b) row 20
and subtracting the entries in the eighth
row. Verify your prediction.
c) row 25
c) Predict the result for the nth row.
d) row (n − 1)
6. a) Predict the sum of the squares of the
4. Determine the row number for each of the
terms in the nth row of Pascal’s triangle.
following row sums from Pascal’s triangle.
b) Predict the result of alternately adding
a) 256 b) 2048
and subtracting the squares of the terms
c) 16 384 d) 65 536 in the nth row of Pascal’s triangle.


7. Communication 11. Application Oranges can be piled in a
a) Compare the first four powers of 11 with tetrahedral shape as shown. The first pile
entries in Pascal’s triangle. Describe any contains one orange, the second contains
pattern you notice. four oranges, the third contains ten oranges,
and so on. The numbers of items in such
b) Explain how you could express row 5 as
stacks are known as tetrahedral numbers.
a power of 11 by regrouping the entries.
c) Demonstrate how to express rows 6 and
7 as powers of 11 using the regrouping
method from part b). Describe your
method clearly.

8. a) How many diagonals are there in
a) Relate the number of oranges in the nth
i) a quadrilateral? pile to entries in Pascal’s triangle.
ii) a pentagon? b) What is the 12th tetrahedral number?
iii) a hexagon?
12. a) Relate the sum of the squares of the first
b) Find a relationship between entries in
n positive integers to entries in Pascal’s
Pascal’s triangle and the maximum
triangle.
number of diagonals in an n-sided
polygon. b) Use part a) to predict the sum of the
squares of the first ten positive integers.
c) Use part b) to predict how many
Verify your prediction by adding the
diagonals are in a heptagon and an
numbers.
octagon. Verify your prediction by
drawing these polygons and counting the 13. Inquiry/Problem Solving A straight line
number of possible diagonals in each. drawn through a circle divides it into two
regions.
9. Make a conjecture about the divisibility
of the terms in prime-numbered rows a) Determine the maximum number of
of Pascal’s triangle. Confirm that your regions formed by n straight lines drawn
conjecture is valid up to row 11. through a circle. Use Pascal’s triangle to
help develop a formula.
10. a) Which rows of Pascal’s triangle contain
only odd numbers? Is there a pattern to
these rows?
b) Are there any rows that have only even
numbers?
c) Are there more even or odd entries in
Pascal’s triangle? Explain how you b) What is the maximum number of regions
arrived at your answer. inside a circle cut by 15 lines?

14. Describe how you would set up a
spreadsheet to calculate the entries in
Pascal’s triangle.


18. a) Write the first 20 rows of Pascal’s
C
triangle on a sheet of graph paper,
15. The Fibonacci sequence is 1, 1, 2, 3, 5, 8,
placing each entry in a separate square.
13, 21, … . Each term is the sum of the
previous two terms. Find a relationship b) Shade in all the squares containing
between the Fibonacci sequence and the numbers divisible by 2.
following version of Pascal’s triangle. c) Describe, in detail, the patterns
1 produced.
1 1 d) Repeat this process for entries divisible
1 2 1 by other whole numbers. Observe the
1 3 3 1 resulting patterns and make a conjecture
1 4 6 4 1 about the divisibility of the terms in
1 5 10 10 5 1 Pascal’s triangle by various whole
1 6 15 20 15 6 1
numbers.
1 7 21 35 35 21 7 1
… 19. Communication

16. Application Toothpicks are laid out to a) Describe the iterative process used to
form triangles as shown below. The first generate the terms in the triangle below.
triangle contains 3 toothpicks, the second 1
ᎏᎏ
contains 9 toothpicks, the third 1
contains 18 toothpicks, and so on. 1 1
ᎏᎏ ᎏᎏ
2 2
1 1 1
ᎏᎏ ᎏᎏ ᎏᎏ
3 6 3
1 1 1 1
ᎏᎏ ᎏᎏ ᎏᎏ ᎏᎏ
4 12 12 4
1 1 1 1 1
ᎏᎏ ᎏᎏ ᎏᎏ ᎏᎏ ᎏᎏ
a) Relate the number of toothpicks in the 5 20 30 20 5
nth triangle to entries in Pascal’s triangle. 1 1 1 1 1 1
ᎏᎏ ᎏᎏ ᎏᎏ ᎏᎏ ᎏᎏ ᎏᎏ
b) How many toothpicks would the 6 30 60 60 30 6
10th triangle contain?
b) Write the entries for the next two rows.
17. Design a 3-dimensional version of Pascal’s c) Describe three patterns in this triangle.
triangle. Use your own criteria for the
d) Research why this triangle is called the
layers. The base may be any regular
harmonic triangle. Briefly explain the
geometric shape, but each successive layer
origin of the name, listing your source(s).
must have larger dimensions than the one
above it.


4.5 Applying Pascal’s Method

The iterative process that generates the terms in Pascal’s triangle can also
be applied to counting paths or routes between two points. Consider
1
water being poured into the top bucket in the diagram. You can use
Pascal’s method to count the different paths that water overﬂowing
from the top bucket could take to each of the buckets in the
bottom row. 1 1

The water has one path to each of the buckets in the second
row. There is one path to each outer bucket of the third
2 1
row, but two paths to the middle bucket, and so on. 1

The numbers in the diagram match those in Pascal’s
triangle because they were derived using the same
1 3 3 1
method—Pascal’s method.

I N V E S T I G AT E & I N Q U I R E : C o u n t i n g R o u t e s

Suppose you are standing at the corner of Pythagoras Kovalevsky Avenue

Sierpinski Street
Street and Kovalevsky Avenue, and want to reach the

Germain Street
corner of Fibonacci Terrace and Euler Boulevard. To de Fermat Drive
avoid going out of your way, you would travel only

Euler Boulevard
Pythagoras Street

east and south. Notice that you could start out by

Gauss Street
going to the corner of either Euclid Street and Agnes Road
Kovalevsky Avenue or Pythagoras Street and de
Descartes Street
Euclid Street

Fermat Drive. Hypatia Street

1. How many routes are possible to the corner of
Euclid Street and de Fermat Drive from your Wiles Lane
starting point? Sketch the street grid and mark
the number of routes onto it. Fibonacci Terrace
2. a) Continue to travel only east or south. How
many routes are possible from the start to the corner of
i) Descartes Street and Kovalevsky Avenue?
ii) Pythagoras Street and Agnes Road?
iii) Euclid Street and Agnes Road?
iv) Descartes Street and de Fermat Drive?
v) Descartes Street and Agnes Road?
b) List the routes you counted in part a).


3. Consider your method and the resulting numbers. How do they relate to
Pascal’s triangle?
4. Continue to mark the number of routes possible on your sketch until you
have reached the corner of Fibonacci Terrace and Euler Boulevard. How
many different routes are possible?
5. Describe the process you used to ﬁnd the number of routes from Pythagoras
Street and Kovalevsky Avenue to Fibonacci Terrace and Euler Boulevard.

Example 1 Counting Paths in an Array

Determine how many different paths will spell PASCAL if you start at the
top and proceed to the next row by moving diagonally left or right.
P
A A
S S S
C C C C
A A A
L L

Solution
Starting at the top, record the number of possible paths moving diagonally P
to the left and right as you proceed to each different letter. For instance, 1
A A
1
there is one path from P to the left A and one path from P to the right A. 1 2 1
There is one path from an A to the left S, two paths from an A to the S S S
1 3 3 1
middle S, and one path from an A to the right S. C C C C
4 6 4
Continuing with this counting reveals that there are 10 different paths A A A
leading to each L. Therefore, a total of 20 paths spell PASCAL.
10 L L10

Example 2 Counting Paths on a Checkerboard

On the checkerboard shown, the checker can travel only diagonally upward.
It cannot move through a square containing an X. Determine the number of
paths from the checker’s current position to the top of the board.

x

4.5 Applying Pascal’s Method • MHR 255

Solution

Use Pascal’s method to find the number of paths to each successive 5 9 8 8
position. There is one path possible into each of the squares diagonally 5 4 4 4
adjacent to the checker’s starting position. From the second row there 1 4 x 4
are four paths to the third row: one path to the third square from the
1 3 3 1
left, two to the fifth square, and one to the seventh square. Continue
1 2 1
this process for the remaining four rows. The square containing an X
gets a zero or no number since there are no paths through this blocked 1 1
square.

From left to right, there are 5, 9, 8, and 8 paths to the white squares at
the top of the board, making a total of 30 paths.

Key Concepts

• Pascal’s method involves adding two neighbouring terms in order to find the
term below.

• Pascal’s method can be applied to counting paths in a variety of arrays and
grids.


1. Suggest a context in which you could apply Pascal’s method, other than those
in the examples above.

2. Which of the numbers along the perimeter of a map tallying possible routes
are always 1? Explain.

Practise 2. In the following arrangements of letters,
start from the top and proceed to the next
A row by moving diagonally left or right. How
1. Fill in the missing numbers using Pascal’s many different paths will spell each word?
method. a) P
495 A A
825 T T T
3003 2112 T T T T
E E E E E
R R R R R R
N N N N N N N
S S S S S S S S


b) M 5. Sung is three blocks east and five blocks
A A south of her friend’s home. How many
T T T different routes are possible if she walks
H H H H only west or north?
E E E E E
M M M M M M 6. Ryan lives four blocks north and five blocks
A A A A A A A west of his school. Is it possible for him to
T T T T T T take a different route to school each day,
I I I I I walking only south and east? Assume that
C C C C there are 194 school days in a year.
S S S
c) T 7. A checker is placed on a checkerboard as
R R shown. The checker may move diagonally
I I I upward. Although it cannot move into a
A A A A square with an X, the checker may jump over
N N N the X into the diagonally opposite square.
G G
L L L
E E E E
x
3. The first nine terms of a row of Pascal’s
x
triangle are shown below. Determine the first
nine terms of the previous and next rows.
1 16 120 560 1820 4368 8008 11 440 12 870

a) How many paths are there to the top of
Apply, Solve, Communicate the board?
B b) How many paths would there be if the
4. Determine the number of possible routes checker could move both diagonally and
from A to B if you travel only south or east. straight upward?
a) A
a) If a checker is placed as shown below,
how many possible paths are there for
that checker to reach the top of the game
B
board? Recall that checkers can travel
b) A
only diagonally on the white squares, one
square at a time, moving upward.

B
c) A

B 1 2 3 4


b) When a checker reaches the opposite 11. Communication A popular game show uses a
side, it becomes a “king.” If the starting more elaborate version of the Plinko board
squares are labelled 1 to 4, from left to shown below. Contestants drop a peg into
right, from which starting square does a one of the slots at the top of the upright
checker have the most routes to become board. The peg is equally likely to go left
a king? Verify your statement. or right at each post it encounters.

9. Application The following diagrams represent 1 2 3 4 5 6
communication networks between a
company’s computer centres in various cities.
Thunder Bay Charlottetown
Sudbury Halifax

North Bay
Toronto
Ottawa Montréal
Kitchener
Kingston $100 $1000 $0 $5000 $0 $1000 $100

Winnipeg
a) Into which slot should contestants drop
Hamilton
Windsor Saskatoon
their pegs to maximize their chances of
winning the $5000 prize? Which slot
Edmonton
gives contestants the least chance of
Vancouver
winning this prize? Justify your answers.
a) How many routes are there from
b) Suppose you dropped 100 pegs into the
Windsor to Thunder Bay?
slots randomly, one at a time. Sketch a
b) How many routes are there from graph of the number of pegs likely to wind
Ottawa to Sudbury? up in each compartment at the bottom of
c) How many routes are there from the board. How is this graph related to
Montréal to Saskatoon? those described in earlier chapters?
d) How many routes are there from
Vancouver to Charlottetown?
a) Build a new version of Pascal’s triangle,
e) If the direction were reversed, would the
using the formula for tn,r on page 247,
number of routes be the same for parts a)
but start with t0,0 = 2.
to d)? Explain.
b) Investigate this triangle and state a
10. To outfox the Big Bad Wolf, Little Red conjecture about its terms.
Riding Hood mapped all the paths through c) State a conjecture about the sum of the
the woods to Grandma’s house. How many terms in each row.
different routes could she take, assuming she
always travels from left to right? 13. Inquiry/Problem Solving Develop a formula
relating tn,r of Pascal’s triangle to the terms
Little Red
Riding Hood's in row n − 3.
House Grandma's
House


ACHIEVEMENT CHECK 17. Inquiry/Problem Solving Water is poured
into the top bucket of a triangular stack of
Communication Application 2-L buckets. When each bucket is full, the
water overflows equally on both sides into
14. The grid below shows the streets in Anya’s
the buckets immediately below. How much
neighbourhood. water will have been poured into the top
B bucket when at least one of the buckets in
the bottom row is full?
D

C

A

a) If she only travels east and north, how
many different routes can Anya take
from her house at intersection A to her
friend’s house at intersection B?
b) How many of the routes in part a) have
only one change of direction?
c) Suppose another friend lives at
intersection C. How many ways can A B C D E F
Anya travel from A to B, meeting her
friend at C along the way?
18. Application Is it possible to arrange a
d) How many ways can she travel to B pyramid of buckets such that the bottom
without passing through C? Explain layer will fill evenly when water overflows
your reasoning. from the bucket at the top of the pyramid?
e) If Anya takes any route from A to B, is she
more likely to pass through intersection C 19. Application Enya is standing in the centre
or D? Explain your reasoning. square of a 9 by 9 grid. She travels outward
one square at a time, moving diagonally or
along a row or column. How many different
paths can Enya follow to the perimeter?
C
15. Develop a general formula to determine the 20. Communication Describe how a chessboard
number of possible routes to travel n blocks path activity involving Pascal’s method is
north and m blocks west. related to network diagrams like those in
section 1.5. Would network diagrams for
16. Inquiry/Problem Solving In chess, a knight such activities be planar? Explain.
moves in L-shaped jumps consisting of two
squares along a row or column plus one
square at a right angle. On a standard 8 × 8
chessboard, the starting position for a knight
is the second square of the bottom row. If
the knight travels upward on every move,
how many routes can it take to the top of
the board?



4.1 Organized Counting 4.3 Permutations With Some Identical
Refer to the Key Concepts on page 228. Items
Refer to the Key Concepts on page 244.
1. A restaurant has a daily special with soup
or salad for an appetizer; fish, chicken, or a 8. How many different ten-digit telephone
vegetarian dish for the entrée; and cake, ice numbers contain four 2s, three 3s, and
cream, or fruit salad for dessert. Use a tree three 7s?
diagram to illustrate all the different meals
9. a) How many permutations are there of
possible with this special.
the letters in the word baseball?
2. A theatre company has a half-price offer for b) How many begin with the letter a?
students who buy tickets for at least three of c) How many end with the letter e?
the eight plays presented this season. How
many choices of three plays would a student 10. Find the number of 4 × 4 patterns you can
have? make using eight white, four grey, and four
blue floor tiles.
3. In how many different orders can a
photographer pose a row of six people
without having the tallest person beside 4.4 Pascal’s Triangle
the shortest one? Refer to the Key Concepts on page 251.

4. A transporter truck has three compact cars, a 11. Write out the first five rows of Pascal’s
station wagon, and a minivan on its trailer. triangle.
In how many ways can the driver load the
shipment so that one of the heavier vehicles 12. What is the sum of the entries in the
is directly over the rear axle of the trailer? seventh row of Pascal’s triangle?

13. Describe three patterns in Pascal’s triangle.
4.2 Factorials and Permutations
Refer to the Key Concepts on page 238. 4.5 Applying Pascal’s Method
5. For what values of n is n! less than 2 ? n Refer to the Key Concepts on page 256.
Justify your answer. 14. Explain why Pascal’s method can be
6. A band has recorded five hit singles. In how considered an iterative process.
many different orders could the band play 15. How many paths through S
three of these five songs at a concert? the array shown will spell I I
E E E
7. In how many ways could a chairperson, SIERPINSKI? R R
treasurer, and secretary be chosen from a P P P
I I I I
12-member board of directors? N N N
S S S S
K K K
I I


Chapter Test

ACHIEVEMENT CHART

Questions All 4, 7, 8 1, 3, 8 3, 4, 5, 6, 8

1. Natasha tosses four coins one after the other. 4. a) How many four-digit numbers can you
a) In how many different orders could form with the digits 1, 2, 3, 4, 5, 6, and 7
heads or tails occur. if no digit is repeated?
b) Draw a tree diagram to illustrate all the b) How many of these four-digit numbers
possible results. are odd numbers?
c) Explain how your tree diagram c) How many of them are even numbers?
corresponds to your calculation in part a).
5. How many ways are there to roll either a
2. Evaluate the following by ﬁrst expressing 6 or a 12 with two dice?
each in terms of factorials.
a) 15P6 b) P(6, 2) c) 7P3 6. How many permutations are there of the
letters of each of the following words?
d) 9P9 e) P(7, 0)
a) data b) management c) microwave
3. Suppose you are designing a remote control
that uses short, medium, or long pulses of 7. A number of long, thin sticks are lying in a
infrared light to send control signals to a pile at odd angles such that the sticks cross
device. each other.
a) How many different control codes can a) Relate the maximum number of
you deﬁne using intersection points of n sticks to entries
i) three pulses? in Pascal’s triangle.
ii) one, two, or three pulses? b) What is the maximum number of
b) Explain how the multiplicative and intersection points with six overlapping
additive counting principles apply in sticks?
your calculations for part a).

ACHIEVEMENT CHECK

8. At a banquet, four couples are sitting along one side of a table with men and women
alternating.
a) How many seating arrangements are possible for these eight people?
b) How many arrangements are possible if each couple sits together? Explain your reasoning.
c) How many arrangements are possible if no one is sitting beside his or her partner?
d) Explain why the answers from parts b) and c) do not add up to the answer from part a).


5
PT ER
Combinations and the
CHA

Binomial Theorem


Use Venn diagrams as a tool for organizing information in counting 5.1
problems.

Solve introductory counting problems involving the additive and 5.1, 5.2, 5.3
multiplicative counting principles.

Express answers to permutation and combination problems, using 5.1, 5.2, 5.3
standard combinatorial symbols.

Evaluate expressions involving factorial notation, using appropriate 5.2, 5.3
methods.

Solve problems, using techniques for counting combinations. 5.2, 5.3

Identify patterns in Pascal’s triangle and relate the terms of Pascal’s 5.4
triangle to values of ΂n΃, to the expansion of a binomial, and to the
r
solution of related problems.

Communicate clearly, coherently, and precisely the solutions to counting 5.1, 5.2, 5.3,
problems. 5.4

Chapter Problem
Radio Programming 2. In how many ways can he program the
Jeffrey works as a DJ at a local radio second hour if he chooses at least
station. He does the drive shift from 16 00 10 songs that are in positions 15 to 40
to 20 00, Monday to Friday. Before going on the charts?
on the air, he must choose the music he will
3. Over his 4-h shift, he will play at least
play during these four hours.
48 songs from the top 100. In how
The station has a few rules that Jeffrey many ways can he choose these songs?
must follow, but he is allowed quite a bit
In these questions, Jeffrey can play the
of leeway. Jeffrey must choose all his music
songs in any order. Such questions can be
from the top 100 songs for the week and he
answered with the help of combinatorics,
must play at least 12 songs an hour. In his
the branch of mathematics introduced in
ﬁrst hour, all his choices must be from the
Chapter 4. However, the permutations in
top-20 list.
Chapter 4 dealt with situations where the
1. In how many ways can Jeffrey choose order of items was important. Now, you
the music for his ﬁrst hour? will learn techniques you can apply in
situations where order is not important.



1. Factorials (section 4.2) Evaluate. 7. Exponent laws Use the exponent laws to
8! simplify each of the following.
a) 8! b) ᎏᎏ
5! a) (−3y)0 b) (−4x)3
24!
c) ᎏᎏ d) 3! × 4!
΂ ΃
1 5
22! c) 15(7x)4(4y)2 d) 21(x3)2 ᎏ2
x
2. Permutations (section 4.2) Evaluate mentally.
΂΃
1 4
e) (4x0y)2(3x2y)3 f) ᎏᎏ (3x2)(2y)3
a) 5 P5 b) 10
P2 2

΂΃
c) P1 d) 7 P3 1 0
12 g) (−3x y)(−5x2y)2 h) ᎏᎏ (−2xy)3
3
3. Permutations (section 4.2) Evaluate manually.
a) P5 b) P(16, 2) 8. Simplifying expressions Expand and simplify.
10
c) P10 d) P(8, 5) a) (x − 5)2 b) (5x − y)2
10
c) (x2 + 5)2 d) (x + 3)(x − 5)2
4. Permutations (section 4.2) Evaluate using
e) (x2 − y)2 f) (2x + 3)2
software or a calculator.
g) (x − 4)2(x − 2) h) (2x2 + 3y)2
a) P25 b) P(37, 16)
i) (2x + 1)2(x − 2) j) (x + y)(x − 2y)2
50
c) 29
P29 d) P
46 23
9. Sigma notation Rewrite the following
5. Organized counting (section 4.1) Every
using sigma notation.
Canadian aircraft has five letters in its
registration. The first letter must be C, the a) 1 + 2 + 4 + 8 + 16
second letter must be F or G, and the last b) x +2x2 +3x3 + 4x4 + 5x5
three letters have no restrictions. If repeated 1 1 1 1
c) ᎏ + ᎏ + ᎏ + ᎏ + …
letters are allowed, how many aircraft can be 2 3 4 5
registered with this system?
10. Sigma notation Expand.
5
6. Applying permutations (Chapter 4)
a) How many arrangements are there of
a) Α 2n
n=2
three different letters from the word 4
xn
kings? b) Αᎏ
n=1 n!
b) How many arrangements are there of 5
all the letters of the word management? c) Α (2
n=1
n
+ n2)
c) How many ways could first, second, and
third prizes be awarded to 12 entrants in
a mathematics contest?

264 MHR • Combinations and the Binomial Theorem

5.1 Organized Counting With Venn Diagrams

In Chapter 4, you used tree diagrams as a tool for counting items when the
order of the items was important. This section introduces a type of diagram that
helps you organize data about groups of items when the order of the items is
not important.

I N V E S T I G AT E & I N Q U I R E : V isualizing Relationships Between Groups

A group of students meet regularly to plan the dances at Vennville High School.
Amar, Belinda, Charles, and Danica are on the dance committee, and Belinda,
Charles, Edith, Franco, and Geoff are on the students’ council. Hans and Irena
are not members of either group, but they attend meetings as reporters for the
school newspaper.

1. Draw two circles to represent the
dance committee and the students’
council. Where on the diagram
would you put initials
representing the students who are
a) on the dance committee?
b) on the students’ council?
c) on the dance committee and
the students’ council?
d) not on either the dance
committee or the students’
council?
2. Redraw your diagram marking on
it the number of initials in each
region. What relationships can
you see among these numbers?

Your sketch representing the dance committee and the students’ council is a simple
example of a Venn diagram. The English logician John Venn (1834−1923)
introduced such diagrams as a tool for analysing situations where there is some
overlap among groups of items, or sets. Circles represent different sets and a
rectangular box around the circles represents the universal set, S, from which all
the items are drawn. This box is usually labelled with an S in the top left corner.

5.1 Organized Counting With Venn Diagrams • MHR 265

The items in a set are often called the elements or members of the set. The
size of a circle in a Venn diagram does not have to be proportional to the
number of elements in the set the circle represents. When some items in a set
are also elements of another set, these items are common elements and the sets
are shown as overlapping circles. If all elements of a set C are also elements of
set A, then C is a subset of A. A Venn diagram would show this set C as a
region contained within the circle for set A.

S A B


To learn more about Venn diagrams, visit the
above web site and follow the links. Describe an
Common elements example of how Venn diagrams can be used
of A and B to organize information.
The common elements are a subset of both A and B.

You can use Venn diagrams to organize information for situations in which the
number of items in a group are important but the order of the items is not.

Example 1 Common Elements

There are 10 students on the volleyball team and 15 on the basketball team.
When planning a ﬁeld trip with both teams, the coach has to arrange
transportation for a total of only 19 students.
a) Use a Venn diagram to illustrate this situation.
b) Explain why you cannot use the additive counting principle to ﬁnd the
total number of students on the teams.
c) Determine how many students are on both teams.
d) Determine the number of students in the remaining regions of your
diagram and explain what these regions represent.

Solution

a) Some students must be on both the volleyball and the basketball S VB BB
team. Draw a box with an S in the top left-hand corner. Draw
and label two overlapping circles to represent the volleyball and
basketball teams.


b) The additive counting principle (or rule of sum) applies only to mutually
exclusive events or items. However, it is possible for students to be on both
teams. If you simply add the 10 students on the volleyball team to 15
students on the basketball team, you get a total of 25 students because the
students who play on both teams have been counted twice.

c) The difference between the total in part b) and the total number of students
actually on the two teams is equal to the number of students who are
members of both teams. Thus, 25 − 19 = 6 students play on both teams. In
the Venn diagram, these 6 students are represented by the area where the
two circles overlap.

d) There are 10 − 6 = 4 students in the section of the VB circle that S VB BB
does not overlap with the BB circle. These are the students who
play only on the volleyball team. Similarly, the non-overlapping 4 9
portion of the BB circle represents the 15 − 6 = 9 students who 6
play only on the basketball team.

Example 1 illustrates the principle of inclusion and exclusion. If you are
counting the total number of elements in two groups or sets that have common
elements, you must subtract the common elements so that they are not included
twice.
Principle of Inclusion and Exclusion for Two Sets
For sets A and B, the total number of elements in either A or B is the number
in A plus the number in B minus the number in both A and B.

n(A or B) = n(A) + n(B) − n(A and B),
where n(X ) represents the numbers of elements in a set X.

The set of all elements in either set A or set B is the union of A and B, which is
often written as A ∪ B. Similarly, the set of all elements in both A and B is the
intersection of A and B, written as A ∩ B. Thus the principle of inclusion and
exclusion for two sets can also be stated as
n(A ∪ B) = n(A) + n(B) − n( A ∩ B)

Note that the additive counting principle (or rule of sum) could be considered
a special case of the principle of inclusion and exclusion that applies only when
sets A and B have no elements in common, so that n(A and B) = 0. The
principle of inclusion and exclusion can also be applied to three or more sets.


Example 2 Applying the Principle of Inclusion and Exclusion

A drama club is putting on two one-act plays. There are 11 actors in the
Feydeau farce and 7 in the Molière piece.
a) If 3 actors are in both plays, how many actors are there in all?
b) Use a Venn diagram to calculate how many students are in only one
of the two plays.

Solution

a) Calculate the number of students in both plays using the principle of
inclusion and exclusion.

n(total) = n(Feydeau) + n(Molière) − n(Feydeau and Molière)
= 11 + 7 − 3
= 15

There are 15 students involved in the two one-act plays.

b) There are 3 students in the overlap between the two circles. So,
S F M
there must be 11 − 3 = 8 students in the region for Feydeau only
and 7 − 3 = 4 students in the region for Molière only.
8 4
3
Thus, a total of 8 + 4 = 12 students are in only one of the two
plays.

As in the ﬁrst example, using a Venn diagram can clarify the relationships
between several sets and subsets.

Example 3 Working With Three Sets

Of the 140 grade 12 students at Vennville High School, 52 have signed up for
biology, 71 for chemistry, and 40 for physics. The science students include 15
who are taking both biology and chemistry, 8 who are taking chemistry and
physics, 11 who are taking biology and physics, and 2 who are taking all three
science courses.
a) How many students are not taking any of these three science courses?
b) Illustrate the enrolments with a Venn diagram.

Solution

a) Extend the principle of inclusion and exclusion to three sets. Total the
numbers of students in each course, subtract the numbers of students taking
two courses, then add the number taking all three. This procedure subtracts
out the students who have been counted twice because they are in two


courses, and then adds back those who were subtracted twice because they
were in all three courses.

For simplicity, let B stand for biology, C stand for chemistry, and P stand for
physics. Then, the total number of students taking at least one of these
three courses is

n(total) = n(B) + n(C) + n(P) − n(B and C) − n(C and P) − n(B and P) + n(B and C and P)
= 52 + 71 + 40 − 15 − 8 − 11 + 2
= 131

There are 131 students taking one or more of the three science courses.
To ﬁnd the number of grade 12 students who are not taking any of these
science courses, subtract 131 from the total number of grade 12 students.

Thus, 140 − 131 = 9 students are not taking any of these three science
courses in grade 12.

b) For this example, it is easiest to start with the overlap among the S B C
three courses and then work outward. Since there are 2 students
taking all three courses, mark 2 in the centre of the diagram where
the three circles overlap. 2

Next, consider the adjacent regions representing the students who
are taking exactly two of the three courses.
P
Biology and chemistry: Of the 15 students taking these two courses,
S B C
2 are also taking physics, so 13 students are taking only biology
and chemistry. 13
Chemistry and physics: 8 students less the 2 in the centre region 2
9 6
leaves 6.
Biology and physics: 11 − 2 = 9.
P
Now, consider the regions representing students taking only one
of the science courses.

Biology: Of the 52 students taking this course, 13 + 2 + 9 = 24 are S B C
in the regions overlapping with the other two courses, leaving
28 students who are taking biology only. 28 13 50
Chemistry: 71 students less the 13 + 2 + 6 leaves 50. 2
9 6
Physics: 40 − (9 + 2 + 6) = 23.
23
Adding all the numbers within the circles gives a total of 131. 9
P
Thus, there must be 140 − 131 = 9 grade 12 students who are
not taking any of the three science courses, which agrees with
the answer found in part a).


Key Concepts

• Venn diagrams can help you visualize the relationships between sets of items,
especially when the sets have some items in common.

• The principle of inclusion and exclusion gives a formula for ﬁnding the
number of items in the union of two or more sets. For two sets, the formula
is n(A or B) = n(A) + n(B) − n(A and B).


1. Describe the principal use of Venn diagrams.

2. Is the universal set the same for all Venn diagrams? Explain why or
why not.

3. Explain why the additive counting principle can be used in place of the
principle of inclusion and exclusion for mutually exclusive sets.

Practise c) List all subsets containing exactly two
elements for
A i) A
1. Let set A consist of an apple, an orange,
ii) B
and a pear and set B consist of the apple
and a banana. iii) A ∪ B

a) List the elements of 2. A recent survey of a group of students found
i) A and B that many participate in baseball, football,
ii) A or B and soccer. The Venn diagram below shows
the results of the survey.
iii) S
iv) S ∩ B S Baseball
v) A ∪ B ∪ S
Football 27 8 10
b) List the value of 6
3 4
i) n(A) + n(B)
19
ii) n(A or B) 5
Soccer
iii) n(S)
iv) n(A ∪ B )
v) n(S ∩ A )


a) How many students participated in the 5. Suppose the Canadian Embassy in the
survey? Netherlands has 32 employees, all of whom
b) How many of these students play both speak both French and English. In addition,
soccer and baseball? 22 of the employees speak German and 15
speak Dutch. If there are 10 who speak both
c) How many play only one sport?
German and Dutch, how many of the
d) How many play football and soccer? employees speak neither German nor
e) How many play all three sports? Dutch? Illustrate your answer with a Venn
f) How many do not play soccer? diagram.

Apply, Solve, Communicate 6. Application There are 900 employees at
CantoCrafts Inc. Of these, 615 are female,
B 345 are under 35 years old, 482 are single,
3. Of the 220 graduating students in a school, 295 are single females, 187 are singles
110 attended the semi-formal dance and 150 under 35 years old, 190 are females
attended the formal dance. If 58 students under 35 years old, and 120 are single
attended both events, how many graduating females under 35 years old. Use a Venn
students did not attend either of the two diagram to determine how many employees
dances? Illustrate your answer with a Venn are married males who are at least
diagram. 35 years old.

4. Application A survey of 1000 television 7. Communication A survey of 100 people who
viewers conducted by a local television volunteered information about their reading
station produced the following data: habits showed that
• 40% watch the news at 12 00 • 75 read newspapers daily
• 60% watch the news at 18 00 • 35 read books at least once a week
• 50% watch the news at 23 00 • 45 read magazines regularly
• 25% watch the news at 12 00 and at 18 00 • 25 read both newspapers and books
• 20% watch the news at 12 00 and at 23 00 • 15 read both books and magazines
• 20% watch the news at 18 00 and at 23 00 • 10 read newspapers, books, and
• 10% watch all three news broadcasts magazines
a) What percent of those surveyed watch at a) Construct a Venn diagram to determine
least one of these programs? the maximum number of people in the
b) What percent watch none of these news
survey who read both newspapers and
broadcasts? magazines.
b) Explain why you cannot determine
c) What percent view the news at 12 00 and
at 18 00, but not at 23 00? exactly how many of the people
surveyed read both newspapers and
d) What percent view only one of these
magazines.
shows?
e) What percent view exactly two of these
shows?


8. Jeffrey works as a DJ at a local radio station.
pte
C
ha On occasion, he chooses some of the songs
9. Inquiry/Problem Solving The Vennville
C

r

he will play based on the phone-in requests
m

junior hockey team has 12 members who
P

r
oble
received by the switchboard the previous
can play forward, 8 who can play defence,
day. Jeffrey’s list of 200 possible selections
and 2 who can be goalies. What is the
includes
smallest possible size of the team if
• all the songs in the top 100
a) no one plays more than one position?
• 134 hard-rock songs
• 50 phone-in requests b) no one plays both defence and forward?
• 45 hard-rock songs in the top 100 c) three of the players are able to play
• 20 phone-in requests in the top 100 defence or forward?
• 24 phone-in requests for hard-rock songs d) both the goalies can play forward but not
Use a Venn diagram to determine defence?
a) how many phone-in requests were for
10. Inquiry/Problem Solving Use the principle of
hard-rock songs in the top 100
inclusion and exclusion to develop a formula
b) how many of the songs in the top 100 for the number of elements in
were neither phone-in requests nor hard-
a) three sets b) four sets c) n sets
rock selections

Career Connection
Forensic Scientist
The field of forensic science could be attractive to those with a
mathematics and science background. The job of a forensic scientist is to
identify, analyse, and match items collected from crime scenes.

Forensic scientists most often work in a forensic laboratory. Such
laboratories examine and analyse physical evidence, including controlled
substances, biological materials, firearms and ammunition components,
and DNA samples.

Forensic scientists may have specialities such as fingerprints, bullistics,
clothing and fibres, footprints, tire tracks,
DNA profiling, or crime scene analysis.

Modern forensic science combines www.mcgrawhill.ca/links/MDM12
mathematics and computers. A forensic
scientist should have a background in For more information about forensic science
combinatorics, biology, and the physical and other careers related to mathematics, visit
the above web site and follow the links. Write a
sciences. Forensic scientists work for a wide
brief description of how combinatorics could be
variety of organizations including police forces,
used by forensic scientists.
government offices, and the military.


5.2 Combinations

In Chapter 4, you learned about permutations—arrangements in which the
order of the items is specified. However, in many situations, order does not
matter. For example, in many card games, what is in your hand is important, but
the order in which it was dealt is not.

I N V E S T I G AT E & I N Q U I R E : S t u d e n t s ’ C o u n c i l

Suppose the students at a secondary school elect a council of eight
members, two from each grade. This council then chooses two of its
members as co-chairpersons. How could you calculate the number of
different pairs of members who could be chosen as the co-chairs?

Choose someone in the class to record your answers to the following
questions on a blackboard or an overhead projector.

a) Start with the simplest case. Choose two students to stand at the front
of the class. In how many ways can you choose two co-chairs from this
pair of students?
b) Choose three students to be at the front of the class. In how many ways
can you choose two co-chairs from this trio?
c) In how many ways can you choose two co-chairs from a group of four
students?
d) In how many ways can you choose two
co-chairs from a group of five students?
Do you see a pattern developing? If so,
what is it? If not, try choosing from a
group of six students and then from a
group of seven students while
continuing to look for a pattern.
e) When you see a pattern, predict the
number of ways two co-chairs can be
chosen from a group of eight students.
f) Can you suggest how you could find
the answers for this investigation from
the numbers of permutations you
found in the investigation in section
4.2?

5.2 Combinations • MHR 273

In the investigation on the previous page, you were dealing with a situation in
which you were selecting two people from a group, but the order in which you
chose the two did not matter. In a permutation, there is a difference between
selecting, say, Bob as president and Margot as vice-president as opposed to
selecting Margot as president and Bob as vice-president. If you select Bob and
Margot as co-chairs, the order in which you select them does not matter since
they will both have the same job.

A selection from a group of items without regard to order is called a
combination.

Example 1 Comparing Permutations and Combinations

a) In how many ways could Alana, Barbara, Carl, Domenic, and Edward fill
the positions of president, vice-president, and secretary?
b) In how many ways could these same five people form a committee with
three members? List the ways.
c) How are the numbers of ways in parts a) and b) related?

Solution

a) Since the positions are different, order is important. Use a permutation, nPr.
There are five people to choose from, so n = 5. There are three people
being chosen, so r = 3. The number of permutations is 5P3 = 60.

There are 60 ways Alana, Barbara, Carl, Domenic, and Edward could fill
the positions of president, vice-president, and secretary.

b) The easiest way to find all committee combinations is to write them in an
ordered fashion. Let A represent Alana, B represent Barbara, C represent
Carl, D represent Domenic, and E represent Edward.

The possible combinations are:
ABC ABD ABE ACD ACE
ADE BCD BCE BDE CDE

All other possible arrangements include the same three people as one of the
combinations listed above. For example, ABC is the same as ACB, BAC,
BCA, CAB, and CBA since order is not important.

So, there are only ten ways Alana, Barbara, Carl, Domenic and Edward can
form a three-person committee.


c) In part a), there were 60 possible permutations, while in part b), there were
10 possible combinations. The difference is a factor of 6. This factor is
P = 3!, the number of possible arrangements of the three people in each
3 3
combination. Thus,
number of permutations
number of combinations = ᎏᎏᎏᎏᎏᎏ
number of permutations of the objects selected
P
= ᎏ5 3
3!
60
= ᎏ
6
= 10

Combinations of n distinct objects taken r at a time
The number of combinations of r objects chosen from a set of n distinct objects is
P
n
Cr = ᎏ
n r
r!
n!
ᎏᎏ
(n − r)!
= ᎏᎏ
r!
n!
= ᎏ
(n − r)!r!

The notations nCr, C(n, r), and ΂ n ΃ are all equivalent. Many people prefer the form
r
n when a number of combinations are multiplied together. The symbol C is used
΂r΃ n r

most often in this text since it is what appears on most scientific and graphing calculators.

Example 2 Applying the Combinations Formula
How many different sampler dishes with 3 different flavours could you get
at an ice-cream shop with 31 different flavours?

Solution
There are 31 flavours, so n = 31. The sampler dish has 3 flavours, so r = 3.
31!
C3 = ᎏᎏ
31
(31 − 3)! 3
31!
= ᎏᎏ
28!3!
31 × 30 × 29
= ᎏᎏ
3×2
= 4495
There are 4495 possible sampler combinations.


Note that the number of combinations in Example 2 was easy to calculate
because the number of items chosen, r, was quite small.

Example 3 Calculating Numbers of Combinations Manually

A ballet choreographer wants 18 dancers for a scene.
a) In how many ways can the choreographer choose the dancers if the
company has 20 dancers? 24 dancers?
b) How would you advise the choreographer to choose the dancers?

Solution

a) When n and r are close in value, nCr can be calculated mentally.
With n = 20 and r = 18,
20!
C18 = ᎏᎏ
20
(20 − 18)!18!
20 × 19
= ᎏᎏ 20 ÷ 2 = 10
2!
= 190 Then, 10 × 19 = 190

The choreographer could choose from 190 different combinations of the
20 dancers.

With n = 24 and r = 18, nCr can be calculated manually or with a basic
calculator once you have divided out the common terms in the factorials.
24!
C18 = ᎏᎏ
24
(24 − 18)!18!
24 × 23 × 22 × 21 × 20 × 19
= ᎏᎏᎏ
6!
24 × 23 × 22 × 21 × 20 × 19
= ᎏᎏᎏ
6×5×4×3×2×1
= 23 × 11 × 7 × 4 × 19
= 134 596

With the 4 additional dancers, the choreographer now has a choice of
134 596 combinations.

b) From part a), you can see that it would be impractical for the choreographer
to try every possible combination. Instead the choreographer could use an
indirect method and try to decide which dancers are least likely to be
suitable for the scene.


Even though there are fewer permutations of n objects than there are combinations,
the numbers of combinations are often still too large to calculate manually.

Example 4 Using Technology to Calculate Numbers of Combinations

Each player in a bridge game receives a hand of 13 cards dealt from a standard For details of
deck. How many different bridge hands are possible? calculator and
software functions,
Solution 1 Using a Graphing Calculator refer to Appendix B.

Here, the order in which the player receives the cards does not matter. What
you want to determine is the number of different combinations of cards a player
could have once the dealing is complete. So, the answer is simply 52C13. You
can evaluate 52C13 by using the nCr function on the MATH PRB menu of a
graphing calculator. This function is similar to the nPr function used for
permutations.

There are about 635 billion possible bridge hands.


Most spreadsheet programs have a combinations function for calculating numbers of
combinations. In Microsoft® Excel, this function is the COMBIN(n,r) function. In
Corel® Quattro® Pro, this function is the COMB(r,n) function.

You now have a variety of methods for finding numbers of combinations—
Project
paper-and-pencil calculations, factorials, scientific or graphing calculators, and
Prep
software. When appropriate, you can also apply both of the counting
principles described in Chapter 4. Techniques for
calculating
Example 5 Using the Counting Principles With Combinations numbers of
combinations
Ursula runs a small landscaping business. She has on hand 12 kinds of rose could be helpful
bushes, 16 kinds of small shrubs, 11 kinds of evergreen seedlings, and 18 kinds for designing the
of flowering lilies. In how many ways can Ursula fill an order if a customer game in your
wants probability project,
a) 15 different varieties consisting of 4 roses, 3 shrubs, 2 evergreens, especially if your
and 6 lilies? game uses cards.
b) either 4 different roses or 6 different lilies?


Solution

a) The order in which Ursula chooses the plants does not matter.

The number of ways of choosing the roses is 12C4.
The number of ways of choosing the shrubs is 16C3.
The number of ways of choosing the evergreens is 11C2.
The number of ways of choosing the lilies is 18C6.

Since varying the rose selection for each different selection of the shrubs,
evergreens, and lilies produces a different choice of plants, you can apply
the fundamental (multiplicative) counting principle. Multiply the series of
combinations to find the total number of possibilities.

12
C4 × 16C3 × 11C2 × 18C6 = 495 × 560 × 55 × 18 564
= 2.830 267 44 × 1011

Ursula has over
283 billion ways
of choosing the
plants for her
customer.

b) Ursula can choose the 4 rose bushes in 12
C4 ways.

She can choose the 6 lilies in 18C6 ways.

Since the customer wants either the rose bushes or the lilies, you can apply
the additive counting principle to find the total number of possibilities.

12
C4 + 18C6 = 495 + 18 564
= 19 059

Ursula can fill the order for either roses or lilies in 19 059 ways.

As you can see, even relatively simple situations can produce very large
numbers of combinations.

Key Concepts

• A combination is a selection of objects in which order is not important.

• The number of combinations of n distinct objects taken r at a time is denoted
n!
as nCr , C(n, r), or ΂ n ΃ and is equal to ᎏᎏ .
r (n − r)! r!
• The multiplicative and additive counting principles can be applied to problems
involving combinations.



1. Explain why n objects have more possible permutations than combinations.
Use a simple example to illustrate your explanation.

2. Explain whether you would use combinations, permutations, or another
method to calculate the number of ways of choosing
a) three items from a menu of ten items
b) an appetizer, an entrée, and a dessert from a menu with three appetizers,
four entrées, and ﬁve desserts

3. Give an example of a combination expression you could calculate
a) by hand
b) algebraically
c) only with a calculator or computer

Practise 4. How many ways can 4 cards be chosen from
a deck of 52, if the order in which they are
A chosen does not matter?
1. Evaluate using a variety of methods.
Explain why you chose each method. 5. How many groups of 3 toys can a child
choose to take on a vacation from a toy box
a) C19 b) C28
21 30 containing 11 toys?
c) 18
C5 d) 16
C3
e) C4 f) C20 6. How many sets of 6 questions for a test can
19 25
be chosen from a list of 22 questions?
2. Evaluate the following pairs of combinations
and compare their values. 7. In how many ways can a teacher select
5 students from a class of 23 to make a
a) C1, 11C10
11 bulletin-board display? Explain your
b) 11
C2, 11C9 reasoning.
c) 11
C3, 11C8
8. As a promotion, a video store decides to give
Apply, Solve, Communicate away posters for recently released movies.
a) If posters are available for 27 recent
B releases, in how many ways could the
3. Communication In how many ways could you video-store owner choose 8 different
choose 2 red jellybeans from a package of posters for the promotion?
15 red jellybeans? Explain your reasoning.
b) Are you able to calculate the number
of ways mentally? Why or why not?


9. Communication A club has 11 members. 14. Jeffrey, a DJ at a local radio station, is
pte
ha choosing the music he will play on his shift.
a) How many different 2-member

C

r
committees could the club form? He must choose all his music from the top

m
P
r
oble
100 songs for the week and he must play at
b) How many different 3-member
least 12 songs an hour. In his first hour, all
committees could the club form?
his choices must be from the top-20 list.
c) In how many ways can a club president,
a) In how many ways can Jeffrey choose the
treasurer, and secretary be chosen?
songs for his first hour if he wants to
d) By what factor do the answers in parts b) choose exactly 12 songs?
and c) differ? How do you account for
b) In how many ways can Jeffrey choose
this difference?
the 12 songs if he wants to pick 8 of the
10. Fritz has a deck of 52 cards, and he may top 10 and 4 from the songs listed from
choose any number of these cards, from 11 to 20 on the chart?
none to all. Use a spreadsheet or Fathom™ c) In how many ways can Jeffrey choose
to calculate and graph the number of either 12 or 13 songs to play in the first
combinations for each of Fritz’s choices. hour of his shift?
d) In how many ways can Jeffrey choose the
11. Application A track club, a swim club, and a
songs if he wants to play up to 15 songs
cycling club are forming a joint committee
in the first hour?
to organize a triathlon. The committee will
have two members from each club. In how 15. The game of euchre uses only 24 of the
many ways can the committee be formed if cards from a standard deck. How many
ten runners, eight swimmers, and seven different five-card euchre hands are
cyclists volunteer to serve on it? possible?
12. In how many ways can a jury of 6 men and 16. Application A taxi is shuttling 11 students
6 women be chosen from a group of 10 men to a concert. The taxi can hold only 4
and 15 women? students. In how many ways can 4 students
be chosen for
13. Inquiry/Problem Solving There are
15 technicians and 11 chemists working in a) the taxi’s first trip?
a research laboratory. In how many ways b) the taxi’s second trip?
could they form a 5-member safety
committee if the committee 17. Diane is making a quilt. She needs three
pieces with a yellow undertone, two pieces
a) may be chosen in any way?
with a blue undertone, and four pieces with
b) must have exactly one technician? a white undertone. If she has six squares
c) must have exactly one chemist? with a yellow undertone, five with a blue
d) must have exactly two chemists? undertone, and eight with a white undertone
to choose from, in how many ways can she
e) may be all technicians or all chemists?
choose the squares for the quilt?


18. Inquiry/Problem Solving At a family reunion, 20. In the game of bridge, each player is dealt
everyone greets each other with a a hand of 13 cards from a standard deck of
handshake. If there are 20 people at the 52 cards.
reunion, how many handshakes take place? a) By what factor does the number of
possible bridge hands differ from the
number of ways a bridge hand could be
ACHIEVEMENT CHECK dealt to a player? Explain your reasoning.
b) Use combinations to write an expression
Understanding Problem Solving for the number of bridge hands that have
19. A basketball team consists of five players— exactly five clubs, two spades, three
one centre, two forwards, and two guards. diamonds, and three hearts.
The senior squad at Vennville Central c) Use combinations to write an expression
High School has two centres, six forwards, for the number of bridge hands that have
and four guards. exactly five hearts.
a) How many ways can the coach pick the d) Use software or a calculator to evaluate
two starting guards for a game? the expressions in parts b) and c).
b) How many different starting lineups are C
possible if all team members play their 21. There are 18 students involved in the class
specified positions? production of Arsenic and Old Lace.
c) How many of these starting lineups a) In how many ways can the teacher cast
include Dana, the team’s 185-cm the play if there are five male roles and
centre? seven female roles and the class has nine
d) Some coaches designate the forwards as male and nine female students?
power forward and small forward. If all b) In how many ways can the teacher cast
six forwards are adept in either position, the play if Jean will play the young
how would this designation affect the female part only if Jovane plays the male
number of possible starting lineups? lead?
e) As the league final approaches, the c) In how many ways can the teacher cast
centre Dana, forward Ashlee, and guard the play if all the roles could be played
Hollie are all down with a nasty flu. by either a male or a female student?
Fortunately, the five healthy forwards
can also play the guard position. If the 22. A large sack contains six basketballs and
coach can assign these players as either five volleyballs. Find the number of
forwards or guards, will the number of combinations of four balls that can be
possible starting lineups be close to the chosen from the sack if
number in part b)? Support your answer a) they may be any type of ball
mathematically. b) two must be volleyballs and two must
f) Is the same result achieved if the be basketballs
forwards play their regular positions but c) all four must be volleyballs
the guards can play as either forwards
d) none may be volleyballs
or guards? Explain your answer.


5.3 Problem Solving With Combinations

In the last section, you considered the number of ways of choosing r items from
a set of n distinct items. This section will examine situations where you want to
know the total number of possible combinations of any size that you could
choose from a given number of items, some of which may be identical.

I N V E S T I G AT E & I N Q U I R E : Combinations of Coins

1. a) How many different sums of money can you create with a
penny and a nickel? List these sums.
b) How many different sums can you create
with a penny, a nickel, and a dime? List
them.
c) Predict how many different sums you
can create with a penny, a nickel,
a dime, and a quarter. Test your
conjecture by listing the possible sums.
2. a) How many different sums of money
can you create with two pennies
and a dime? List them.
b) How many different sums can you
create with three pennies and a dime?
c) Predict how many sums you can create with
four pennies and a dime. Test your conjecture.
Can you see a pattern developing? If so, what is it?

Example 1 All Possible Combinations of Distinct Items

An artist has an apple, an orange, and a pear in his refrigerator. In how many
ways can the artist choose one or more pieces of fruit for a still-life painting?

Solution
The artist has two choices for each piece of fruit: either include it in the
painting or leave it out. Thus, the artist has a total of 2 × 2 × 2 = 8 choices.
Note that one of these choices is to leave out the apple, the orange, and the
pear. However, the artist wants at least one piece of fruit in his painting.
Thus, he has 23 − 1 = 7 combinations to choose from.


You can apply the same logic to any group of distinct items.

The total number of combinations containing at least one item chosen
from a group of n distinct items is 2n − 1.

Remember that combinations are subsets of the group of n objects.
A null set is a set that has no elements. Thus,

A set with n distinct elements has 2n subsets including the null set.

Example 2 Applying the Formula for Numbers of Subsets

In how many ways can a committee with at least one member be appointed
from a board with six members?

Solution

The board could choose 1, 2, 3, 4, 5, or 6 people for the committee, so n = 6.
Since the committee must have at least one member, use the formula that
excludes the null set.

26 − 1 = 64 − 1
= 63

There are 63 ways to choose a committee of at least one person from a
six-member board.

Example 3 All Possible Combinations With Some Identical Items

Kate is responsible for stocking the coffee room at her ofﬁce. She can purchase
up to three cases of cookies, four cases of soft drinks, and two cases of coffee
packets without having to send the order through the accounting department.
How many different direct purchases can Kate make?

Solution

Kate can order more than one of each kind of item, so this situation involves
combinations in which some items are alike.
• Kate may choose to buy three or two or one or no cases of cookies, so she
has four ways to choose cookies.
• Kate may choose to buy four or three or two or one or no cases of soft
drinks, so she has ﬁve ways to choose soft drinks.
• Kate may choose to buy two or one or no cases of coffee packets, so she has
three ways to choose coffee.

5.3 Problem Solving With Combinations • MHR 283

Cookies 0 1 2 3

Soft Drinks 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4

Coffee 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2

As shown on the first branch of the diagram above, one of these choices is
purchasing no cookies, no soft drinks, and no coffee. Since this choice is not
a purchase at all, subtract it from the total number of choices.

Thus, Kate can make 4 × 5 × 3 − 1 = 59 different direct purchases.

In a situation where you can choose all, some, or none of the p items available,
you have p + 1 choices. You can then apply the fundamental (multiplicative)
counting principle if you have successive choices of different kinds of items.
Always consider whether the choice of not picking any items makes sense. If it
does not, subtract 1 from the total.

Combinations of Items in Which Some are Alike

If at least one item is chosen, the total number of selections that can be
made from p items of one kind, q items of another kind, r items of
another kind, and so on, is ( p + 1)(q + 1)(r + 1) … − 1

Having identical elements in a set reduces the number of possible combinations
when you choose r items from that set. You cannot calculate this number by
simply dividing by a factorial as you did with permutations in section 4.3. Often,
you have to consider a large number of cases individually. However, some
situations have restrictive conditions that make it much easier to count the
number of possible combinations.

Example 4 Combinations With Some Identical Items

The director of a short documentary has found five rock songs, two blues tunes,
and three jazz pieces that suit the theme of the film. In how many ways can the
director choose three pieces for the soundtrack if she wants the film to include
some jazz?


Solution 1 Counting Cases

The director can select exactly one, two, or three jazz pieces.
Case 1: One jazz piece
The director can choose the one jazz piece in 3C1 ways and two of
the seven non-jazz pieces in 7C2 ways. Thus, there are 3C1 × 7C2 = 63
combinations of music with one jazz piece.
Case 2: Two jazz pieces
The director can choose the two jazz pieces in 3C2 ways and one of
the seven non-jazz pieces in 7C1 ways. There are 3C2 × 7C1 = 21
combinations with two jazz pieces.
Case 3: Three jazz pieces
The director can choose the three jazz pieces and none of the seven
non-jazz pieces in only one way: 3C3 × 7C0 = 1.

The total number of combinations with at least one jazz piece is 63 + 21 + 1 = 85.

Solution 2 Indirect Method
You can find the total number of possible combinations of three pieces of music
and subtract those that do not have any jazz.

The total number of ways of choosing any three pieces from the ten available is
C = 120. The number of ways of not picking any jazz, that is, choosing only
10 3
from the non-jazz pieces is 7C3 = 35.

Thus, the number of ways of choosing at least one jazz piece is 120 − 35 = 85.

Here is a summary of ways to approach questions involving choosing or
selecting objects.
Is order important?
Yes: Use permutations. Can the same No: Use combinations. Are you choosing exactly
objects be selected more than once r objects?
(like digits for a telephone number)? Yes: Could some of the objects be identical?
Yes: Use the fundamental counting Yes: Count the individual cases.
principle. n!
No: Use nCr = ᎏ
No: Are some of the objects identical? (n − r)!r!
n! No: Are some of the objects identical?
Yes: Use the formula ᎏ .
a!b!c!… Yes: Use ( p + 1)(q + 1)(r + 1) − 1 to find
n! the total number of combinations
No: Use nPr = ᎏ . with at least one object.
(n − r)!
No: Use 2n to find the total number of
combinations; subtract 1 if you do
not want to include the null set.


Key Concepts

• Use the formula ( p + 1)(q + 1)(r + 1) … − 1 to find the total number of
selections of at least one item that can be made from p items of one kind,
q of a second kind, r of a third kind, and so on.

• A set with n distinct elements has 2n subsets including the null set.

• For combinations with some identical elements, you often have to consider
all possible cases individually.

• In a situation where you must choose at least one particular item, either
consider the total number of choices available minus the number without the
desired item or add all the cases in which it is possible to have the desired
item.


1. Give an example of a situation where you would use the formula
( p + 1)(q + 1)(r + 1) … − 1. Explain why this formula applies.

2. Give an example of a situation in which you would use the expression 2n − 1.

3. Using examples, describe two different ways to solve a problem where at least one
particular item must be chosen. Explain why both methods give the same answer.

Practise 4. In how many ways can a committee with
eight members form a subcommittee with
A at least one person on it?
1. How many different sums of money can you
make with a penny, a dime, a one-dollar B
coin, and a two-dollar coin? 5. Determine whether the following questions
involve permutations or combinations and
2. How many different sums of money can be list any formulas that would apply.
made with one $5 bill, two $10 bills, and a) How many committees of 3 students can
one $50 bill? be formed from 12 students?
3. How many subsets are there for a set with b) In how many ways can 12 runners finish
a) two distinct elements? first, second, and third in a race?
b) four distinct elements? c) How many outfits can you assemble from
c) seven distinct elements? three pairs of pants, four shirts, and two
pairs of shoes?
d) How many two-letter arrangements can
be formed from the word star?


Apply, Solve, Communicate 11. The number 5880 can be factored into
prime divisors as 2 × 2 × 2 × 3 × 5 × 7 × 7.
6. Seven managers and eight sales representatives
a) Determine the total number of divisors
volunteer to attend a trade show. Their of 5880.
company can afford to send five people. In
b) How many of the divisors are even?
how many ways could they be selected
c) How many of the divisors are odd?
a) without any restriction?
b) if there must be at least one manager and 12. Application A theme park has a variety of
one sales representative chosen? rides. There are seven roller coasters, four
water rides, and nine story rides. If
7. Application A cookie jar contains three
Stephanie wants to try one of each type
chocolate-chip, two peanut-butter, one of ride, how many different combinations
lemon, one almond, and five raisin cookies. of rides could she choose?
a) In how many ways can you reach into
the jar and select some cookies? 13. Shuwei finds 11 shirts in his size at a
b) In how many ways can you select some
clearance sale. How many different
cookies, if you must include at least one purchases could Shuwei make?
chocolate-chip cookie? 14. Communication Using the summary on
8. A project team of 6 students is to be selected
page 285, draw a flow chart for solving
from a class of 30. counting problems.
a) How many different teams can be selected? 15. a) How many different teams of 4 students
b) Pierre, Gregory, and Miguel are students could be chosen from the 15 students in
in this class. How many of the teams the grade-12 Mathematics League?
would include these 3 students? b) How many of the possible teams would
c) How many teams would not include include the youngest student in the league?
Pierre, Gregory, and Miguel? c) How many of the possible teams would
exclude the youngest student?
9. The game of euchre uses only the 9s, 10s,
jacks, queens, kings, and aces from a standard 16. Inquiry/Problem Solving
deck of cards. How many five-card hands have a) Use combinations to determine how
a) all red cards? many diagonals there are in
b) at least two red cards? i) a pentagon ii) a hexagon
c) at most two red cards? b) Draw sketches to verify your answers in
part a).
10. If you are dealing from a standard deck of
52 cards, 17. A school is trying to decide on new school
a) how many different 4-card hands could colours. The students can choose three
have at least one card from each suit? colours from gold, black, green, blue, red,
b) how many different 5-card hands could
and white, but they know that another
have at least one spade? school has already chosen black, gold, and
red. How many different combinations of
c) how many different 5-card hands could
three colours can the students choose?
have at least two face cards (jacks,
queens, or kings?

18. Application The social convenor has 21. Communication
12 volunteers to work at a school dance. a) How many possible combinations are
Each dance requires 2 volunteers at the there for the letters in the three circles
door, 4 volunteers on the floor, and for each of the clue words in this puzzle?
6 floaters. Joe and Jim have not volunteered
b) Explain why you cannot answer part a)
before, so the social convenor does not want
with a single nCr calculation for each word.
to assign them to work together. In how
many ways can the volunteers be assigned?

19. Jeffrey is a DJ at a local radio station. For
pt
ha e the second hour of his shift, he must choose
C

r

all his music from the top 100 songs for the
m
P

r
oble
week. Jeffery will play exactly 12 songs
during this hour.
a) How many different stacks of discs could
Jeffrey pull from the station’s collection if
he chooses at least 10 songs that are in
positions 15 to 40 on the charts?
b) Jeffrey wants to start his second hour with
a hard-rock song and finish with a pop
classic. How many different play lists can
Jeffrey prepare if he has chosen 4 hard rock
songs, 5 soul pieces, and 3 pop classics? © Tribune Media Services, Inc. All Rights Reserved. Reprinted with Permission.

c) Jeffrey has 8 favourite songs currently on
22. Determine the number of ways of selecting
the top 100 list. How many different four letters, without regard for order, from
subsets of these songs could he choose to the word parallelogram.
play during his shift?
C
ACHIEVEMENT CHECK
23. Inquiry/Problem Solving Suppose the artist
Communication Application in Example 1 of this section had two apples,
two oranges, and two pears in his
20. There are 52 white keys on a piano. The refrigerator. How many combinations does
lowest key is A. The keys are designated A, he have to choose from if he wants to paint
B, C, D, E, F, and G in succession, and a still-life with
then the sequence of letters repeats, ending a) two pieces of fruit?
with a C for the highest key.
b) three pieces of fruit?
a) If five notes are played simultaneously,
c) four pieces of fruit?
in how many ways could the notes all be
i) As? ii) Gs? 24. How many different sums of money can be
iii) the same letter? iv) different letters? formed from one $2 bill, three $5 bills, two
$10 bills, and one $20 bill?
b) If the five keys are played in order, how
would your answers in part a) change?


5.4 The Binomial Theorem

Recall that a binomial is a polynomial
with just two terms, so it has the form
a + b. Expanding (a + b)n becomes very
laborious as n increases. This section
introduces a method for expanding
powers of binomials. This method is
useful both as an algebraic tool and
for probability calculations, as you
will see in later chapters.

Blaise Pascal

I N V E S T I G AT E & I N Q U I R E : Patterns in the Binomial Expansion

1. Expand each of the following and simplify fully.
a) (a + b)1 b) (a + b)2 c) (a + b)3
d) (a + b)4 e) (a + b)5
2. Study the terms in each of these expansions. Describe how the degree
of each term relates to the power of the binomial.
3. Compare the terms in Pascal’s triangle to the expansions in question 1.
Describe any pattern you ﬁnd.
4. Predict the terms in the expansion of (a + b)6.

In section 4.4, you found a number of patterns in Pascal’s triangle. Now
that you are familiar with combinations, there is another important
pattern that you can recognize. Each term in Pascal’s triangle corresponds
to a value of nCr.

1 0
C0
1 1 1
C0 1
C1
1 2 1 C
2 0
C
2 1 2
C2
1 3 3 1 3
C0 3
C1 3
C2 3
C3
1 4 6 4 1 C
4 0
C
4 1
C
4 2
C
4 3 4
C4
1 5 10 10 5 1 C0
5 5
C1 5
C2 5
C3 5
C4 C5
5

5.4 The Binomial Theorem • MHR 289

Comparing the two triangles shown on page 289, you can see that tn,r = nCr.
Recall that Pascal’s method for creating his triangle uses the relationship

tn,r = tn−1, r−1 + tn−1, r.

So, this relationship must apply to combinations as well.

Pascal’s Formula

n
Cr = n−1Cr−1 + n−1Cr

Proof:
(n − 1)! (n − 1)!
Cr−1 + n−1Cr = ᎏᎏ + ᎏᎏ
n−1
(r − 1)!(n − r)! r!(n − r − 1)!
r(n − 1)! (n − 1)!(n − r)
= ᎏᎏ + ᎏᎏᎏ
r(r − 1)!(n − r)! r!(n − r)(n − r − 1)!
r(n − 1)! (n − 1)!(n − r)
= ᎏ + ᎏᎏ
r!(n − r)! r!(n − r)!
(n − 1)!
= ᎏ [r + (n − r)]
r!(n − r)!
(n − 1)! × n
= ᎏᎏ
r!(n − r)!
n!
= ᎏ
r!(n − r)!
= nCr
This proof shows that the values of nCr do indeed follow the pattern that
creates Pascal’s triangle. It follows that nCr = tn,r for all the terms in Pascal’s
triangle.

Example 1 Applying Pascal’s Formula to Combinations

Rewrite each of the following using Pascal’s formula.
a) C
12 8
b) 19C5 + 19C6

Solution

a) 12
C8 = 11C7 + 11C8 b) 19
C5 + 19C6 = 20C6

As you might expect from the investigation at the beginning of this section,
the coefﬁcients of each term in the expansion of (a + b)n correspond to the
terms in row n of Pascal’s triangle. Thus, you can write these coefﬁcients in
combinatorial form.


The Binomial Theorem

(a + b)n = nC0 a n + nC1a n−1b + nC2a n−2b 2 + … + nCr a n−rb r + … + nCnb n
n
or (a + b)n = Α nCr a n−rb r
r=0

Example 2 Applying the Binomial Theorem

Use combinations to expand (a + b)6.

Solution
6
(a + b)6 = Α 6Cr a 6−rb r
r=0
= 6C0a6 + 6C1a5b + 6C2a4b 2 + 6C3a3b3 + 6C4a2b4 + 6C5ab5 + 6C6b6
= a6 + 6a5b + 15a4b 2 + 20a3b3 + 15a2b4 + 6ab5 + b6

Example 3 Binomial Expansions Using Pascal’s Triangle

Use Pascal’s triangle to expand
a) (2x −1)4
b) (3x − 2y)5

Solution

a) Substitute 2x for a and −1 for b. Since the exponent is 4, use the terms
in row 4 of Pascal’s triangle as the coefﬁcients: 1, 4, 6, 4, and 1. Thus,

(2x − 1)4 = 1(2x)4 + 4(2x)3( − 1) + 6(2x)2( − 1)2 + 4(2x)( − 1)3 + 1( − 1)4
= 16x4 + 4(8x3)( − 1) + 6(4x2)(1) + 4(2x)( − 1) + 1
= 16x4 − 32x3 + 24x2 − 8x + 1

b) Substitute 3x for a and −2y for b, and use the terms from row 5 as coefﬁcients.

(3x − 2y)5 = 1(3x)5 + 5(3x)4( − 2y) + 10(3x)3( − 2y)2 + 10(3x)2( − 2y)3 + 5(3x)( − 2y)4 + 1( − 2y)5
= 243x5 − 810x4 y + 1080x3y2 − 720x2y3 + 240xy4 − 32y5

Example 4 Expanding Binomials Containing Negative Exponents

΂ ΃
2 4
Use the binomial theorem to expand and simplify x + ᎏ .
x2


Solution
2
Substitute x for a and ᎏ for b.
x2
4

΂ ΃ ΂ ΃
2 4 2 r
x+ ᎏ
x2
= Α
r=0
Cr x 4−r ᎏᎏ
4
x2

΂ ΃ ΂ ΃ ΂ ΃ ΂ ΃
2 2 2 2 3 2 4
= 4C0 x4 + 4C1x3 ᎏ + 4C2x2 ᎏ + 4C3x ᎏ + 4C4 ᎏ
x2 x2 x2 x2

= 1x + 4x ΂ ᎏ ΃ + 6x ΂ ᎏ ΃ + 4x΂ ᎏ ΃ + 1΂ ᎏ ΃
4 2 3 4 2 8 16
2 4 6 8
x x x x
= x4 + 8x + 24x −2 + 32x −5 + 16x −8

Example 5 Patterns With Combinations

Using the patterns in Pascal’s triangle from the investigation and Example 4 in
section 4.4, write each of the following in combinatorial form.
a) the sum of the terms in row 5 and row 6
b) the sum of the terms in row n
c) the ﬁrst 5 triangular numbers
d) the nth triangular number

Solution

a) Row 5: Row 6:
1 + 5 + 10 + 10 + 5 + 1 1 + 6 + 15 + 20 + 15 + 6 + 1
= 5C0 + 5C1 + 5C2 + 5C3 + 5C4 + 5C5 = 6C0 + 6C1 + 6C2 + 6C3 + 6C4 + 6C5 + 6C6
= 32 = 64
= 25 = 26

b) From part a) it appears that nC0 + nC1 + … + nCn = 2n.

Using the binomial theorem,
2n = (1 + 1)n
= nC0 × 1n + nC1 × 1n−1 × 1 + … + nCn × 1n
= nC0 + nC1 + … + nCn

c) n Triangular Numbers Combinatorial Form
1 1 C
2 2

2 3 3
C2
3 6 4
C2
4 10 5
C2
5 15 6
C2

d) The nth triangular number is C2.
n+1


Example 6 Factoring Using the Binomial Theorem

Rewrite 1 + 10x2 + 40x4 + 80x6 + 80x8 + 32x10 in the form (a + b)n.

Solution

There are six terms, so the exponent must be 5.
The first term of a binomial expansion is an, so a must be 1.
The final term is 32x10 = (2x2)5, so b = 2x2.
Therefore, 1 + 10x2 + 40x4 + 80x6 + 80x8 + 32x10 = (1 + 2x2)5

Key Concepts

• The coefficients of the terms in the expansion of (a + b)n correspond to the
terms in row n of Pascal’s triangle.

• The binomial (a + b)n can also be expanded using combinatorial symbols:
n
(a + b)n = nC0 a n + nC1 a n−1b + nC2 a n−2b 2 + … + nCn b n or Α
r=0
Cr a n−rb r
n

• The degree of each term in the binomial expansion of (a + b)n is n.

• Patterns in Pascal’s triangle can be summarized using combinatorial symbols.


1. Describe how Pascal’s triangle and the binomial theorem are related.

2. a) Describe how you would use Pascal’s triangle to expand (2x + 5y)9.
b) Describe how you would use the binomial theorem to expand (2x + 5y)9.

3. Relate the sum of the terms in the nth row of Pascal’s triangle to the total
number of subsets of a set of n elements. Explain the relationship.

Practise 2. Determine the value of k in each of these
terms from the binomial expansion of (a + b)10.
A a) 210a 6b k b) 45a kb8 c) 252a kb k
1. Rewrite each of the following using Pascal’s
formula. 3. How many terms would be in the expansion
a) C11 b) C36 of the following binomials?
17 43
c) Cr+1 d) C4 + 32C5 a) (x + y)12 b) (2x − 3y)5 c) (5x − 2)20
n+1 32
e) 15
C10 + 15C9 f) nCr + nCr+1 4. For the following terms from the expansion
g) C − 17C9
18 9
h) C − 23C7
24 8
of (a + b)11, state the coefficient in both nCr
i) nCr − Cr−1 and numeric form.
n−1
a) a 2b 9 b) a11 c) a 6b 5

Apply, Solve, Communicate 10. Communication
a) Find and simplify the first five terms of
B the expansion of (3x + y)10.
5. Using the binomial theorem and patterns in
b) Find and simplify the first five terms of
Pascal’s triangle, simplify each of the
the expansion of (3x − y)10.
following.
c) Describe any similarities and differences
a) 9C0 + 9C1 + … + 9C9
between the terms in parts a) and b).
b) 12
C0 − 12C1 + 12C2 − … − 12C11 + 12C12
15 n 11. Use the binomial theorem to expand and
c) Α
r=0
15
Cr d) ΑC
r=0
n r
simplify the following.

΂ ΃ ΂ ΃
1 5 3 4
a) x2 − ᎏ b) 2y + ᎏ
n x y2
6. If Α nCr = 16 384, determine the value of n.
΂ ΃
r=0 k 5
c) (͙x + 2x2)6
ෆ d) k + ᎏ2
7. a) Write formulas in combinatorial form for m
the following. (Refer to section 4.4, if
΂ ΃ ΂ ΃
2 7 2 4
necessary.) e) ͙y − ᎏ
ෆ f) 2 3m2 − ᎏ
͙yෆ ͙ෆm
i) the sum of the squares of the terms in
the nth row of Pascal’s triangle 12. Application Rewrite the following expansions
ii) the result of alternately adding and in the form (a + b)n, where n is a positive
subtracting the squares of the terms in integer.
the nth row of Pascal’s triangle a) x6 + 6x5y + 15x4 y2 + 20 x3y3 + 15x2y4
iii) the number of diagonals in an n-sided + 6xy5 + y6
polygon b) y12 + 8y9 + 24y6 + 32y3 + 16
b) Use your formulas from part a) to c) 243a5 − 405a4b + 270a3b 2 − 90a2b3
determine + 15ab4 − b5
i) the sum of the squares of the terms in
13. Communication Use the binomial theorem to
row 15 of Pascal’s triangle
simplify each of the following. Explain your
ii) the result of alternately adding and results.
subtracting the squares of the terms in
΂ ΃ ΂ ΃ ΂ ΃ ΂ ΃
1 5 1 5 1 5 1 5
row 12 of Pascal’s triangle a) ᎏ + 5 ᎏ + 10 ᎏ + 10 ᎏ
2 2 2 2
iii) the number of diagonals in a 14-sided
΂2΃ ΂2΃
1 1 5 5
polygon +5 ᎏ + ᎏ

8. How many terms would be in the expansion b) (0.7)7 + 7(0.7)6(0.3) + 21(0.7)5(0.3)2 + …
of (x2 + x)8? + (0.3)7
c) 79 − 9 × 78 + 36 × 77 − … − 70
9. Use the binomial theorem to expand and
simplify the following.
΂ ΃ and compare it with
2 4
14. a) Expand x + ᎏᎏ
a) (x + y) 7
b) (2x + 3y) 6 x
c) (2x − 5y) d) (x2 + 5)4 1
5
the expansion of ᎏ (x2 + 2)4.
x4
e) (3a2 + 4c)7 f) 5(2p − 6c2)5
b) Explain your results.


15. Use your knowledge of algebra and the 20. Inquiry/Problem Solving
binomial theorem to expand and simplify a) Use the binomial theorem to expand
each of the following. (x + y + z)2 by first rewriting it as
a) (25x2 + 30xy + 9y2)3 [x + ( y + z)]2.
b) (3x − 2y)5(3x + 2y)5 b) Repeat part a) with (x + y + z)3.
c) Using parts a) and b), predict the
16. Application
expansion of (x + y + z)4. Verify your
a) Calculate an approximation for (1.2)9 by
prediction by using the binomial
expanding (1 + 0.2)9. theorem to expand (x + y + z)4.
b) How many terms do you have to evaluate
d) Write a formula for (x + y + z)n.
to get an approximation accurate to two
e) Use your formula to expand and simplify
decimal places?
(x + y + z)5.
17. In a trivia contest, Adam has drawn a topic he
21. a) In the expansion of (x + y)5, replace x
knows nothing about, so he makes random
and y with B and G, respectively. Expand
guesses for the ten true/false questions. Use
and simplify.
the binomial theorem to help find
b) Assume that a couple has an equal
a) the number of ways that Adam can
chance of having a boy or a girl. How
answer the test using exactly four trues
would the expansion in part a) help find
b) the number of ways that Adam can
the number of ways of having k girls in a
answer the test using at least one true family with five children?
c) In how many ways could a family with
ACHIEVEMENT CHECK
five children have exactly three girls?
Communication Application d) In how many ways could they have
exactly four boys?
18. a) Expand (h + t)5.
22. A simple code consists of a string of five
b) Explain how this expansion can be used to
symbols that represent different letters of
determine the number of ways of getting
the alphabet. Each symbol is either a dot (•)
exactly h heads when five coins are tossed.
or a dash (–).
c) How would your answer in part b)
a) How many different letters are possible
change if six coins are being tossed? How
using this code?
would it change for n coins? Explain.
b) How many coded letters will contain
exactly two dots?
C c) How many different coded letters will
contain at least one dash?
19. Find the first three terms, ranked by degree
of the terms, in each expansion.
a) (x + 3)(2x + 5)4
b) (2x + 1)2(4x − 3)5
c) (x2 − 5)9(x3 + 2)6



5.1 Organized Counting With Venn b) Use a Venn diagram to find the
Diagrams proportion of households in each
Refer to the Key Concepts on page 270. of these categories.

1. Which regions in the diagram below
5.2 Combinations
correspond to
a) the union of sets A and B?
b) the intersection of sets B and C? 4. Evaluate the following and indicate any
calculations that could be done manually.
c) A ∩ C?
a) 41
C8 b) 33
C15
d) either B or S?
c) 25
C17 d) 50
C10
S A B
e) 10
C8 f) 15
C13
R2 R6 R3 g) 5C4 h) C24
25
R1 R8
R5 R7 i) 15
C11 j) 25
C20
R4 k) 16
C8 l) 30
C26

C 5. A track and field club has 12 members who
are runners and 10 members who specialize
2. a) Write the equation for the number of
in field events. The club has been invited to
elements contained in either of two sets. send a team of 3 runners and 2 field athletes
b) Explain why the principle of inclusion to an out-of-town meet. How many
and exclusion subtracts the last term in different teams could the club send?
this equation.
6. A bridge hand consists of 13 cards. How
c) Give a simple example to illustrate your
explanation. many bridge hands include 5 cards of one
suit, 6 cards of a second, and 2 cards of a
3. A survey of households in a major city found third?
that
7. Explain why combination locks should really
• 96% had colour televisions be called permutation locks.
• 65% had computers
• 51% had dishwashers 5.3 Problem Solving With Combinations
• 63% had colour televisions and computers Refer to the Key Concepts on page 286.
• 49% had colour televisions and 8. At Subs Galore, you have a choice of lettuce,
dishwashers onions, tomatoes, green peppers,
• 31% had computers and dishwashers mushrooms, cheese, olives, cucumbers, and
• 30% had all three hot peppers on your submarine sandwich.
a) List the categories of households not
How many ways can you “dress” your
included in these survey results. sandwich?


9. Ballots for municipal elections usually list 16. Use the binomial theorem to expand
candidates for several different positions. a) (x + y)6
If a resident can vote for a mayor, two
b) (6x − 5y)4
councillors, a school trustee, and a hydro
commissioner, how many combinations of c) (5x + 2y)5
positions could the resident choose to mark d) (3x − 2)6
on the ballot?
17. Write the first three terms of the expansion
10. There are 12 questions on an examination, of
and each student must answer 8 questions a) (2x + 5y)7
including at least 4 of the first 5 questions. b) (4x − y)6
How many different combinations of
questions could a student choose to answer? 18. Describe the steps in the binomial expansion
of (2x − 3y)6.
11. Naomi invites eight friends to a party on
short notice, so they may not all be able to 19. Find the last term in the binomial expansion

΂ᎏxᎏ + 2x΃ .
come. How many combinations of guests 1 5
of 2
could attend the party?

12. In how many ways could 15 different books 20. Find the middle term in the binomial

΂ ΃
be divided equally among 3 people? 5 8
expansion of ͙x + ᎏ .
ෆ
͙ෆ
x
13. The camera club has five members, and the
mathematics club has eight. There is only 21. In the expansion of (a + x)6, the first three
one member common to both clubs. In how terms are 1 + 3 + 3.75. Find the values
many ways could a committee of four people of a and x.
be formed with at least one member from
22. Use the binomial theorem to expand and
each club?
simplify ( y2 − 2)6( y2 + 2)6.

23. Write 1024x10 − 3840x8 + 5760x6 − 4320x4 +
5.4 The Binomial Theorem 1620x2 − 243 in the form (a + b)n. Explain
your steps.
14. Without expanding (x + y)5, determine
a) the number of terms in the expansion
b) the value of k in the term 10x k y2

15. Use Pascal’s triangle to expand
a) (x + y)8
b) (4x − y)6
c) (2x + 5y)4
d) (7x − 3)5


Chapter Test

ACHIEVEMENT CHART

Questions All 12 6, 12 5, 6, 7, 8, 9

1. Evaluate each of the following. List any 6. A track club has 20 members.
calculations that require a calculator. a) In how many ways can the club choose
a) C
25 25
3 members to help ofﬁciate at a meet?
b) 52
C1 b) In how many ways can the club choose
c) C3 a starter, a marshal, and a timer?
12
d) C15 c) Should your answers to parts a) and b)
40
be the same? Explain why or why not.
2. Rewrite each of the following as a single
combination. 7. Statistics on the grade-12 courses taken
by students graduating from a secondary
a) 10
C7 + 10C8
school showed that
b) 23
C15 − 22C14 • 85 of the graduates had taken a science
3. Use Pascal’s triangle to expand
course
• 75 of the graduates had taken a second
a) (3x − 4)4
language
b) (2x + 3y)7 • 41 of the graduates had taken
mathematics
4. Use the binomial theorem to expand
• 43 studied both science and a second
a) (8x − 3)5 language
b) (2x − 5y)6 • 32 studied both science and mathematics
• 27 had studied both a second language
5. A student fundraising committee has 14 and mathematics
members, including 7 from grade 12. In how • 19 had studied all three subjects
many ways can a 4-member subcommittee
a) Use a Venn diagram to determine the
for commencement awards be formed if
minimum number of students who could
a) there are no restrictions? be in this graduating class.
b) the subcommittee must be all grade-12 b) How many students studied
students? mathematics, but neither science nor
c) the subcommittee must have 2 students a second language?
from grade 12 and 2 from other grades?
d) the subcommittee must have no more
than 3 grade-12 students?


8. A ﬁeld-hockey team played seven games and b) The restaurant also has a lunch special
won four of them. There were no ties. with your choice of one item from each
a) How many arrangements of the four group. How many choices do you have
wins and three losses are possible? with this special?
b) In how many of these arrangements 10. In the expansion of (1 + x)n, the ﬁrst three
would the team have at least two wins terms are 1 − 0.9 + 0.36. Find the values of x
in a row? and n.

9. A restaurant offers an all-you-can-eat 11. Use the binomial theorem to expand and
Chinese buffet with the following items: simplify (4x2 − 12x + 9)3.
• egg roll, wonton soup 12. A small transit bus has 8 window seats and
• chicken wings, chicken balls, beef, pork 12 aisle seats. Ten passengers board the bus
• steamed rice, fried rice, chow mein and select seats at random. How many
• chop suey, mixed vegetables, salad seating arrangements have all the window
• fruit salad, custard tart, almond cookie seats occupied if which passenger is in a seat
a) How many different combinations of a) does not matter? b) matters?
items could you have?

ACHIEVEMENT CHECK

13. The students’ council is having pizza at their next meeting. There are 20
council members, 6 of whom are vegetarian. A committee of 3 will order six
pizzas from a pizza shop that has a special price for large pizzas with up to
three toppings. The shop offers ten different toppings.
a) How many different pizza committees can the council choose if there must
be at least one vegetarian and one non-vegetarian on the committee?
b) In how many ways could the committee choose exactly three toppings for a
pizza?
c) In how many ways could the committee choose up to three toppings for a
pizza?
d) The committee wants as much variety as possible in the toppings. They
decide to order each topping exactly once and to have at least one topping
on each pizza. Describe the different cases possible when distributing the
toppings in this way.
e) For one of these cases, determine the number of ways of choosing and
distributing the ten toppings.


6
PT ER
Introduction to Probability
CHA


Use Venn diagrams as a tool for organizing information in counting 6.5
problems.

Solve problems, using techniques for counting permutations where some 6.3
objects may be alike.

Solve problems, using techniques for counting combinations. 6.3

Solve probability problems involving combinations of simple events, 6.3, 6.4, 6.5,
using counting techniques. 6.6

Interpret probability statements, including statements about odds, from a 6.1, 6.2, 6.3,
variety of sources. 6.4, 6.5, 6.6

Design and carry out simulations to estimate probabilities in situations 6.3
for which the calculation of the theoretical probabilities is difﬁcult or
impossible.

Assess the validity of some simulation results by comparing them with the 6.3
theoretical probabilities, using the probability concepts developed in the
course.

Represent complex tasks or issues, using diagrams. 6.1, 6.5

Represent numerical data, using matrices, and demonstrate an
understanding of terminology and notation related to matrices. 6.6

Demonstrate proﬁciency in matrix operations, including addition, scalar
multiplication, matrix multiplication, the calculation of row sums, and the 6.6
calculation of column sums, as necessary to solve problems, with and
without the aid of technology.

Solve problems drawn from a variety of applications, using matrix
methods. 6.6

Chapter Problem
Genetic Probabilities themselves, while offspring of healthy does
Biologists are studying a deer population have only a 20% likelihood of developing
in a provincial conservation area. The it. Currently, 30% of the does have bald
biologists know that many of the bucks patches.
(male deer) in the area have an unusual 1. Out of ten deer randomly captured,
“cross-hatched” antler structure, which how many would you expect to have
seems to be genetic in origin. Of either cross-hatched antlers or bald
48 randomly tagged deer, 26 were does patches?
(females), 22 were bucks, and 7 of the
bucks had cross-hatched antlers. 2. Do you think that the proportion of
does with the bald patches will increase,
Several of the does have small bald patches
decrease, or remain relatively stable?
on their hides. This condition also seems to
have some genetic element. Careful long- In this chapter, you will learn methods that
term study has found that female offspring the biologists could use to calculate
of does with bald patches have a 65% probabilities from their samples and to
likelihood of developing the condition make predictions about the deer population.



1. Fractions, percents, decimals Express each 6. Tree diagrams In the game of backgammon,
decimal as a percent. you roll two dice to determine how you can
a) 0.35 move your counters. Suppose you roll ﬁrst
one die and then the other and you need to
b) 0.04
roll 9 or more to move a counter to safety.
c) 0.95 Use a tree diagram to list the different
d) 0.008 rolls in which
e) 0.085 a) you make at least 9
f) 0.375 b) you fail to move your counter to safety

2. Fractions, percents, decimals Express each 7. Fundamental counting principle (section 4.1)
percent as a decimal. Benoit is going skating on a cold wintry day.
a) 15% b) 3% He has a toque, a watch cap, a beret, a heavy
c) 85% d) 6.5%
scarf, a light scarf, leather gloves, and wool
gloves. In how many different ways can
e) 26.5% f) 75.2%
Benoit dress for the cold weather?
3. Fractions, percents, decimals Express each
8. Additive counting principle (section 4.1) How
percent as a fraction in simplest form. many 13-card bridge hands include either
a) 12% b) 35% seven hearts or eight diamonds?
c) 67% d) 4%
9. Venn diagrams (section 5.1)
e) 0.5% f) 98%
a) List the elements for each of the
4. Fractions, percents, decimals Express each following sets for whole numbers from
fraction as a percent. Round answers to the 1 to 10 inclusive.
nearest tenth, if necessary. i) E, the set of even numbers
1 13 ii) O, the set of odd numbers
a) ᎏᎏ b) ᎏᎏ
4 15 iii) C, the set of composite numbers
11 7
c) ᎏᎏ d) ᎏᎏ iv) P, the set of perfect squares
14 10
4 13 b) Draw a diagram to illustrate how the
e) ᎏᎏ f) ᎏᎏ
9 20 following sets are related.
i) E and O
5. Tree diagrams A coin is ﬂipped three times.
Draw a tree diagram to illustrate all possible ii) E and C
outcomes. iii) O and P
iv) E, C, and P


10. Principle of inclusion and exclusion 16. Combinations (section 5.2) A pizza shop has
(section 5.1) nine toppings available. How many different
a) Explain the principle of inclusion and three-topping pizzas are possible if each
exclusion. topping is selected no more than once?
b) A gift store stocks baseball hats in red or 17.Combinations (section 5.3) A construction
green colours. Of the 35 hats on display crew has 12 carpenters and 5 drywallers.
on a given day, 20 are green. As well, How many different safety committees could
18 of the hats have a grasshopper logo they form if the members of this committee
on the brim. Suppose 11 of the red hats are
have logos. How many hats are red, or
have logos, or both? a) any 5 of the crew?
b) 3 carpenters and 2 drywallers?
11. Factorials (section 4.2) Evaluate.
a) 6! b) 0! 18. Matrices (section 1.6) Identify any square
16! 12! matrices among the following. Also identify
c) ᎏ d) ᎏ any column or row matrices.
14! 9! 3!

e) ᎏᎏ
100!
98!
f) ᎏᎏ
16!
10! × 8!
a)
΄
3 4
0 1 ΅ b) [0.4 0.3 0.2]

΄ ΅ ΄ ΅
1 0 −2 3 9
12. Permutations (section 4.2) Evaluate.
c) 0.5 0.5 d) 0 11 −4
a) 5P3 b) P
7 1 0.8 0.6 3 6 −1
c) P(6, 2) d) P

΄ ΅ ΄ ΅
9 9
49 63 8
e) 100 1
P f) P(100, 2)
e) 25 14 f) 16
13. Permutations (section 4.2) A baseball team 72 9 32
has 13 members. If a batting line-up consists
19. Matrices (section 1.7) Given A = [0.3 0.7]
of 9 players, how many different batting
line-ups are possible? and B =
΄ 0.4
0.55
0.45 ΅
0.6 , perform the

14. Permutations (section 4.2) What is the following matrix operations, if possible. If
maximum number of three-digit area codes the operation is not possible, explain why.
possible if the area codes cannot start with
a) A × B b) B × A
either 1 or 0?
2
c) B d) B3
15. Combinations (section 5.2) Evaluate these e) A 2 f) A × A t
expressions.
a) 6C3 b) C(4, 3)
c) 8C8 d) 11
C0

΂4΃ × ΂5 ΃ ΂1΃
6 7 100
e) f)

g) 20
C2 h) 20
C18
Review of Prerequisite Skills • MHR 303

6.1 Basic Probability Concepts

How likely is rain tomorrow? What are the
chances that you will pass your driving test on
the first attempt? What are the odds that the
flight will be on time when you go to meet
someone at the airport?

Probability is the branch of mathematics that
attempts to predict answers to questions like
these. As the word probability suggests, you can
often predict only what might happen.
However, you may be able to calculate how
likely it is. For example, if the weather report
forecasts a 90% chance of rain, there is still
that slight possibility that sunny skies will
prevail. While there are no sure answers, in
this case it probably will rain.

I N V E S T I G AT E & I N Q U I R E : A Number Game

Work with a partner. Have each partner take three identical slips of paper,
number them 1, 2, and 3, and place them in a hat, bag, or other container.
For each trial, both partners will randomly select one of their three slips of
paper. Replace the slips after each trial. Score points as follows:
• If the product of the two numbers shown is less than the sum, Player A gets
a point.
• If the product is greater than the sum, Player B gets a point.
• If the product and sum are equal, neither player gets a point.

1. Predict who has the advantage in this game. Explain why you think so.

2. Decide who will be Player A by flipping a coin or using the random
number generator on a graphing calculator. Organize your results in a
table like the one below.
Trial 1 2 3 4 5 6 7 8 9 10
Number drawn by A
Number drawn by B
Product
Sum
Point awarded to:


3. a) Record the results for 10 trials. Total the points and determine the winner.
Do the results confirm your prediction? Have you changed your opinion on
who has the advantage? Explain.
b) To estimate a probability for each player getting a point, divide the number
of points each player earned by the total number of trials.

4. a) Perform 10 additional trials and record point totals for each player over all
20 trials. Estimate the probabilities for each player, as before.
b) Are the results for 20 trials consistent with the results for 10 trials? Explain.
c) Are your results consistent with those of your classmates? Comment on
your findings.

5. Based on your results for 20 trials, predict how many points each player will
have after 50 trials.

6. Describe how you could alter the game so that the other player has the advantage.

The investigation you have just completed is an example of a probability experiment.
In probability, an experiment is a well-defined process consisting of a number of trials
in which clearly distinguishable outcomes, or possible results, are observed.

The sample space, S, of an experiment is the set of all possible outcomes. For the
sum/product game in the investigation, the outcomes are all the possible pairings of
slips drawn by the two players. For example, if Player A draws 1 and Player B draws 2,
you can label this outcome (1, 2). In this particular game, the result is the same for the
outcomes (1, 2) and (2, 1), but with different rules it might be important who draws
which number, so it makes sense to view the two outcomes as different.

Outcomes are often equally likely. In the sum/product game, each possible pairing of
numbers is as likely as any other. Outcomes are often grouped into events. An
example of an event is drawing slips for which the product is greater than the sum, and
there are several outcomes in which this event happens. Different events often have
different chances of occurring. Events are usually labelled with capital letters.

Example 1 Outcomes and Events

Let event A be a point awarded to Player A in the sum/product game.
List the outcomes that make up event A.

Solution
Player A earns a point if the sum of the two numbers is greater than the
product. This event is sometimes written as event A = {sum > product}.
A useful technique in probability is to tabulate the possible outcomes.

6.1 Basic Probability Concepts • MHR 305

Sums Products
Player A Player A
1 2 3 1 2 3
Player B 1 2 3 4 Player B 1 1 2 3
2 3 4 5 2 2 4 6
3 4 5 6 3 3 6 9

Use the tables shown to list the outcomes where the sum is greater than the product:
(1, 1), (1, 2), (1, 3), (2, 1), (3, 1)
These outcomes make up event A. Using this list, you can also write event A as
event A = {(1, 1), (1, 2), (1, 3), (2, 1), (3, 1)}

The probability of event A, P(A), is a quantified measure of the likelihood that
the event will occur. The probability of an event is always a value between 0 and 1.
A probability of 0 indicates that the event is impossible, and 1 signifies that the
event is a certainty. Most events in probability studies fall somewhere between
these extreme values. Probabilities less than 0 or greater than 1 have no
meaning. Probability can be expressed as fractions, decimals, or percents.
Probabilities expressed as percents are always between 0% and 100%. For
example, a 70% chance of rain tomorrow means the same as a probability of 0.7,
7
or ᎏᎏ, that it will rain.
10
The three basic types of probability are
• empirical probability, based on direct observation or experiment
• theoretical probability, based on mathematical analysis
• subjective probability, based on informed guesswork

The empirical probability of a particular event (also called experimental or
relative frequency probability) is determined by dividing the number of times
that the event actually occurs in an experiment by the number of trials. In the
sum/product investigation, you were calculating empirical probabilities. For
example, if you had found that in the first ten trials, the product was greater
than the sum four times, then the empirical probability of this event would be
4
P(A) = ᎏ
10
2
= ᎏᎏ or 0.4
5
The theoretical probability of a particular event is deduced from analysis of
the possible outcomes. Theoretical probability is also called classical or a priori
probability. A priori is Latin for “from the preceding,” meaning based on
analysis rather than experiment.


For example, if all possible outcomes are equally likely, then Project
Prep
n(A)
P(A) = ᎏᎏ
n(S) You will need to
where n(A) is the number of outcomes in which event A can occur, and n(S) determine theoretical
is the total number of possible outcomes. You used tables to list the probabilities to
outcomes for A in Example 1, and this technique allows you to ﬁnd the design and analyse
theoretical probability P(A) by counting n(A) = 5 and n(S) = 9. Another way your game in the
to determine the values of n(A) and n(S) is by organizing the information in probability project.
a tree diagram.

Example 2 Using a Tree Diagram to Calculate Probability

Determine the theoretical probabilities for each key event in the sum/product
game.

Solution
product sum
The tree diagram shows the nine possible outcomes, each 1 1 < 2
equally likely, for the sum/product game. 1 2 2 < 3
3 3 < 4
Let event A be a point for Player A, event B a point 1 2 < 3
for Player B, and event C a tie between sum and 2 2 4 = 4
product. From the tree diagram, ﬁve of the nine possible 3 6 > 5
outcomes have the sum greater than the product. 1 3 < 4
Therefore, the theoretical probability of this event is 3 2 6 > 5
3 9 > 6
n(A)
P(A) = ᎏ
n(S)
5
= ᎏᎏ
9
Similarly,
n(B) n(C)
P(B) = ᎏ and P(C ) = ᎏ
n(S) n(S)
3 1
= ᎏᎏ = ᎏᎏ
9 9

In Example 2, you know that one, and only one, of the three events will occur.
The sum of the probabilities of all possible events always equals 1.
5 3 1
P(A) + P(B) + P(C) = ᎏᎏ + ᎏᎏ + ᎏᎏ
9 9 9
=1
Here, the numerator in each fraction represents the number of ways that each
event can occur. The total of these numerators is the total number of possible
outcomes, which is equal to the denominator.


Empirical probabilities may differ sharply from theoretical probabilities when
only a few trials are made. Such statistical fluctuation can result in an event
occurring more frequently or less frequently than theoretical probability
suggests. Over a large number of trials, however, statistical fluctuations tend to
cancel each other out, and empirical probabilities usually approach theoretical
values. Statistical fluctuations often appear in sports, for example, where a team
can enjoy a temporary winning streak that is not sustainable over an entire
season.

In most problems, you will be determining theoretical probability. Therefore,
from now on you may take the term probability to mean theoretical probability
unless stated otherwise.

Example 3 Dice Probabilities

Many board games involve a roll of two six-sided dice to see how far you may
move your pieces or counters. What is the probability of rolling a total of 7?

Solution
The table shows the totals for all possible rolls of two dice.
First Die
1 2 3 4 5 6
1 2 3 4 5 6 7
2
Second Die

3 4 5 6 7 8
3 4 5 6 7 8 9
4 5 6 7 8 9 10
5 6 7 8 9 10 11
6 7 8 9 10 11 12

To calculate the probability of a particular total, count the number of times it
appears in the table. For event A = {rolling 7},
n(A)
P(A) = ᎏ
n(S)
n(rolls totalling 7)
= ᎏᎏᎏ
n(all possible rolls)
6
= ᎏᎏ
36
1
= ᎏᎏ
6
1
The probability of rolling a total of 7 is ᎏᎏ.
6


A useful and important concept in probability is the complement of an event.
The complement of event A, A′ or ~A, is the event that “event A does not
happen.” Thus, whichever outcomes make up A, all the other outcomes make
up A′. Because A and A′ together include all possible outcomes, the sum of their
probabilities must be 1. Thus,
P(A) + P(AЈ) = 1 and P(AЈ) = 1 − P(A)

A'
A

The event A′ is usually called “A-prime,” or sometimes “not-A”; ~A is called
“tilde-A.”

Example 4 The Complement of an Event

What is the probability that a randomly drawn integer between 1 and 40 is not a
perfect square?

Solution

Let event A = {a perfect square}. Then, the complement of A is the event
A′ = {not a perfect square}. In this case, you need to calculate P(A′), but it is easier
to do this by ﬁnding P(A) ﬁrst. There are six perfect squares between 1 and
40: 1, 4, 9, 16, 25, and 36. The probability of a perfect square is, therefore,
n(A)
P(A) = ᎏ
n(S)
6
= ᎏᎏ
40
3
= ᎏᎏ
20
Thus,
P(A′) = 1 − P(A)
3
= 1 − ᎏᎏ
20
17
= ᎏᎏ
20
17
There is a ᎏᎏ or 85% chance that a random integer between 1 and 40 will not
20
be a perfect square.


Subjective probability, the third basic type of probability, is an estimate of
likelihood based on intuition and experience—an educated guess. For example,
a well-prepared student may be 90% confident of passing the next data
management test. Subjective probabilities often figure in everyday speech in
expressions such as “I think the team has only a 10% chance of making the
finals this year.”

Example 5 Determining Subjective Probability

Estimate the probability that
a) the next pair of shoes you buy will be the same size as the last pair you
bought
b) an expansion baseball team will win the World Series in their first season
c) the next person to enter a certain coffee shop will be male

Solution

a) There is a small chance that the size of your feet has changed significantly
or that different styles of shoes may fit you differently, so 80–90% would be
a reasonable subjective probability that your next pair of shoes will be the
same size as your last pair.

b) Expansion teams rarely do well during their first
season, and even strong teams have difficulty
winning the World Series. The subjective
probability of a brand-new team winning the
World Series is close to zero. For some interesting baseball statistics, visit the
above web site and follow the links. Write a
c) Without more information about the coffee shop problem that could be solved using
in question, your best estimate is to assume that probabilities.
the shop’s patrons are representative of the general
population. This assumption gives a subjective probability of
50% that the next customer will be male.

Note that the answers in Example 5 contain estimates, assumptions, and, in some
cases, probability ranges. While not as rigorous a measure as theoretical or
empirical probability, subjective probabilities based on educated guesswork can
still prove useful in some situations.


Key Concepts

• A probability experiment is a well-defined process in which clearly identifiable
outcomes are measured for each trial.

• An event is a collection of outcomes satisfying a particular condition. The
probability of an event can range between 0 (impossible) and 1 or 100%
(certain).

• The empirical probability of an event is the number of times the event occurs
divided by the total number of trials.
n(A)
• The theoretical probability of an event A is given by P(A) = ᎏ , where
n(S)
n(A) is the number of outcomes making up A, n(S ) is the total number
of outcomes in the sample space S, and all outcomes are equally likely
to occur.

• A subjective probability is based on intuition and previous experience.

• If the probability of event A is given by P(A), then the probability of the
complement of A is given by P(AЈ) = 1 − P(A).


1. Give two synonyms for the word probability.

2. a) Explain why P(A) + P(AЈ) = 1.
b) Explain why probabilities less than 0 or greater than 1 have no meaning.

3. Explain the difference between theoretical, empirical, and subjective
probability. Give an example of how you would determine each type.

4. Describe three situations in which statistical fluctuations occur.

5. a) Describe a situation in which you might determine the probability of
event A indirectly by calculating P(AЈ) first.
b) Will this method always yield the same result as calculating P(A) directly?
c) Defend your answer to part b) using an explanation or proof, supported
by an example.


Practise Determine the following probabilities.
a) P(resident owns home)
A
b) P(resident rents and has lived at present
1. Determine the probability of
address less than two years)
a) tossing heads with a single coin
c) P(homeowner has lived at present
b) tossing two heads with two coins address more than two years)
c) tossing at least one head with three coins
B
d) rolling a composite number with one die
5. Application Suppose your school’s basketball
e) not rolling a perfect square with two dice
team is playing a four-game series against
f) drawing a face card from a standard deck another school. So far this season, each team
of cards has won three of the six games in which they
faced each other.
2. Estimate a subjective probability of each of
the following events. Provide a rationale for a) Draw a tree diagram to illustrate all
each estimate. possible outcomes of the series.
a) the sun rising tomorrow b) Use your tree diagram to determine the
probability of your school winning
b) it never raining again
exactly two games.
c) your passing this course
c) What is the probability of your school
d) your getting the next job you apply for sweeping the series (winning all four
3. Recall the sum/product game at the
games)?
beginning of this section. Suppose that the d) Discuss any assumptions you made in the
game were altered so that the slips of paper calculations in parts b) and c).
showed the numbers 2, 3, and 4, instead of
6. Application Suppose that a graphing calculator
1, 2, and 3.
is programmed to generate a random natural
a) Identify all the outcomes that will number between 1 and 10 inclusive. What is
produce each of the three possible events the probability that the number will be prime?
i) p>s ii) p < s iii) p = s
7. Communication
b) Which player has the advantage in this
situation? a) A game involves rolling two dice. Player
A wins if the throw totals 5, 7, or 9.
Apply, Solve, Communicate Player B wins if any other total is
thrown. Which player has the advantage?
4. The town planning department surveyed Explain.
residents of a town about home ownership.
b) Suppose the game is changed so that
The table shows the results of the survey.
Player A wins if 5, 7, or doubles (both dice
At Address At Address showing the same number) are thrown.
Less Than More Than Total for
Residents 2 Years 2 Years Category Who has the advantage now? Explain.
Owners 2000 8000 10 000 c) Design a similar game in which each
Renters 4500 1500 6 000 player has an equal chance of winning.
Total 6500 9500 16 000


8. a) Based on the randomly tagged sample, 11. Communication Prior to a municipal
pte
ha what is the empirical probability that a election, a public-opinion poll determined
C

r

deer captured at random will be a doe? that the probability of each of the four
m
P

r
oble
b) If ten deer are captured at random, candidates winning was as follows:
how many would you expect to be Jonsson 10%
bucks? Trimble 32%
C Yakamoto 21%
9. Inquiry/Problem Solving Refer to the prime Audette 37%
number experiment in question 6. What a) How will these probabilities change if
happens to the probability if you change the Jonsson withdraws from the race after
upper limit of the sample space? Use a ballots are cast?
graphing calculator or appropriate computer b) How will these probabilities change if
software to investigate this problem. Let A Jonsson withdraws from the race before
be the event that the random natural ballots are cast?
number will be a prime number. Let the
c) Explain why your answers to a) and b)
random number be between 1 and n
are different.
inclusive. Predict what you think will
happen to P(A) as n increases. Investigate 12. Inquiry/Problem Solving It is known from
P(A) as a function of n, and reflect on your studying past tests that the correct answers
hypothesis. Did you observe what you to a certain university professor’s multiple-
expected? Why or why not? choice tests exhibit the following pattern.
10. Suppose that the Toronto Blue Jays face the Correct Answer Percent of Questions
New York Yankees in the division final. In A 15%
this best-of-five series, the winner is the first B 25%
team to win three games. The games are C 30%
played in Toronto and in New York, with D 15%
Toronto hosting the first, second, and if E 15%
needed, fifth games. The consensus among
experts is that Toronto has a 65% chance of a) Devise a strategy for guessing that would
winning at home and a 40% chance of maximize a student’s chances for success,
winning in New York. assuming that the student has no idea of
the correct answers. Explain your
a) Construct a tree diagram to illustrate all
method.
the possible outcomes.
b) Suppose that the study of past tests
b) What is the chance of Toronto winning
revealed that the correct answer choice
in three straight games?
for any given question was the same as
c) For each outcome, add to your tree that of the immediately preceding
diagram the probability of that outcome. question only 10% of the time. How
d) Communication Explain how you found would you use this information to adjust
your answers to parts b) and c). your strategy in part a)? Explain your
reasoning.


6.2 Odds

Odds are another way to
express a level of conﬁdence
about an outcome. Odds are
commonly used in sports and
other areas. Odds are often
used when the probability of
an event versus its complement
is of interest, for example
whether a sprinter will win
or lose a race or whether a
basketball team will make it
to the ﬁnals.

I N V E S T I G AT E & I N Q U I R E : Te n n i s To u r n a m e n t

For an upcoming tennis tournament, a television commentator estimates
that the top-seeded (highest-ranked) player has “a 25% probability of
winning, but her odds of winning are only 1 to 3.”

1. a) If event A is the top-seeded player winning the tournament, what is A′?
b) Determine P(A′).

2. a) How are the odds of the top-seeded player winning related to P(A) and
P(A′)?
b) Should the television commentator be surprised that the odds were only
1 to 3? Why or why not?

3. a) What factors might the commentator consider when
estimating the probability of the top-seeded player
winning the tournament?
b) How accurate do you think the
commentator’s estimate is likely to be? For more information about tennis rankings and
Would you consider such an estimate other tennis statistics, visit the above web site and
primarily a classical, an empirical, or a follow the links. Locate some statistics about a
subjective probability? Explain. tennis player of your choice. Use odds to
describe these statistics.


The odds in favour of an event’s occurring are given by the ratio of the
probability that the event will occur to the probability that it will not occur.
P(A)
odds in favour of A = ᎏᎏ
P(AЈ)
Giving odds in favour of an event is a common way to express a probability.

Example 1 Determining Odds

A messy drawer contains three red socks, ﬁve white socks, and four black socks.
What are the odds in favour of randomly drawing a red sock?

Solution

Let the event A be drawing a red sock. The probability of this event is
3
P(A) = ᎏ
12
1
= ᎏᎏ
4 Project
The probability of not drawing a red sock is Prep
P(AЈ) = 1 − P(A)
A useful feature you
3
= ᎏᎏ could include in your
4
probability project is a
Using the deﬁnition of odds,
calculation of the odds
P(A)
odds in favour of A = ᎏ of winning your game.
P(AЈ)
1
ᎏᎏ
4
= ᎏ
3
ᎏᎏ
4
1
= ᎏᎏ
3
1
Therefore, the odds in favour of drawing a red sock are ᎏᎏ, or less than 1. You
3
are more likely not to draw a red sock. These odds are commonly written as
1:3, which is read as “one to three” or “one in three.”

Notice in Example 1 that the ratio of red socks to other socks is 3:9, which
is the same as the odds in favour of drawing a red sock. In fact, the odds in
favour of an event A can also be found using
n(A)
odds in favour of A = ᎏᎏ
n(AЈ)

6.2 Odds • MHR 315

A common variation on the theme of odds is to express the odds against an event
happening.

P(AЈ)
odds against A = ᎏᎏ
P(A)

Example 2 Odds Against an Event

If the chance of a snowstorm in Windsor, Ontario, in January is estimated at
0.4, what are the odds against Windsor’s having a snowstorm next January? Is a
January snowstorm more likely than not?

Solution

Let event A = {snowstorm in January}.
Since P(A) + P(AЈ) = 1,
P(AЈ)
odds against A = ᎏ
P(A)
1 − P(A)
= ᎏ
P(A)
1 − 0.4
= ᎏ
0.4
0.6
= ᎏ
0.4
3
= ᎏᎏ
2
The odds against a snowstorm are 3:2, which is greater than 1:1. So a
snowstorm is less likely to occur than not.

Sometimes, you might need to convert an expression of odds into a probability.
You can do this conversion by expressing P(AЈ) in terms of P(A).

Example 3 Probability From Odds

A university professor, in an effort to promote good attendance habits, states
that the odds of passing her course are 8 to 1 when a student misses fewer
than ﬁve classes. What is the probability that a student with good attendance
will pass?

Solution

Let the event A be that a student with good attendance passes. Since
P(A)
odds in favour of A = ᎏ ,
P(AЈ)


8 P(A)
ᎏ = ᎏ
1 P(AЈ)
P(A)
= ᎏ
1 − P(A)
8 − 8P(A) = P(A)
8 = 9P(A)
8
P(A) = ᎏᎏ
9
8
The probability that a student with good attendance will pass is ᎏᎏ, or
9
approximately 89%.

h h
In general, it can be shown that if the odds in favour of A = ᎏᎏ, then P(A) = ᎏᎏ.
k h+k

Example 4 Using the Odds-Probability Formula

The odds of Rico’s hitting a home run are 2:7. What is the probability of Rico’s
hitting a home run?

Solution

Let A be the event that Rico hits a home run. Then, h = 2 and k = 7, and
h
P(A) = ᎏ
h+k
2
= ᎏ
2+7
2
= ᎏᎏ
9
Rico has approximately a 22% chance of hitting a home run.

Key Concepts
P(A)
• The odds in favour of A are given by the ratio ᎏ .
P(AЈ)

P(AЈ)
• The odds against A are given by the ratio ᎏ .
P(A)
h h
• If the odds in favour of A are ᎏ , then P(A) = ᎏ .
k h+k

6.2 Odds • MHR 317


1. Explain why the terms odds and probability have different meanings. Give an
example to illustrate your answer.

2. Would you prefer the odds in favour of passing your next data management
test to be 1:3 or 3:1? Explain your choice.

3. Explain why odds can be greater than 1, but probabilities must be between
0 and 1.

A B
1. Suppose the odds in favour of good weather 5. Greta’s T-shirt drawer contains three tank
tomorrow are 3:2. tops, six V-neck T-shirts, and two sleeveless
a) What are the odds against good weather shirts. If she randomly draws a shirt from
tomorrow? the drawer, what are the odds that she will
b) What is the probability of good weather a) draw a V-neck T-shirt?
tomorrow? b) not draw a tank top?

2. The odds against the Toronto Argonauts 6. Application If the odds in favour of Boris
winning the Grey Cup are estimated at 19:1. beating Elena in a chess game are 5 to 4,
What is the probability that the Argos will what is the probability that Elena will win
win the cup? an upset victory in a best-of-ﬁve chess
tournament?
3. Determine the odds in favour of rolling
each of the following sums with a standard 7. a) Based on the randomly tagged sample,
pte
pair of dice. ha what are the odds in favour of a captured
C

r

a) 12 b) 5 or less deer being a cross-hatched buck?
m
P

r
oble
c) a prime number d) 1 b) What are the odds against capturing a
doe?
4. Calculate the odds in favour of each event.
a) New Year’s Day falling on a Friday
b) tossing three tails with three coins
c) not tossing exactly two heads with three
coins Visit the above web site and follow the links
for more information about Canadian
d) randomly drawing a black 6 from a
wildlife.
complete deck of 52 cards
e) a random number from 1 to 9 inclusive
being even


8. The odds against A, by deﬁnition, are 12. George estimates that there is a 30% chance
equivalent to the odds in favour of A′. Use of rain the next day if he waters the lawn, a
this deﬁnition to show that the odds against 40% chance if he washes the car, and a 50%
A are equal to the reciprocal of the odds in chance if he plans a trip to the beach.
favour of A. Assuming George’s estimates are accurate,
what are the odds
9. Application Suppose the odds of the Toronto
a) in favour of rain tomorrow if he waters
Maple Leafs winning the Stanley Cup are
the lawn?
1:5, while the odds of the Montréal
Canadiens winning the Stanley Cup are b) in favour of rain tomorrow if he washes
2:13. What are the odds in favour of either the car?
Toronto or Montréal winning the Stanley c) against rain tomorrow if he plans a trip
Cup? to the beach?

10. What are the odds against drawing C
a) a face card from a standard deck? 13. Communication A volleyball coach claims
b) two face cards? that at the next game, the odds of her team
winning are 3:1, the odds against losing are
ACHIEVEMENT CHECK
5:1, and the odds against a tie are 7:1. Are
these odds possible? Explain your reasoning.
14. Inquiry/Problem Solving Aki is a participant
11. Mike has a loaded (or unfair) six-sided die. on a trivia-based game show. He has an
He rolls the die 200 times and determines equal likelihood on any given trial of being
the following probabilities for each score: asked a question from one of six categories:
P(1) = 0.11 Hollywood, Strange Places, Number Fun,
P(2) = 0.02 Who?, Having a Ball, and Write On! Aki
P(3) = 0.18 feels that he has a 50/50 chance of getting
P(4) = 0.21 Having a Ball or Strange Places questions
P(5) = 0.40 correct, but thinks he has a 90% probability
of getting any of the other questions right. If
a) What is P(6)?
Aki has to get two of three questions
b) Mike claims that the odds in favour of correct, what are his odds of winning?
tossing a prime number with this die are
the same as with a fair die. Do you agree 15. Inquiry/Problem Solving Use logic and
with his claim? mathematical reasoning to show that if
c) Using Mike’s die, devise a game with the odds in favour of A are given by
odds in Mike’s favour that an h h
ᎏᎏ, then P(A) = ᎏᎏ. Support your
unsuspecting person would be tempted k h+k
to play. Use probabilities to show that reasoning with an example.
the game is in Mike’s favour. Explain
why a person who does not realize that
the die is loaded might be tempted by
this game.

6.2 Odds • MHR 319

6.3 Probabilities Using Counting Techniques

How likely is it that, in a game of cards, you will be dealt just the hand that you
need? Most card players accept this question as an unknown, enjoying the
unpredictability of the game, but it can also be interesting to apply counting
analysis to such problems.

In some situations, the possible outcomes are not easy or convenient to count
individually. In many such cases, the counting techniques of permutations and
combinations (see Chapters 4 and 5, respectively) can be helpful for calculating
theoretical probabilities, or you can use a simulation to determine an empirical
probability.

I N V E S T I G AT E & I N Q U I R E : Fishing Simulation

Suppose a pond has only three types of fish: catfish, trout, and
bass, in the ratio 5:2:3. There are 50 fish in total. Assuming you
are allowed to catch only three fish before throwing them back,
consider the following two events:
• event A = {catching three trout}
• event B = {catching the three types of fish, in alphabetical order}

1. Carry out the following probability experiment, independently
or with a partner. You can use a hat or paper bag to represent
the pond, and some differently coloured chips or markers to
represent the fish. How many of each type of fish should you
release into the pond? Count out the appropriate numbers
and shake the container to simulate the fish swimming
around.

2. Draw a tree diagram to illustrate the different possible outcomes of this
experiment.

3. Catch three fish, one at a time, and record the results in a table. Replace all
three fish and shake the container enough to ensure that they are randomly
distributed. Repeat this process for a total of ten trials.

4. Based on these ten trials, determine the empirical probability of event A,
catching three trout. How accurate do you think this value is? Compare your
results with those of the rest of the class. How can you obtain a more accurate
empirical probability?

5. Repeat step 4 for event B, which is to catch a bass, catfish, and trout in order.


6. Perform step 3 again for 10 new trials. Calculate the empirical probabilities
of events A and B, based on your 20 trials. Do you think these probabilities
are more accurate than those from 10 trials? Explain why or why not.

7. If you were to repeat the experiment for 50 or 100 trials, would your results
be more accurate? Why or why not?

8. In this investigation, you knew exactly how many of each type of fish were in
the pond because they were counted out at the beginning. Describe how you
could use the techniques of this investigation to estimate the ratios of
different species in a real pond.

This section examines methods for determining the theoretical probabilities of
successive or multiple events.

Example 1 Using Permutations

Two brothers enter a race with five friends. The racers draw lots to determine
their starting positions. What is the probability that the older brother will start
in lane 1 with his brother beside him in lane 2?

Solution

A permutation nPr, or P(n, r), is the number of ways to select r objects from a set
of n objects, in a certain order. (See Chapter 4 for more about permutations.)
The sample space is the total number of ways the first two lanes can be
occupied. Thus,
n(S) = 7P2
7!
= ᎏ
(7 − 2)!
7!
= ᎏᎏ
5!
7 × 6 × (5!)
= ᎏᎏ
5!
= 42

The specific outcome of the older brother starting in lane 1 and the younger
brother starting in lane 2 can only happen one way, so n(A) = 1. Therefore,
n(A)
P(A) = ᎏ
n(S)
1
= ᎏᎏ
42
The probability that the older brother will start in lane 1 next to his brother in
1
lane 2 is ᎏᎏ, or approximately 2.3%.
42

6.3 Probabilities Using Counting Techniques • MHR 321

Example 2 Probability Using Combinations

A focus group of three members is to be randomly selected from a medical
team consisting of ﬁve doctors and seven technicians.
a) What is the probability that the focus group will be comprised of
doctors only?
b) What is the probability that the focus group will not be comprised of
doctors only?

Solution

΂ r ΃, is the number of ways to
n
a) A combination nCr, also written C(n, r) or

select r objects from a set of n objects, in any order. (See Chapter 5 for more
about combinations.) Let event A be selecting three doctors to form the
focus group. The number of possible ways to make this selection is

n(A) = 5C3
5!
= ᎏᎏ
3!(5 − 3)!
5 × 4 × 3!
= ᎏᎏ
3! × 2!
20
= ᎏᎏ
2
= 10

However, the focus group can consist of any three people from the team of 12.
n(S) = 12C3
12!
= ᎏᎏ
3!(12 − 3)!
12 × 11 × 10 × 9!
= ᎏᎏ
3! × 9!
1320
= ᎏᎏ
6
= 220

The probability of selecting a focus group of doctors only is
n(A)
P(A) = ᎏ
n(S)
10
= ᎏᎏ
220
1
= ᎏᎏ
22
1
The probability of selecting a focus group consisting of three doctors is ᎏᎏ,
22
or approximately 0.045.


b) Either the focus group is comprised of doctors only, or it is not. Project
Therefore, the probability of the complement of A, P(A′), gives Prep
the desired result.
When you determine the
P(A′ ) = 1 − P(A) classical probabilities for
1 your probability project,
= 1 − ᎏᎏ
22 you may need to apply
21 the counting techniques
= ᎏ
22 of permutations and
So, the probability of selecting a focus group not comprised of combinations.
21
doctors only is ᎏᎏ, or approximately 0.955.
22

Example 3 Probability Using the Fundamental Counting Principle

What is the probability that two or more students out of a class of 24 will have
the same birthday? Assume that no students were born on February 29.

The simplest method is to ﬁnd the probability of the complementary event that
no two people in the class have the same birthday.

Pick two students at random. The second student has a different birthday than
the ﬁrst for 364 of the 365 possible birthdays. Thus, the probability that the
364
two students have different birthdays is ᎏᎏ. Now add a third student. Since
365
there are 363 ways this person can have a different birthday from the other
two students, the probability that all three students have different birthdays
364 363
is ᎏᎏ × ᎏᎏ. Continuing this process, the probability that none of the
365 365
24 people have the same birthday is
n(A′)
P(A′) = ᎏ
n(S)
364 363 362 342
= ᎏ × ᎏ × ᎏ ×…× ᎏ
365 365 365 365
=⋅ 0.462

P(A) = 1 − P(A′)
= 1 − 0.462
= 0.538
The probability that at least two people in the group have the same birthday
is approximately 0.538.



Use the iterative functions of a graphing calculator to evaluate the
formula above much more easily. The prod( function on the LIST
MATH menu will find the product of a series of numbers. The
seq( function on the LIST OPS menu generates a sequence for the
range you specify. Combining these two functions allows you to
calculate the probability in a single step.

Key Concepts

• In probability experiments with many possible outcomes, you can apply the
fundamental counting principle and techniques using permutations and
combinations.

• Permutations are useful when order is important in the outcomes;
combinations are useful when order is not important.


1. In the game of bridge, each player is dealt 13 cards out of the deck of 52.
Explain how you would determine the probability of a player receiving
a) all hearts b) all hearts in ascending order

2. a) When should you apply permutations in solving probability problems,
and when should you apply combinations?
b) Provide an example of a situation where you would apply permutations
to solve a probability problem, other than those in this section.
c) Provide an example of a situation where you would apply combinations
to solve a probability problem, other than those in this section.

Practise 3. A fruit basket contains five red apples and
three green apples. Without looking, you
A randomly select two apples. What is the
1. Four friends, two females and two males, are probability that
playing contract bridge. Partners are a) you will select two red apples?
randomly assigned for each game. What is b) you will not select two green apples?
the probability that the two females will be
partners for the first game? 4. Refer to Example 1. What is the probability
that the two brothers will start beside each
2. What is the probability that at least two
other in any pair of lanes?
out of a group of eight friends will have the
same birthday?


Apply, Solve, Communicate b) What is the probability that the friends
will arrive in order of ascending age?
B
c) What assumptions must be made in parts
5. An athletic committee with three members a) and b)?
is to be randomly selected from a group of
six gymnasts, four weightlifters, and eight 9. A hockey team has two goalies, six defenders,
long-distance runners. Determine the eight wingers, and four centres. If the team
probability that randomly selects four players to attend a
a) the committee is comprised entirely of
charity function, what is the likelihood that
runners a) they are all wingers?
b) the committee is represented by each of b) no goalies or centres are selected?
the three types of athletes
10. Application A lottery promises to award
6. A messy drawer contains three black socks, ten grand-prize trips to Hawaii and sells
five blue socks, and eight white socks, none 5 400 000 tickets.
of which are paired up. If the owner grabs a) Determine the probability of winning a
two socks without looking, what is the grand prize if you buy
probability that both will be white? i) 1 ticket
7. a) A family of nine has a tradition of ii) 10 tickets
drawing two names from a hat to see iii) 100 tickets
whom they will each buy presents for. If b) Communication How many tickets do
there are three sisters in the family, and you need to buy in order to have a 5%
the youngest sister is always allowed the chance of winning a grand prize? Do you
first draw, determine the probability that think this strategy is sensible? Why or
the youngest sister will draw both of the why not?
other two sisters’ names. If she draws her
c) How many tickets do you need to ensure
own name, she replaces it and draws
a 50% chance of winning?
another.
b) Suppose that the tradition is modified 11. Suki is enrolled in one data-management
one year, so that the first person whose class at her school and Leo is in another. A
name is drawn is to receive a “main” school quiz team will have four volunteers,
present, and the second a less expensive, two randomly selected from each of the two
“fun” present. Determine the probability classes. Suki is one of five volunteers from
that the youngest sister will give a main her class, and Leo is one of four volunteers
present to the middle sister and a fun from his. Calculate the probability of the
present to the eldest sister. two being on the team and explain the steps
in your calculation.
8. Application
a) Laura, Dave, Monique, Marcus, and
Sarah are going to a party. What is the
probability that two of the girls will
arrive first?


12. a) Suppose 4 of the 22 tagged bucks are c) Could the random-number generator of
pte
ha randomly chosen for a behaviour study. a graphing calculator be used to simulate
C

r

What is the probability that this investigation? If so, explain how. If
m
P

r
oble
i) all four bucks have the cross-hatched not, explain why.
antlers? d) Outline the steps you would use to
ii) at least one buck has cross-hatched model this problem with software such
antlers? as FathomTM or a spreadsheet.
b) If two of the seven cross-hatched males e) Is the assumption that the fish are
are randomly selected for a health study, randomly distributed likely to be
what is the probability that the eldest of completely correct? Explain. What other
the seven will be selected first, followed assumptions might affect the accuracy of
by the second eldest? the calculated probabilities?

15. A network of city streets forms square
ACHIEVEMENT CHECK
blocks as shown in the diagram.
Knowledge/ Thinking/Inquiry/ Library

13. Suppose a bag contains the letters to spell
probability.
a) How many four-letter arrangements are
possible using these letters?
Pool
b) What is the probability that Barb
chooses four letters from the bag in the Jeanine leaves the library and walks toward
order that spell her name? the pool at the same time as Miguel leaves
c) Pick another four-letter arrangement the pool and walks toward the library.
and calculate the probability that it is Neither person follows a particular route,
chosen. except that both are always moving toward
their destination. What is the probability
d) What four-letter arrangement would be
that they will meet if they both walk at the
most likely to be picked? Explain your
same rate?
reasoning.
16. Inquiry/Problem Solving A committee is
C formed by randomly selecting from eight
14. Communication Refer to the fishing
nurses and two doctors. What is the
investigation at the beginning of this section. minimum committee size that ensures at
least a 90% probability that it will not be
a) Determine the theoretical probability of
comprised of nurses only?
i) catching three trout
ii) catching a bass, catfish, and trout in
alphabetical order
b) How do these results compare with the
empirical probabilities from the
investigation? How do you account for
any differences?


6.4 Dependent and Independent Events

If you have two examinations next Tuesday, what is the probability that you will
pass both of them? How can you predict the risk that a critical network server
and its backup will both fail? If you flip an ordinary coin repeatedly and get
heads 99 times in a row, is the next toss almost certain to come up tails?

In such situations, you are dealing with compound events involving two or
more separate events.

I N V E S T I G AT E & I N Q U I R E : G e t t i n g O u t o f J a i l i n M O N O P O LY ®

While playing MONOPOLY® for the first
time, Kenny finds himself in jail. To get out
of jail, he needs to roll doubles on a pair of
standard dice.

1. Determine the probability that Kenny
will roll doubles on his first try.

2. Suppose that Kenny fails to roll doubles
on his first two turns in jail. He reasons
that on his next turn, his odds are now
50/50 that he will get out of jail. Explain
how Kenny has reasoned this.

3. Do you agree or disagree with Kenny’s
reasoning? Explain.

4. What is the probability that Kenny will
get out of jail on his third attempt?

5. After how many turns is Kenny certain to
roll doubles? Explain.

6. Kenny’s opponent, Roberta, explains to
Kenny that each roll of the dice is an
independent event and that, since the
relatively low probability of rolling doubles never changes from trial to trial,
Kenny may never get out of jail and may as well just forfeit the game. Explain
the flaws in Roberta’s analysis.

6.4 Dependent and Independent Events • MHR 327

In some situations involving compound events, the occurrence of one event
has no effect on the occurrence of another. In such cases, the events are
independent.

Example 1 Simple Independent Events

a) A coin is flipped and turns up heads. What is the probability that the
second flip will turn up heads?
b) A coin is flipped four times and turns up heads each time. What is the
probability that the fifth trial will be heads?
c) Find the probability of tossing five heads in a row.
d) Comment on any difference between your answers to parts b) and c).

Solution
a) Because these events are independent, the outcome of the first toss has no
effect on the outcome of the second toss. Therefore, the probability of
tossing heads the second time is 0.5.

b) Although you might think “tails has to come up sometime,” there is still a
50/50 chance on each independent toss. The coin has no memory of the
past four trials! Therefore, the fifth toss still has just a 0.5 probability of
coming up heads.

c) Construct a tree diagram to represent five tosses of the coin.

H
H
T
H H
T
H T
H
H
T T
H
T There is an equal
T number of outcomes
H
H in which the first flip
H turns up tails.
T
H H
T
T T
H
H
T T
H
T
T


The number of outcomes doubles with each trial. After the fifth toss,
there are 25 or 32 possible outcomes, only one of which is five heads in a
row. So, the probability of five heads in a row, prior to any coin tosses,
1
is ᎏᎏ or 0.031 25.
32
d) The probability in part c) is much less than in part b). In part b), you
calculate only the probability for the fifth trial on its own. In part c), you
are finding the probability that every one of five separate events actually
happens.

Example 2 Probability of Two Different Independent Events

A coin is flipped while a die is rolled. What is the probability of flipping
heads and rolling 5 in a single trial?

Solution

Here, two independent events occur in a single trial. Let A be the event of
flipping heads, and B be the event of rolling 5. The notation P(A and B)
represents the compound, or joint, probability that both events, A and B,
will occur simultaneously. For independent events, the probabilities can
simply be multiplied together.

P(A and B) = P(A) × P(B)
1 1
= ᎏᎏ × ᎏᎏ
2 6
1
= ᎏᎏ
12
1
The probability of simultaneously flipping heads while rolling 5 is ᎏᎏ or
12
approximately 8.3%

In general, the compound probability of two independent events can be
calculated using the product rule for independent events:

P(A and B) = P(A) × P(B)

From the example above, you can see that the product rule for independent
events agrees with common sense. The product rule can also be derived
mathematically from the fundamental counting principle (see Chapter 4).


Proof:
A and B are separate events and so they correspond to separate sample spaces,
SA and SB.

Their probabilities are thus
n(A) n(B)
P(A) = ᎏ and P(B) = ᎏ .
n(SA ) n(SB )

Call the sample space for the compound event S, as usual.

You know that
n(A and B)
P(A and B) = ᎏᎏ (1)
n(S)
Because A and B are independent, you can apply the fundamental counting
principle to get an expression for n(A and B).

n(A and B) = n(A) × n(B) (2)

Similarly, you can also apply the fundamental counting principle to get an
expression for n(S).

n(S) = n(SA ) × n(SB ) (3)

Substitute equations (2) and (3) into equation (1).

n(A)n(B)
P(A and B) = ᎏᎏ
n(SA )n(SB )
n(A) n(B)
= ᎏ × ᎏ
n(SA ) n(SB )
= P(A) × P(B)

Example 3 Applying the Product Rule for Independent Events

Soo-Ling travels the same route to work every day. She has determined that
there is a 0.7 probability that she will wait for at least one red light and that
there is a 0.4 probability that she will hear her favourite new song on her way
to work.
a) What is the probability that Soo-Ling will not have to wait at a red light
and will hear her favourite song?
b) What are the odds in favour of Soo-Ling having to wait at a red light and
not hearing her favourite song?


Solution

a) Let A be the event of Soo-Ling having to wait at a red light, and B be the
event of hearing her favourite song. Assume A and B to be independent events.
In this case, you are interested in the combination A′ and B.
P(A′ and B) = P(A′) × P(B)
= (1 − P(A)) × P(B)
= (1 − 0.7) × 0.4
= 0.12
There is a 12% chance that Soo-Ling will hear her favourite song and not
have to wait at a red light on her way to work.

b) P(A and B′) = P(A) × P(B′)
= P(A) × (1 − P(B))
= 0.7 × (1 − 0.4)
= 0.42
The probability of Soo-Ling having to wait at a red light and not hearing her
favourite song is 42%.
The odds in favour of this happening are
P(A and B′)
odds in favour = ᎏᎏ
1 − P(A and B′)
42%
= ᎏᎏ
100% − 42%
42
= ᎏᎏ
58
21
= ᎏᎏ
29
The odds in favour of Soo-Ling having to wait at a red light and not hearing
her favourite song are 21:29.

In some cases, the probable outcome of an event, B, depends directly on the
outcome of another event, A. When this happens, the events are said to be
dependent. The conditional probability of B, P(B | A), is the probability that
B occurs, given that A has already occurred.

Example 4 Probability of Two Dependent Events

A professional hockey team has eight wingers. Three of these wingers are 30-goal
scorers, or “snipers.” Every fall the team plays an exhibition match with the club’s
farm team. In order to make the match more interesting for the fans, the coaches
agree to select two wingers at random from the pro team to play for the farm
team. What is the probability that two snipers will play for the farm team?


Solution

Let A = {first winger is a sniper} and B = {second winger is a sniper}. Three
of the eight wingers are snipers, so the probability of the first winger selected
being a sniper is
3
P(A) = ᎏᎏ
8
If the first winger selected is a sniper, then there are seven remaining wingers
to choose from, two of whom are snipers. Therefore,
2
P(B | A) = ᎏᎏ
7
Applying the fundamental counting principle, the probability of randomly
selecting two snipers for the farm team is the number of ways of selecting two
snipers divided by the number of ways of selecting any two wingers.
3×2
P(A and B) = ᎏ
8×7
3
= ᎏ
28
3
There is a ᎏᎏ or 10.7% probability that two professional snipers will play for
28
the farm team in the exhibition game.

Notice in Example 4 that, when two events A and B are Project
dependent, you can still multiply probabilities to find the Prep
probability that they both happen. However, you must use
the conditional probability for the second event. Thus, When designing your game for
the probability that both events will occur is given by the the probability project, you may
product rule for dependent events: decide to include situations
involving independent or
P(A and B) = P(A) × P(B | A) dependent events. If so, you will
need to apply the appropriate
This reads as: “The probability that both A and B will occur product rule in order to
equals the probability of A times the probability of B given determine classical probabilities.
that A has occurred.”

Example 5 Conditional Probability From Compound Probability

Serena’s computer sometimes crashes while she is trying to use her e-mail
program, OutTake. When OutTake “hangs” (stops responding to commands),
Serena is usually able to close OutTake without a system crash. In a computer
magazine, she reads that the probability of OutTake hanging in any 15-min
period is 2.5%, while the chance of OutTake and the operating system failing
together in any 15-min period is 1%. If OutTake is hanging, what is the
probability that the operating system will crash?


Solution

Let event A be OutTake hanging, and event B be an operating system failure.
Since event A can trigger event B, the two events are dependent. In fact,
you need to ﬁnd the conditional probability P(B | A). The data from the
magazine tells you that P(A) = 2.5%, and P(A and B) = 1%. Therefore,
P(A and B) = P(A) × P(B | A)
1% = 2.5% × P(B | A)
1%
P(B | A) = ᎏ
2.5%
= 0.4
There is a 40% chance that the operating system will crash when OutTake is
hanging.

Example 5 suggests a useful rearrangement of the product rule for dependent events.

P(A and B)
P(B | A ) = ᎏᎏ
P(A)
This equation is sometimes used to deﬁne the conditional probability P(B | A ).

Key Concepts

• If A and B are independent events, then the probability of both occurring is
given by P(A and B) = P(A) × P(B).

• If event B is dependent on event A, then the conditional probability of B given
A is P(B | A). In this case, the probability of both events occurring is given by
P(A and B) = P(A) × P(B | A).


1. Consider the probability of randomly drawing an ace from a standard deck
of cards. Discuss whether or not successive trials of this experiment are
independent or dependent events. Consider cases in which drawn cards are
a) replaced after each trial
b) not replaced after each trial

2. Suppose that for two particular events A and B, it is true that P(B A) = P(B).
|
What does this imply about the two events? (Hint: Try substituting this
equation into the product rule for dependent events.)


Practise 5. a) Rocco and Biff are two koala bears
participating in a series of animal
A behaviour tests. They each have 10 min
1. Classify each of the following as to solve a maze. Rocco has an 85%
independent or dependent events. probability of succeeding if he can smell
First Event Second Event the eucalyptus treat at the other end. He
a) Attending a rock Passing a final can smell the treat 60% of the time. Biff
concert on Tuesday examination the has a 70% chance of smelling the treat,
night following but when he does, he can solve the maze
Wednesday morning only 75% of the time. Neither bear will
b) Eating chocolate Winning at checkers try to solve the maze unless he smells the
c) Having blue eyes Having poor hearing eucalyptus. Determine which koala bear
d) Attending an Improving personal is more likely to enjoy a tasty treat on
employee training productivity any given trial.
session
e) Graduating from b) Communication Explain how you arrived
Running a marathon
university at your conclusion.
f) Going to a mall Purchasing a new
shirt 6. Shy Tenzin’s friends assure him that if he
asks Mikala out on a date, there is an 85%
2. Amitesh estimates that he has a 70% chance
chance that she will say yes. If there is a
of making the basketball team and a 20%
60% chance that Tenzin will summon the
chance of having failed his last geometry
courage to ask Mikala out to the dance next
quiz. He defines a “really bad day” as one in
week, what are the odds that they will be
which he gets cut from the team and fails his
seen at the dance together?
quiz. Assuming that Amitesh will receive
both pieces of news tomorrow, how likely is 7. When Ume’s hockey team uses a “rocket
it that he will have a really bad day? launch” breakout, she has a 55% likelihood
of receiving a cross-ice pass while at full
3. In the popular dice game Yahtzee®, a
speed. When she receives such a pass, the
Yahtzee occurs when five identical numbers 1
turn up on a set of five standard dice. What probability of getting her slapshot away is ᎏᎏ.
3
is the probability of rolling a Yahtzee on one Ume’s slapshot scores 22% of the time.
roll of the five dice? What is the probability of Ume scoring with
her slapshot when her team tries a rocket
Apply, Solve, Communicate launch?
B
8. Inquiry/Problem Solving Show that if A and
4. There are two tests for a particular antibody. B are dependent events, then the conditional
T A gives a correct result 95% of the time.
est probability P(A | B) is given by
T B is accurate 89% of the time. If a patient
est P(A and B)
is given both tests, find the probability that P(A | B) = ᎏᎏ .
P(B)
a) both tests give the correct result
b) neither test gives the correct result
c) at least one of the tests gives the correct
result


9. A consultant’s study found Megatran’s call 14. Application A critical circuit in a
centre had a 5% chance of transferring a communication network relies on a set of
call about schedules to the lost articles eight identical relays. If any one of the relays
department by mistake. The same study fails, it will disrupt the entire network. The
shows that, 1% of the time, customers design engineer must ensure a 90%
calling for schedules have to wait on hold, probability that the network will not fail
only to discover that they have been over a five-year period. What is the
mistakenly transferred to the lost articles maximum tolerable probability of failure for
department. What are the chances that a each relay?
customer transferred to lost articles will be
C
put on hold?
15. a) Show that if a coin is tossed n times, the
10. Pinder has examinations coming up in data probability of tossing n heads is given by
management and biology. He estimates that
΂΃
1 n
his odds in favour of passing the data- P(A) = ᎏᎏ .
2
management examination are 17:3 and his
b) What is the probability of getting at least
odds against passing the biology examination
one tail in seven tosses?
are 3:7. Assume these to be independent
events. 16. What is the probability of not throwing 7 or
a) What is the probability that Pinder will doubles for six consecutive throws with a
pass both exams? pair of dice?
b) What are the odds in favour of Pinder
17. Laurie, an avid golfer, gives herself a 70%
failing both exams? chance of breaking par (scoring less than 72
c) What factors could make these two on a round of 18 holes) if the weather is
events dependent? calm, but only a 15% chance of breaking par
on windy days. The weather forecast gives a
11. Inquiry/Problem Solving How likely is it for
40% probability of high winds tomorrow.
a group of five friends to have the same birth
What is the likelihood that Laurie will break
month? State any assumptions you make for
par tomorrow, assuming that she plays one
your calculation.
round of golf?
12. Determine the probability that a captured
pte
18. Application The Tigers are leading the
ha deer has the bald patch condition.
Storm one game to none in a best-of-five
C

r

playoff series. After a playoff win, the
m
P

r
oble

13. Communication Five different CD-ROM probability of the Tigers winning the next
games, Garble, Trapster, Zoom!, Bungie, game is 60%, while after a loss, their
and Blast ’Em, are offered as a promotion probability of winning the next game drops
by SugarRush cereals. One game is by 5%. The first team to win three games
randomly included with each box of cereal. takes the series. Assume there are no ties.
What is the probability of the Storm coming
a) Determine the probability of getting all
back to win the series?
5 games if 12 boxes are purchased.
b) Explain the steps in your solution.
c) Discuss any assumptions that you make
in your analysis.

6.5 Mutually Exclusive Events

The phone rings. Jacques is really hoping that it is one of his friends calling
about either softball or band practice. Could the call be about both?

In such situations, more than one event could occur during a single trial. You
need to compare the events in terms of the outcomes that make them up. What
is the chance that at least one of the events happens? Is the situation “either/or,”
or can both events occur?

I N V E S T I G AT E & I N Q U I R E : Baseball Pitches

Marie, at bat for the Coyotes, is facing Anton, who is
pitching for the Power Trippers. Anton uses three pitches: a
fastball, a curveball, and a slider. Marie feels she has a good
chance of making a base hit, or better, if Anton throws either
a fastball or a slider. The count is two strikes and three balls.
In such full-count situations, Anton goes to his curveball one
third of the time, his slider half as often, and his fastball the
rest of the time.

1. Determine the probability of Anton throwing his
a) curveball b) slider c) fastball

2. a) What is the probability that Marie will get the pitch
she does not want?
b) Explain how you can use this information to
determine the probability that Marie will get a pitch
she likes.

3. a) Show another method of determining this probability.
b) Explain your method.

4. What do your answers to questions 2 and 3 suggest
about the probabilities of events that cannot happen
simultaneously?

The possible events in this investigation are said to be mutually exclusive (or
disjoint) since they cannot occur at the same time. The pitch could not be both a
fastball and a slider, for example. In this particular problem, you were interested in
the probability of either of two favourable events. You can use the notation P(A or B)
to stand for the probability of either A or B occurring.


Example 1 Probability of Mutually Exclusive Events

Teri attends a fundraiser at which 15 T-shirts are being given away as door
prizes. Door prize winners are randomly given a shirt from a stock of 2 black
shirts, 4 blue shirts, and 9 white shirts. Teri really likes the black and blue shirts,
but is not too keen on the white ones. Assuming that Teri wins the ﬁrst door
prize, what is the probability that she will get a shirt that she likes?

Solution

Let A be the event that Teri wins a black shirt, and B be the event that she
wins a blue shirt.
2 4
P(A) = ᎏ and P(B) = ᎏ
15 15
Teri would be happy if either A or B occurred.
There are 2 + 4 = 6 non-white shirts, so
6
P(A or B) = ᎏ
15
2
= ᎏᎏ
5
2
The probability of Teri winning a shirt that she likes is ᎏᎏ or 40%. Notice that
5
this probability is simply the sum of the probabilities of the two mutually
exclusive events.

When events A and B are mutually exclusive, the probability that A or B
will occur is given by the addition rule for mutually exclusive events:

P(A or B) = P(A) + P(B)

A Venn diagram shows mutually exclusive events as non-overlapping,
S
or disjoint. Thus, you can apply the additive counting principle (see
Chapter 4) to prove this rule. A B

Proof:
If A and B are mutually exclusive events, then

n(A or B)
P(A or B) = ᎏᎏ
n(S)
n(A) + n(B)
= ᎏᎏ A and B are disjoint sets, and thus share no elements.
n(S )
n(A) n(B)
= ᎏ + ᎏ
n(S) n(S)
= P(A) + P(B)

6.5 Mutually Exclusive Events • MHR 337

In some situations, events are non-mutually exclusive, which means Second die
that they can occur simultaneously. For example, consider a board game 1 2 3 4 5 6
in which you need to roll either an 8 or doubles, using two dice.
1 2 3 4 5 6 7
Notice that in one outcome, rolling two fours, both events have 2 3 4 5 6 7 8
occurred simultaneously. Hence, these events are not mutually First 3 4 5 6 7 8 9
exclusive. Counting the outcomes in the diagram shows that the die 4 5 6 7 8 9 10
10 5
probability of rolling either an 8 or doubles is ᎏ or ᎏ . You 5 6 7 8 9 10 11
36 18 6 7 8 9 10 11 12
need to take care not to count the (4, 4) outcome twice. You are
applying the principle of inclusion and exclusion, which was explained
in greater detail in Chapter 5.

Example 2 Probability of Non-Mutually Exclusive Events

A card is randomly selected from a standard deck of cards. What is the
probability that either a heart or a face card (jack, queen, or king) is selected?

Solution

Let event A be that a heart is selected, and event B be that a face card is
selected.
13 12
P(A) = ᎏ and P(B) = ᎏ
52 52
If you add these probabilities, you get
13 12
P(A) + P(B) = ᎏ + ᎏ
52 52
25
= ᎏ
52
However, since the jack, queen, and king of hearts are in both A and B, the
sum P(A) + P(B) actually includes these outcomes twice.
A ♣ 2 ♣ 3 ♣ 4 ♣ 5 ♣ 6 ♣ 7 ♣ 8 ♣ 9 ♣ 10 ♣ J ♣ Q ♣ K ♣

A ♦ 2 ♦ 3 ♦ 4 ♦ 5 ♦ 6 ♦ 7 ♦ 8 ♦ 9 ♦ 10 ♦ J ♦ Q ♦ K ♦

A ♥ 2 ♥ 3 ♥ 4 ♥ 5 ♥ 6 ♥ 7 ♥ 8 ♥ 9 ♥ 10 ♥ J ♥ Q ♥ K ♥

A ♠ 2 ♠ 3 ♠ 4 ♠ 5 ♠ 6 ♠ 7 ♠ 8 ♠ 9 ♠ 10 ♠ J ♠ Q ♠ K ♠

Based on the diagram, the actual theoretical probability of drawing either
22 11
a heart or a face card is ᎏᎏ, or ᎏᎏ. You can ﬁnd the correct value by subtracting
52 26
the probability of selecting the three elements that were counted twice.


13 12 3 S
P(A or B) = ᎏ + ᎏ − ᎏ
52 52 52
22 Hearts Face card
= ᎏᎏ 13 12
52 P = ––– P = –––
52 52
11
= ᎏᎏ
26
Heart and face card
The probability that either a heart 3
P = –––
11 52
or a face card is selected is ᎏᎏ.
26

When events A and B are non-mutually exclusive, the probability that S
A or B will occur is given by the addition rule for non-mutually
exclusive events: A B

P(A or B) = P(A) + P(B) − P(A and B)

A and B

Example 3 Applying the Addition Rule for Project
Non-Mutually Exclusive Events Prep

An electronics manufacturer is testing a new product to see When analysing the possible
whether it requires a surge protector. The tests show that a outcomes for your game in the
voltage spike has a 0.2% probability of damaging the probability project, you may need to
product’s power supply, a 0.6% probability of damaging consider mutually exclusive or non-
downstream components, and a 0.1% probability of mutually exclusive events. If so, you
damaging both the power supply and other components. will need to apply the appropriate
Determine the probability that a voltage spike will damage addition rule to determine theoretical
the product. probabilities.

Solution

Let A be damage to the power supply and C be S
damage to other components.
A C
The overlapping region represents the probability that 0.2 0.1 0.6
a voltage surge damages both the power supply and another
component. The probability that either A or C occurs is
given by
P(A or C) = P(A) + P(C) − P(A and C)
= 0.2% + 0.6% − 0.1%
= 0.7%
There is a 0.7% probability that a voltage spike will damage the product.


Key Concepts

• If A and B are mutually exclusive events, then the probability of either A or B
occurring is given by P(A or B) = P(A) + P(B).

• If A and B are non-mutually exclusive events, then the probability of either
A or B occurring is given by P(A or B) = P(A) + P(B) − P(A and B).


1. Are an event and its complement mutually exclusive? Explain.

2. Explain how to determine the probability of randomly throwing either a
composite number or an odd number using a pair of dice.

3. a) Explain the difference between independent events and mutually
exclusive events.
b) Support your explanation with an example of each.
c) Why do you add probabilities in one case and multiply them in the other?

Practise 2. Nine members of a baseball team are
randomly assigned field positions. There are
A three outfielders, four infielders, a pitcher,
1. Classify each pair of events as mutually and a catcher. Troy is happy to play any
exclusive or non-mutually exclusive. position except catcher or outfielder.
Determine the probability that Troy will
Event A Event B
Randomly drawing Randomly drawing
be assigned to play
a)
a grey sock from a a wool sock from a a) catcher
drawer drawer
b) outfielder
b) Randomly selecting Randomly selecting
a student with a student on the c) a position he does not like
brown eyes honour roll
c) Having an even Having an odd 3. A car dealership analysed its customer
number of students number of students database and discovered that in the last
in your class in your class model year, 28% of its customers chose a
d) Rolling a six with a Rolling a prime 2-door model, 46% chose a 4-door model,
die number with a die 19% chose a minivan, and 7% chose a
e) Your birthday Your birthday
falling on a falling on a
4-by-4 vehicle. If a customer was selected
Saturday next year weekend next year randomly from this database, what is the
f) Getting an A on the Passing the next test probability that the customer
next test a) bought a 4-by-4 vehicle?
g) Calm weather at Stormy weather at
noon tomorrow noon tomorrow b) did not buy a minivan?
h) Sunny weather next Rainy weather next c) bought a 2-door or a 4-door model?
week week
d) bought a minivan or a 4-by-4 vehicle?


Apply, Solve, Communicate 7. Application In an animal-behaviour study,
hamsters were tested with a number
B of intelligence tasks, as shown in the
4. As a promotion, a resort has a draw for free table below.
family day-passes. The resort considers July,
Number of Tests Number of Hamsters
August, March, and December to be
0 10
“vacation months.”
1 6
a) If the free passes are randomly dated,
2 4
what is the probability that a day-pass
3 3
will be dated within
4 or more 5
i) a vacation month?
ii) June, July, or August If a hamster is randomly chosen from this
study group, what is the likelihood that the
b) Draw a Venn diagram of the events in
hamster has participated in
part a).
a) exactly three tests?
5. A certain provincial park has 220 campsites. b) fewer than two tests?
A total of 80 sites have electricity. Of the 52
c) either one or two tests?
sites on the lakeshore, 22 of them have
electricity. If a site is selected at random, what d) no tests or more than three tests?
is the probability that 8. Communication
a) it will be on the lakeshore?
a) Prove that, if A and B are non-mutually
b) it will have electricity? exclusive events, the probability of either
c) it will either have electricity or be on the A or B occurring is given by
lakeshore? P(A or B) = P(A) + P(B) − P(A and B).
d) it will be on the lakeshore and not have b) What can you conclude if P(A and B) = 0?
electricity? Give reasons for your conclusion.

6. A market-research firm monitored 1000 9. Inquiry/Problem Solving Design a game in
television viewers, consisting of 800 adults which the probability of drawing a winning
and 200 children, to evaluate a new comedy card from a standard deck is between 55%
series that aired for the first time last week. and 60%.
Research indicated that 250 adults and
10. Determine the probability that a captured
148 children viewed some or all of the pte
ha deer has either cross-hatched antlers or bald
program. If one of the 1000 viewers was
C

r

patches. Are these events mutually exclusive?
selected, what is the probability that
m
P

r
oble
Why or why not?
a) the viewer was an adult who did not
watch the new program? 11. The eight members of the debating club
b) the viewer was a child who watched pose for a yearbook photograph. If they line
the new program? up randomly, what is the probability that
c) the viewer was an adult or someone a) either Hania will be first in the row or
who watched the new program? Aaron will be last?
b) Hania will be first and Aaron will not be
last?


ACHIEVEMENT CHECK C
Communication Application 13. A grade 12 student is selected at random to
sit on a university liaison committee. Of the
12. Consider a Stanley Cup playoff series in 120 students enrolled in the grade 12
which the Toronto Maple Leafs hockey university-preparation mathematics courses,
team faces the Ottawa Senators. Toronto • 28 are enrolled in data management only
hosts the first, second, and if needed, fifth • 40 are enrolled in calculus only
and seventh games in this best-of-seven • 15 are enrolled in geometry only
contest. The Leafs have a 65% chance of • 16 are enrolled in both data management
beating the Senators at home in the first and calculus
game. After that, they have a 60% chance • 12 are enrolled in both calculus and geometry
of a win at home if they won the previous • 6 are enrolled in both geometry and data
game, but a 70% chance if they are management
bouncing back from a loss. Similarly, the • 3 are enrolled in all three of data
Leafs’ chances of victory in Ottawa are management, calculus, and geometry
40% after a win and 45% after a loss. a) Draw a Venn diagram to illustrate this
a) Construct a tree diagram to illustrate situation.
all the possible outcomes of the first b) Determine the probability that the
three games. student selected will be enrolled in either
b) Consider the following events: data management or calculus.
A = {Leafs lose the first game but go c) Determine the probability that the
on to win the series in the fifth game} student selected will be enrolled in only
B = {Leafs win the series in the fifth one of the three courses.
game}
14. Application For a particular species of cat,
C = {Leafs lose the series in the fifth the odds against a kitten being born with
game} either blue eyes or white spots are 3:1. If the
Identify all the outcomes that make up probability of a kitten exhibiting only one of
each event, using strings of letters, such these traits is equal and the probability of
as LLSLL. Are any pairs from these exhibiting both traits is 10%, what are the
three events mutually exclusive? odds in favour of a kitten having blue eyes?
c) What is the probability of event A in
15. Communication
part b)?
a) A standard deck of cards is shuffled and
d) What is the chance of the Leafs winning
three cards are selected. What is the
in exactly five games?
probability that the third card is either
e) Explain how you found your answers to
a red face card or a king if the king of
parts c) and d). diamonds and the king of spades are
selected as the first two cards?
b) Does this probability change if the first
two cards selected are the queen of
diamonds and the king of spades? Explain.


16. Inquiry/Problem Solving The table below lists v) a male or a graduate in mathematics
the degrees granted by Canadian universities and physical sciences?
from 1994 to 1998 in various ﬁelds of study. b) If a male graduate from 1996 is selected
a) If a Canadian university graduate from at random, what is the probability that
1998 is chosen at random, what is the he is graduating in mathematics and
probability that the student is physical sciences?
i) a male? c) If a mathematics and physical sciences
ii) a graduate in mathematics and graduate is selected at random from the
physical sciences? period 1994 to 1996, what is the
probability that the graduate is a male?
iii) a male graduating in mathematics and
physical sciences? d) Do you think that being a male and
graduating in mathematics and physical
iv) not a male graduating in mathematics
sciences are independent events? Give
and physical sciences?
reasons for your hypothesis.

1994 1995 1996 1997 1998
Canada 178 074 178 066 178 116 173 937 172 076
Male 76 470 76 022 75 106 73 041 71 949
Female 101 604 102 044 103 010 100 896 100 127
Social sciences 69 583 68 685 67 862 66 665 67 019
Male 30 700 29 741 29 029 28 421 27 993
Female 38 883 38 944 38 833 38 244 39 026
Education 30 369 30 643 29 792 27 807 25 956
Male 9093 9400 8693 8036 7565
Female 21 276 21 243 21 099 19 771 18 391
Humanities 23 071 22 511 22 357 21 373 20 816
Male 8427 8428 8277 8034 7589
Female 14 644 14 083 14 080 13 339 13 227
Health professions and occupations 12 183 12 473 12 895 13 073 12 658
Male 3475 3461 3517 3460 3514
Female 8708 9012 9378 9613 9144
Engineering and applied sciences 12 597 12 863 13 068 12 768 12 830
Male 10 285 10 284 10 446 10 125 10 121
Female 2312 2579 2622 2643 2709
Agriculture and biological sciences 10 087 10 501 11 400 11 775 12 209
Male 4309 4399 4756 4780 4779
Female 5778 6102 6644 6995 7430
Mathematics and physical sciences 9551 9879 9786 9738 9992
Male 6697 6941 6726 6749 6876
Female 2854 2938 3060 2989 3116
Fine and applied arts 5308 5240 5201 5206 5256
Male 1773 1740 1780 1706 1735
Female 3535 3500 3421 3500 3521
Arts and sciences 5325 5271 5755 5532 5340
Male 1711 1628 1882 1730 1777
Female 3614 3643 3873 3802 3563


6.6 Applying Matrices to Probability Problems

In some situations, the probability of an
outcome depends on the outcome of the
previous trial. Often this pattern appears in
stock market trends, weather patterns,
athletic performance, and consumer habits.
Dependent probabilities can be calculated
using Markov chains, a powerful probability
model pioneered about a century ago by the
Russian mathematician Andrei Markov.

I N V E S T I G AT E & I N Q U I R E : Running Late

Although Marla tries hard to be punctual, the demands of her home life and the
challenges of commuting sometimes cause her to be late for work. When she is
late, she tries especially hard to be punctual the next day. Suppose that the
following pattern emerges: If Marla is punctual on any given day, then there is
a 70% chance that she will be punctual the next day and a 30% chance that she
will be late. On days she is late, however, there is a 90% chance that she will be
punctual the next day and just a 10% chance that she will be late. Suppose
Marla is punctual on the first day of the work week.

1. Create a tree diagram of the possible outcomes for the second and third
days. Show the probability for each branch.

2. a) Describe two branches in which Marla is punctual on day 3.
b) Use the product rule for dependent events on page 332 to calculate the
compound probability of Marla being punctual on day 2 and on day 3.
c) Find the probability of Marla being late on day 2 and punctual on day 3.
d) Use the results from parts b) and c) to determine the probability that
Marla will be punctual on day 3.

3. Repeat question 2 for the outcome of Marla being late on day 3.

4. a) Create a 1 × 2 matrix A in which the first element is the probability that
Marla is punctual and the second element is the probability that she is
late on day 1. Recall that Marla is punctual on day 1.
b) Create a 2 × 2 matrix B in which the elements in each row represent
conditional probabilities that Marla will be punctual and late. Let the first
row be the probabilities after a day in which Marla was punctual, and the
second row be the probabilities after a day in which she was late.


c) Evaluate A × B and A × B2.
d) Compare the results of part c) with your answers to questions 2 and 3.
Explain what you notice.
e) What does the first row of the matrix B2 represent?

The matrix model you have just developed is an example of a Markov chain,
a probability model in which the outcome of any trial depends directly on the
outcome of the previous trial. Using matrix operations can simplify
probability calculations, especially in determining long-term trends.

The 1 × 2 matrix A in the investigation is an initial probability vector, S (0),
and represents the probabilities of the initial state of a Markov chain. The
2 × 2 matrix B is a transition matrix, P, and represents the probabilities of
moving from any initial state to a new state in any trial.

These matrices have been arranged such that the product S (0) × P generates
the row matrix that gives the probabilities of each state after one trial. This
matrix is called the first-step probability vector, S (1). In general, the nth-
step probability vector, S (n), can be obtained by repeatedly multiplying the
probability vector by P. Sometimes these vectors are also called first-state
and nth-state vectors, respectively.

Notice that each entry in a probability vector or a transition matrix is a
probability and must therefore be between 0 and 1. The possible states in a
Markov chain are always mutually exclusive events, one of which must occur
at each stage. Therefore, the entries in a probability vector must sum to 1, as
must the entries in each row of the transition matrix.

Example 1 Probability Vectors

Two video stores, Video Vic’s and MovieMaster, have just opened in a new
residential area. Initially, they each have half of the market for rented movies.
A customer who rents from Video Vic’s has a 60% probability of renting from
Video Vic’s the next time and a 40% chance of renting from MovieMaster.
On the other hand, a customer initially renting from MovieMaster has only
a 30% likelihood of renting from MovieMaster the next time and a 70%
probability of renting from Video Vic’s.
a) What is the initial probability vector?
b) What is the transition matrix?
c) What is the probability of a customer renting a movie from each store
the second time?
d) What is the probability of a customer renting a movie from each store
the third time?
e) What assumption are you making in part d)? How realistic is it?
6.6 Applying Matrices to Probability Problems • MHR 345

Solution

a) Initially, each store has 50% of the market, so, the initial probability vector is

VV MM
S (0) = [0.5 0.5]

b) The first row of the transition matrix represents the probabilities for the
second rental by customers whose initial choice was Video Vic’s. There is a
60% chance that the customer returns, so the first entry is 0.6. It is 40%
likely that the customer will rent from MovieMaster, so the second entry
is 0.4.
Similarly, the second row of the transition matrix represents the probabilities
for the second rental by customers whose first choice was MovieMaster.
There is a 30% chance that a customer will return on the next visit, and a
70% chance that the customer will try Video Vic’s.
VV MM
P=
΄ 0.6 0.4 VV
0.7 0.3 MM ΅
Regardless of which store the customer chooses the first time, you are
assuming that there are only two choices for the next visit. Hence, the
sum of the probabilities in each row equals one.

c) To find the probabilities of a customer renting from either store on the
second visit, calculate the first-step probability vector, S (1):

S (1) = S (0)P
= [0.5 0.5] ΄ 0.6
0.7
0.4
0.3 ΅
= [0.65 0.35]

This new vector shows that there is a 65% probability that a customer will
rent a movie from Video Vic’s on the second visit to a video store and a 35%
chance that the customer will rent from MovieMaster.

d) To determine the probabilities of which store a customer will pick on the
third visit, calculate the second-step probability vector, S (2):

S (2) = S (1)P
= [0.65 0.35] ΄ 0.6
0.7
0.4
0.3 ΅
= [0.635 0.365]

So, on a third visit, a customer is 63.5% likely to rent from Video Vic’s and
36.5% likely to rent from MovieMaster.


e) To calculate the second-step probabilities, you assume that the conditional
transition probabilities do not change. This assumption might not be
realistic since customers who are 70% likely to switch away from
MovieMaster may not be as much as 40% likely to switch back, unless they
forget why they switched in the ﬁrst place. In other words, Markov chains
have no long-term memory. They recall only the latest state in predicting
the next one.

Note that the result in Example 1d) could be calculated in another way.

S (2) = S (1)P
= (S (0)P)P
= S (0)(PP) since matrix multiplication is associative
= S (0)P 2

Similarly, S (3) = S (0)P 3, and so on. In general, the nth-step probability vector, S (n),
is given by

S (n) = S (0)P n

This result enables you to determine higher-state probability vectors easily
using a graphing calculator or software.

Example 2 Long-Term Market Share

A marketing-research ﬁrm has tracked the sales of three brands of hockey sticks.
Each year, on average,
• Player-One keeps 70% of its customers, but loses 20% to Slapshot and 10%
to Extreme Styx
• Slapshot keeps 65% of its customers, but loses 10% to Extreme Styx and 25%
to Player-One
• Extreme Styx keeps 55% of its customers, but loses 30% to Player-One and
15% to Slapshot
a) What is the transition matrix?
b) Assuming each brand begins with an equal market share, determine the
market share of each brand after one, two, and three years.
c) Determine the long-range market share of each brand.
d) What assumption must you make to answer part c)?



a) The transition matrix is

P S E

΄ ΅
0.7 0.2 0.1 P
P= 0.25 0.65 0.1 S
0.3 0.15 0.55 E
b) Assuming each brand begins with an equal market share, the initial
probability vector is

΄ ΅
1 1 1
S (0) = ᎏᎏ ᎏᎏ ᎏᎏ
3 3 3
To determine the market shares of each brand after one year, compute the
ﬁrst-step probability vector.

S (1) = S (0)P

΄ ΅
0.7 0.2 0.1
΄ ΅
1 1 1
= ᎏᎏ ᎏᎏ ᎏᎏ 0.25 0.65 0.1
3 3 3 0.3 0.15 0.55
– –
= [0.416 0.3 0.25]

So, after one year Player-One will have a market share of approximately
42%, Slapshot will have 33%, and Extreme Styx will have 25%.

Similarly, you can predict the market shares after two years using

S (2) = S (1)P

΄ ΅
0.7 0.2 0.1
– –
= [0.416 0.3 0.25] 0.25 0.65 0.1
0.3 0.15 0.55
= [0.45 0.3375 0.2125]

After two years, Player-One will have approximately 45% of the market,
Slapshot will have 34%, and Extreme Styx will have 21%.

The probabilities after three years are given by

S (3) = S (2)P

΄ ΅
0.7 0.2 0.1
= [0.45 0.3375 0.2125] 0.25 0.65 0.1
0.3 0.15 0.55
= [0.463 0.341 0.196]

After three years, Player-One will have approximately 46% of the market,
Slapshot will have 34%, and Extreme Styx will have 20%.


c) The results from part b) suggest that the relative market shares may be
converging to a steady state over a long period of time. You can test this
hypothesis by calculating higher-state vectors and checking for stability.

For example,
S (10) = S (9)P S (11) = S (10)P
= [0.471 0.347 0.182] = [0.471 0.347 0.182]

The values of S (10) and S (11) are equal. It is easy to verify that they are equal to
all higher orders of S (n) as well. The Markov chain has reached a steady state.
A steady-state vector is a probability vector that remains unchanged when
multiplied by the transition matrix. A steady state has been reached if
S (n) = S (n)P
= S (n+1)
In this case, the steady state vector [0.471 0.347 0.182] indicates that, over a
long period of time, Player-One will have approximately 47% of the market
for hockey sticks, while Slapshot and Extreme Styx will have 35% and 18%,
respectively, based on current trends.

d) The assumption you make in part c) is that the transition matrix does not
change, that is, the market trends stay the same over the long term.


a) Use the MATRX EDIT menu to enter and store a matrix for the transition matrix B.

b) Similarly, enter the initial probability vector as matrix A. Then, use the MATRX
EDIT menu to enter the calculation A × B on the home screen. The resulting
matrix shows the market shares after one year are 42%, 33%, and 25%,
respectively.

To ﬁnd the second-step probability vector use the formula S (2) = S (0)P2.
Enter A × B2 using the MATRX NAMES menu and the 2 key. After
two years, therefore, the market shares are 45%, 34%, and 21%,
respectively.


Similarly, enter A × B 3 to ﬁnd the third-step probability vector. After
three years, the market shares are 46%, 34%, and 20%, respectively.

c) Higher-state probability vectors are easy to determine with a graphing
calculator.

S (10) = S (0)P 10
= [0.471 0.347 0.182]

S (100) = S (0)P 100
= [0.471 0.347 0.182]

S (10) and S (100) are equal. The tiny difference between S (10) and S (100) is
unimportant since the original data has only two signiﬁcant digits. Thus,
[0.471 0.347 0.182] is a steady-state vector, and the long-term market
shares are predicted to be about 47%, 35%, and 18% for Player-One,
Slapshot, and Extreme Styx, respectively.

Regular Markov chains always achieve a steady state. A Markov chain is Project
regular if the transition matrix P or some power of P has no zero entries. Prep
Thus, regular Markov chains are fairly easy to identify. A regular Markov
chain will reach the same steady state regardless of the initial probability vector. In the probability
project, you may
need to use
Example 3 Steady State of a Regular Markov Chain
Markov chains to
Suppose that Player-One and Slapshot initially split most of the market evenly determine long-
between them, and that Extreme Styx, a relatively new company, starts with a term probabilities.
10% market share.
a) Determine each company’s market share after one year.
b) Predict the long-term market shares.

Solution

a) The initial probability vector is
S (0) = [0.45 0.45 0.1]

Using the same transition matrix as in Example 2,
S(1) = S(0)P

΄ ΅
0.7 0.2 0.1
= [0.45 0.45 0.1] 0.25 0.65 0.1
0.3 0.15 0.55
= [0.4575 0.3975 0.145]
– –
These market shares differ from those in Example 2, where S (1) = [0.416 0.3 0.25].


b) S (100) = S (0)P 100
= [0.471 0.347 0.182]

In the long term, the steady state is the same as in Example 2. Notice that
although the short-term results differ as seen in part a), the same steady
state is achieved in the long term.

The steady state of a regular Markov chain can also be determined analytically.

Example 4 Analytic Determination of Steady State

The weather near a certain seaport follows this pattern: If it is a calm day, there
is a 70% chance that the next day will be calm and a 30% chance that it will be
stormy. If it is a stormy day, the chances are 50/50 that the next day will also be
stormy. Determine the long-term probability for the weather at the port.

Solution
The transition matrix for this Markov chain is
C S
P= ΄0.7 0.3 C
0.5 0.5 S ΅
The steady-state vector will be a 1 × 2 matrix, S (n) = [ p q].

The Markov chain will reach a steady state when S (n) = S (n)P, so

[ p q] = [ p q]
΄ 0.7
0.5
0.3
0.5 ΅
= [0.7p + 0.5q 0.3p + 0.5q]

Setting first elements equal and second elements equal gives two equations in
two unknowns. These equations are dependent, so they define only one
relationship between p and q.

p = 0.7p + 0.5q
q = 0.3p + 0.5q

Subtracting the second equation from the first gives

p − q = 0.4p
q = 0.6p


Now, use the fact that the sum of probabilities at any state must equal 1,
p+q=1
p + 0.6p = 1
1
p= ᎏ
1.6
= 0.625
q=1−p
= 0.375
So, the steady-state vector for the weather is [0.625 0.375]. Over the long term,
there will be a 62.5% probability of a calm day and 37.5% chance of a stormy
day at the seaport.

Key Concepts

• The theory of Markov chains can be applied to probability models in which
the outcome of one trial directly affects the outcome of the next trial.

• Regular Markov chains eventually reach a steady state, which can be used to
make long-term predictions.


1. Why must a transition matrix always be square?

2. Given an initial probability vector S (0) = [0.4 0.6] and a transition matrix

P= ΄ 0.5
0.3 0.7 ΅
0.5 , state which of the following equations is easier to use

for determining the third-step probability vector:
S (3) = S (2)P or S (3) = S (0)P 3
Explain your choice.

3. Explain how you can determine whether a Markov chain has reached a
steady state after k trials.

4. What property or properties must events A, B, and C have if they are the
only possible different states of a Markov chain?


A B
1. Which of the following cannot be an initial 4. Refer to question 3.
probability vector? Explain why. a) Which company do you think will
a) [0.2 0.45 0.25] increase its long-term market share,
b) [0.29 0.71] based on the information provided?
Explain why you think so.
c)
΄ 0.4 ΅
0.6 b) Calculate the steady-state vector for the
Markov chain.
d) [0.4 −0.1 0.7]
c) Which company increased its market
e) [0.4 0.2 0.15 0.25] share over the long term?
2. Which of the following cannot be a d) Compare this result with your answer
transition matrix? Explain why. to part a). Explain any differences.

΄ ΅
0.3 0.3 0.4
5. For which of these transition matrices will
a) 0.1 0 0.9
the Markov chain be regular? In each case,
0.2 0.3 0.4
explain why.
b)
΄ 0.2
0.65
0.8
0.35 ΅ a)
΄ 0.2
0.95
0.8
0.05 ΅
c)
΄ 0.5
0.3
0.1
0.22
0.4
0.48 ΅ b)
΄ 1 0΅
0 1
3. Two competing companies, ZapShot and

΄ ΅
0.1 0.6 0.3
E-pics, manufacture and sell digital cameras.
c) 0.33 0.3 0.37
Customer surveys suggest that the
0.5 0 0.5
companies’ market shares can be modelled
using a Markov chain with the following 6. Gina noticed that the performance of her
initial probability vector S(0) and transition baseball team seemed to depend on the
matrix P. outcome of their previous game. When her

΄ 0.6 ΅
0.4 team won, there was a 70% chance that they
S (0) = [0.67 0.33] P=
0.50.5 would win the next game. If they lost,
Assume that the ﬁrst element in the initial however, there was only a 40% chance that
probability vector pertains to ZapShot. they would their next game.
Explain the signiﬁcance of a) What is the transition matrix of the
a) the elements in the initial probability Markov chain for this situation?
vector b) Following a loss, what is the probability
b) each element of the transition matrix that Gina’s team will win two games later?
c) each element of the product S (0)P c) What is the steady-state vector for the
Markov chain, and what does it mean?


7. Application Two popcorn manufacturers, 9. Application On any given day, the stock price
Ready-Pop and ButterPlus, are competing for Bluebird Mutual may rise, fall, or remain
for the same market. Trends indicate that unchanged. These states, R, F, and U, can
65% of consumers who purchase Ready-Pop be modelled by a Markov chain with the
will stay with Ready-Pop the next time, transition matrix:
while 35% will try ButterPlus. Among those R F U
who purchase ButterPlus, 75% will buy

΄ ΅
0.75 0.15 0.1 R
ButterPlus again and 25% will switch to 0.25 0.6 0.15 F
Ready-Pop. Each popcorn producer initially 0.4 0.4 0.2 U
has 50% of the market.
a) What is the initial probability vector? a) If, after a day of trading, the value of
Bluebird’s stock has fallen, what is the
b) What is the transition matrix?
probability that it will rise the next day?
c) Determine the ﬁrst- and second-step
b) If Bluebird’s value has just risen, what is
probability vectors.
the likelihood that it will rise one week
d) What is the long-term probability that a from now?
customer will buy Ready-Pop?
c) Assuming that the behaviour of the
8. Inquiry/Problem Solving The weather Bluebird stock continues to follow this
pattern for a certain region is as follows. On established pattern, would you consider
a sunny day, there is a 50% probability that Bluebird to be a safe investment? Explain
the next day will be sunny, a 30% chance your answer, and justify your reasoning
that the next day will be cloudy, and a 20% with appropriate calculations.
chance that the next day will be rainy. On a
cloudy day, the probability that the next day 10. Assume that each doe produces one female
pt
will be cloudy is 35%, while it is 40% likely ha e offspring. Let the two states be D, a normal
C

r

to be rainy and 25% likely to be sunny the doe, and B, a doe with bald patches.
m
P

r
oble
next day. On a rainy day, there is a 45% Determine
chance that it will be rainy the next day, a a) the initial probability vector
20% chance that the next day will be sunny,
b) the transition matrix for each generation
and a 35% chance that the next day will be
of offspring
cloudy.
c) the long-term probability of a new-born
a) What is the transition matrix?
doe developing bald patches
b) If it is cloudy on Wednesday, what is the
d) Describe the assumptions which are
probability that it will be sunny on
inherent in this analysis. What other
Saturday?
factors could affect the stability of this
c) What is the probability that it will be Markov chain?
sunny four months from today, according
to this model?
d) What assumptions must you make in
part c)? Are they realistic? Why or
why not?


ACHIEVEMENT CHECK
C
12. Communication Refer to Example 4 on
Understanding Problem Solving page 351.
11. When Mazemaster, the mouse, is placed in a) Suppose that the probability of stormy
a maze like the one shown below, he will weather on any day following a calm day
explore the maze by picking the doors at increases by 0.1. Estimate the effect this
random to move from compartment to change will have on the steady state of
compartment. A transition takes place the Markov chain. Explain your
when Mazemaster moves through one of prediction.
the doors into another compartment. Since b) Calculate the new steady-state vector and
all the doors lead to other compartments, compare the result with your prediction.
the probability of moving from a Discuss any difference between your
compartment back to the same estimate and the calculated steady state.
compartment in a single transition is zero. c) Repeat parts a) and b) for the situation in
which the probability of stormy weather
1 6 following either a calm or a stormy day
4 increases by 0.1, compared to the data in
Example 4.
2 7
d) Discuss possible factors that might cause
5
the mathematical model to be altered.
3 8
13. For each of the transition matrices below,
decide whether the Markov chain is regular
a) Construct the transition matrix, P, for
and whether it approaches a steady state.
the Markov chain.
(Hint: An irregular Markov chain could still
b) Use technology to calculate P 2, P 3, and have a steady-state vector.)
P 4.
c) If Mazemaster starts in compartment 1, a)
΄ 0 1΅
1 0
b)
΄0
0.5
1
0.5 ΅
what is the probability that he will be in
compartment 4 after
i) two transitions?
c)
΄1
0.5
0
0.5 ΅
ii) three transitions? 14. Refer to Example 2 on page 347.
iii) four transitions? a) Using a graphing calculator, ﬁnd P 100.
d) Predict where Mazemaster is most likely Describe this matrix.
to be in the long run. Explain the b) Let S (0) = [a b c]. Find an expression for
reasoning for your prediction. the value of S (0)P 100. Does this expression
e) Calculate the steady-state vector. Does depend on S (0), P, or both?
it support your prediction? If not, c) What property of a regular Markov
identify the error in your reasoning in chain can you deduce from your answer
part d). to part b)?


15. Inquiry/Problem Solving The transition
matrix for a Markov chain with steady-state

΄ 7
13 13
6
΅ ΄
vector of ᎏᎏ ᎏᎏ is 0.4 0.6 .
m n ΅
Determine the unknown transition matrix
elements, m and n.
Career Connection
Investment Broker
Many people use the services of an investment
broker to help them invest their earnings. An
investment broker provides advice to clients
on how to invest their money, based on their
individual goals, income, and risk tolerance,
among other factors. An investment broker can
work for a financial institution, such as a bank
or trust company, or a brokerage, which is a
company that specializes in investments. An
investment broker typically buys, sells, and
trades a variety of investment items, including
stocks, bonds, mutual funds, and treasury bills.

An investment broker must be able to read and acceptable substitute. A broker must have
interpret a variety of financial data including a licence from the provincial securities
periodicals and corporate reports. Based on commission and must pass specialized courses
experience and sound mathematical principles, in order to trade in specific investment
the successful investment broker must be able products such as securities, options, and futures
to make reasonable predictions of uncertain contracts. The chartered financial analyst (CFA)
outcomes. designation is recommended for brokers
wishing to enter the mutual-fund field or other
Because of the nature of this industry, earnings financial-planning services.
often depend directly on performance. An
investment broker typically earns a commission,
similar to that for a sales representative. In the
short term, the investment broker can expect
some fluctuations in earnings. In the long www.mcgrawhill.ca/links/MDM12
term, strong performers can expect a very
comfortable living, while weak performers
find out more about an investment broker and
are not likely to last long in the field.
other careers related to mathematics.
Usually, an investment broker requires a
minimum of a bachelor’s degree in economics
or business, although related work experience
in investments or sales is sometimes an



6.1 Basic Probability Concepts Based on this survey, calculate
Refer to the Key Concepts on page 311. a) the odds that a customer visited the
restaurant exactly three times
1. A bag of marbles contains seven whites, five
blacks, and eight cat’s-eyes. Determine the b) the odds in favour of a customer having
probability that a randomly drawn marble is visited the restaurant fewer than three times
a) a white marble c) the odds against a customer having visited
the restaurant more than three times
b) a marble that is not black

2. When a die was rolled 20 times, 4 came up 6.3 Probabilities Using Counting
five times. Techniques
a) Determine the empirical probability of Refer to the Key Concepts on page 324.
rolling a 4 with a die based on the 20 trials. 6. Suppose three marbles are selected at random
b) Determine the theoretical probability of from the bag of marbles in question 1.
rolling a 4 with a die. a) Draw a tree diagram to illustrate all
c) How can you account for the difference possible outcomes.
between the results of parts a) and b)? b) Are all possible outcomes equally likely?

3. Estimate the subjective probability of each
Explain.
event and provide a rationale for your c) Determine the probability that all three
decision. selected marbles are cat’s-eyes.
a) All classes next week will be cancelled. d) Determine the probability that none of

b) At least one severe snow storm will occur
the marbles drawn are cat’s-eyes.
in your area next winter. 7. The Sluggers baseball team has a starting line-
up consisting of nine players, including Tyrone
6.2 Odds and his sister Amanda. If the batting order is
Refer to the Key Concepts on page 317. randomly assigned, what is the probability that
Tyrone will bat first, followed by Amanda?
4. Determine the odds in favour of flipping
three coins and having them all turn up 8. A three-member athletics council is to be
heads. randomly chosen from ten students, five of
whom are runners. The council positions
5. A restaurant owner conducts a study that
are president, secretary, and treasurer.
measures the frequency of customer visits in
Determine the probability that
a given month. The results are recorded in
the following table. a) the committee is comprised of all
runners
Number of Visits Number of Customers
1 4 b) the committee is comprised of the three
2 6 eldest runners
3 7 c) the eldest runner is president, second
4 or more 3 eldest runner is secretary, and third
eldest runner is treasurer

6.4 Dependent and Independent Events 12. During a marketing blitz, a telemarketer
Refer to the Key Concepts on page 333. conducts phone solicitations continuously
from 16 00 until 20 00. Suppose that you
9. Classify each of the following pairs of events have a 20% probability of being called
as independent or dependent. during this blitz. If you generally eat dinner
First Event Second Event between 18 00 and 18 30, how likely is it
a) Hitting a home run Catching a pop fly that the telemarketer will interrupt your
while at bat while in the field dinner?
b) Staying up late Sleeping in the next
day
c) Completing your Passing your 6.5 Mutually Exclusive Events
calculus review calculus exam Refer to the Key Concepts on page 340.
d) Randomly selecting Randomly selecting
13. Classify each pair of events as mutually
a shirt a tie
exclusive or non-mutually exclusive.
10. Bruno has just had job interviews with two First Event Second Event
separate firms: Golden Enterprises and a) Randomly selecting Randomly selecting
Outer Orbit Manufacturing. He estimates a classical CD a rock CD
that he has a 40% chance of receiving a job b) Your next birthday Your next birthday
offer from Golden and a 75% chance of occurring on a occurring on a
Wednesday weekend
receiving an offer from Outer Orbit.
c) Ordering a Ordering a
a) What is the probability that Bruno will hamburger with hamburger with no
receive both job offers? cheese onions
b) Is Bruno applying the concept of d) Rolling a perfect Rolling an even
theoretical, empirical, or subjective square with a die number with a die
probability? Explain.

Mc graw hill - data management

Mc graw hill - data management

More Related Content

Similar to Mc graw hill - data management (20)

Recently uploaded (20)

Mc graw hill - data management