Principles of Counting

Discrete Mathematics- CS 303
Aimhee Martinez

What real life situations would involve probability?
 Probability is the chance that something will
happen - how likely it is that some event will
happen. It is the likelihood that a given event
will occur.
 Sometimes you can measure a probability
with a number: "10% chance of rain", or you
can use words such as impossible, unlikely,
possible, even chance, likely and certain.
Example: "It is unlikely to rain tomorrow".

A scientific guess or estimate
that an event will happen, based
on a frequency of a large
number of trials.
◦ Empirical probability can be thought of as the
most accurate scientific "guess" based on the
results of experiments to collect data about an
event.

Ex. A cone shaped paper cup is tossed 1,000
times and it lands on its side 753 times, the
relative frequency which the cup lands on its
side is…
753/1000
◦ We estimate the probability of the cup landing on it’s side is
about 0.753 . This is a biased object because landing on its side
has a better chance than landing on its base.
Ex. Geologists say that the probability of a
major earthquake occurring the San Francisco
Bay area in the next 30 years is about 90%.

The number of ways that an event can
occur, divided by the total number of
outcomes in the sample space.
Prob. of Event =
s
number of ways event can occur
numberof possible outcome in sample space

1. An outcome is a result of a some activity.
Ex. Rolling a die has six outcomes:
1, 2, 3, 4, 5, 6
2. A sample space is a set of all possible
outcomes for an activity.
Ex. Rolling a die sample space: S = {1,2,3,4,5,6}
Tossing a coin: S = {H,T}
3. An event is any subset of a sample space.
Ex. Rolling a die:
the event of obtaining a 3 is E = {3},
the event of obtaining an odd # is
E = {1,3,5}

Find the probability of rolling an even # on one
toss of a die. (This is an unbiased object)
E = event of rolling even # = {2,4,6}
Number of ways it can happen: n(E) = 3
S = sample space of all outcomes = {1,2,3,4,5,6}
Number of possible outcomes: n(S) = 6
SO…
Probability of rolling an even number is

At a sporting goods store, skateboards
are available in 8 different deck designs.
Each deck design is available with 4
different wheel assemblies.
 How many skateboard choices does the
store offer? Lets make a tree diagram!

What else could you do to find the
solution?
Multiply # of deck designs by the number of
wheel assemblies
4•8 = 32 skateboard choices

If one event can occur in m ways and
another event can occur in n ways,
then the number of ways that both
events can occur together is m•n.
This principle can be extended to
three or more events.

If the possibilities being counted can be
divided into groups with no possibilities
in common, then the total number of
possibilities (outcomes) is the sum of the
numbers of possibilities in each group.

Suppose that we want to buy a computer from
one of two makes, Dell and Apple.
Suppose also that those makes have 12 and 18
different models, respectively. Then how
many models are there altogether to choose
from ?

Since we can choose one of 12 models of Dells or
one of 18 of Apples,
 there are all together 12 + 18 = 30 models to
choose from.
This is the Addition Principle of Counting.
Choosing one from given models of either make
is called an event and the choices for either
event are called the outcomes of the event.
Thus the event "selecting one from make Dell",
for example, has 12 outcomes.

How would we find the probability of choosing a
Dell?
12 Dells = 2
30 total choices 5
The probability of choosing a Dell would be:
2 = .4 = 40%
5

Codes- Every purchase made on a
company’s website is given a
randomly generated confirmation
code. The code consists of 3
symbols (letters and digits). How
many codes can be generated if at
least one letter is used in each?

To find the number of codes, find the sum of the
numbers of possibilities for 1-letter codes, 2-
letter codes and 3-letter codes.
1-letter: 26 choices for each letter and 10
choices for each digit. So 26•10•10 = 2600
letter-digit-digit possibilities. The letter can
be in any of the three positions , so there are
3•2600 = 7800 total possibilities.

2-letter: There are 26•26•10 = 6760 letter-
letter-digit possibilities. The letter can be in
any of the three positions , so there are
3•6760 = 20,280 total possibilities.
3-letter: There are 26•26•26 = 17,576
possibilities
Total Possibilities: 7800 + 20,280 + 17,576 =
45, 656 possible codes

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Principles of Counting

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