4th_Quarter_Mathematics_8 (1).docx

1 ǁ Math 8
Fourth Quarter
Choose and encircle the letter of the correct answer.
1.) He is an Italian physician, mathematician, and gambler who wrote the first theoretical study of
probabilities in gambling.
a.) Gerolamo Cardano
b.) Rene Descartes
c.) Albert Einstein
d.) Leonardo Fibonacci
2.) In probability, _____ is a possible result of the experiment.
a.) sample space
b.) outcome
c.) experiment
d.) event
3.) What is the probability of getting heads in tossing a coin?
a.) 1 b.) 1/2 c.) 1/4 d.) 1/4
4.) It is a process that has a number of distinct possible outcomes in which the result cannot be predicted
with certainty.
a.) outcome
b.) sample space
c.) event
d.) experiment
5.) It is a subset of the sample space.
a.) Sample space
b.) Event
c.) Experiment
d.) Outcome
6.) How many possible outcomes are there in rolling two dice?
a.) 6
b.) 12
c.) 24
d.) 36
7.) It is the set of all possible outcomes of an experiment.
a.) Sample space
b.) Experiment
c.) Event
d.)Outcome
8.) You asked your friend his/her birth month. How many elements of sample space are there?
a.) 6
b.) 8
c.) 10
d.) 12
9.) If one thing can occur in m ways and a second thing can occur in n ways, and the third thing can occur
in p ways, and so on, then the sequence of things can occur m x n x p x … ways. This principle
explains the ______.
a.) Fundamental Counting Principle c.) Principle of Multiplication
b.) Fundamental Operations d.) Basic Principles in Mathematics
10.) What is the probability of getting a jack card in a regular deck playing cards?
a.) 4/52
b.) 2/26
c.) 1/13
d.) 1
Try this!

2 ǁ Math 8
11.) There are two kinds of merienda available at the canteen and there are also three kinds of
beverages. How many possible combinations are there?
a.) 5
b.) 6
c.) 7
d.) 8
12.) What is the probability of a sure event?
a.) 1
b.) 1/2
c.) 1/3
d.) 1/4
13.) What is the probability of an impossible event?
a.) 1
b.) 1/2
c.) 1/3
d.) 0
14.) Which of the following is the correct sample space in tossing a coin?
a.) {H, T}
b.) {H, H}
c.) {T, T}
d.) {H, T, T}
15.) How many possible combinations can be made from a 4-digit pin code given that the digits can
be repeated?
a.) 1,000
b.) 10,000
c.) 720
d.) 5,040

3 ǁ Math 8
Rearrange the letters in each number to identify the following word. Write your answer in the
blank grid.
1. E E U M S R A - __________________
2. E E I O R R T X - __________________
3. E N A L G - __________________
4. I E R R O I T N - __________________
5. I T R A N L E G - __________________
Since the topic is about triangle inequalities, let us start with some inequality axioms:
1. Substitution Axiom: if a > b and a = c, then c > b
2. Transitive Axiom: if a > b and b > c, then a > c
3. Partition Axiom: A whole quantity is greater than anyone of its parts
Recall: Adjacent angles are two angles that have the same vertex and a common side between them.
An exterior angle of a triangle is the angle formed externally between two adjacent sides. It is an angle
outside a triangle that is formed when one of the triangle’s is extended. The angle formed by two adjacent side
is the internal angles of the triangle are the remote interior angles of the exterior angle.
Theorem: (Exterior Angle Inequality Theorem) The measure of an exterior angle is greater than the measure of
any of its remote interior angles.
Lesson 9.1
This section aims to:
1. Identify the interior, remote interior, and exterior angle in a triangle.
2. state and illustrate the exterior angle theorem.
G
O
A
L
S
Chapter 9
Exterior Angle
Engage
yourself
Let’s dig
deeper
Remote Interior Angle

4 ǁ Math 8
A
E B C D
1
2
3
6
5
4
Example 1:
Refer to the given figure:
Exterior angle Remote interior angles Conclusion
⦟ 4 ⦟ 1, ⦟ 2 m⦟4 > m⦟1 and m⦟4 > m⦟2
⦟ 5 ⦟ 1, ⦟ 3 m⦟5 > m⦟1 and m⦟5 > m⦟3
⦟ 6 ⦟ 2, ⦟ 3 m⦟6 > m⦟2 and m⦟6 > m⦟3
B
Example 2:
Consider the given triangle with following angle measurement:
m⦟ABC = 130°, m⦟BCA = 30°, m⦟BAC = 20°.
Exterior angle Remote interior angles Conclusion
⦟ EBA ⦟BAC, ⦟ACB m⦟EBA > m⦟BAC and m⦟EBA > m⦟ACB
⦟ ACD ⦟BAC , ⦟CBA m⦟ACB > m⦟BAC and m⦟ACB > m⦟CBA
Keep this in Mind
The measure of an exterior angle is greater than the measure of any of its
remote interior angles

5 ǁ Math 8
4
6
5
1
2
3
7 8
9
1
6
2
5
3 4
Give the remote interior angles of the given exterior angle.
Exterior angle Remote interior angle
⦟1
⦟3
⦟4
⦟5
⦟8
⦟9
Consider the given triangle with the following angle measurement:
1. If m⦟1 = 40°, m⦟3 = 55°, what is m⦟5?
2. If m⦟2 =90°, m⦟3 = 60°, what is m⦟6?
3. If m⦟1 = 45°, m⦟3 = 60°, what is m⦟5?
Solve the following problem.
1. Give the measures of all the exterior angles of an equilateral triangle.
2. Give the measures of all the exterior angles of a right triangle with one interior angle measuring 18°.
Think about this
Activity 1
Activity 2
Activity 3

6 ǁ Math 8
The pigpen is geometric is a geometric simple substitution cipher that exchanges letters for symbols that are
fragments of a grid.
ANSWER:
Lesson 9.2
1. state and illustrate theorems on triangle inequalities
2. given three measurements, determine if a triangle can be
constructed such that the sides of triangle will have the three given
measurements
G
O
A
L
S
Engage
yourself

7 ǁ Math 8
A
C
B
D
C
A D E B
A
C
B
C
A
D B
Theorem 9: If two sides of a triangle are not congruent, then the larger angle is opposite the longer side.
Given: ABC, AC > BC
Prove: m⦟ABC > m⦟BAC
STATEMENT REASON
1. AC > BC Given
2. Locate point D on CB such that CD = CA (Point Plotting Theorem)
3. ACD is isosceles Definition of Isosceles triangles
4. m⦟DAC = m⦟CDA Isosceles Triangle Theorem
5. m⦟DAC = m⦟DAB + m⦟BAC Angle addition postulate
6. m⦟DAC > m⦟BAC Partition Axiom
7. m⦟ABC > m⦟CDA Exterior Angle Inequality Theorem
8. m⦟ ABC > m⦟DAC Substitution
9. m⦟ ABC > m⦟BAC Transitive Property
Corollary to Theorem 9.
The shortest segment from a point to a line is the perpendicular line segments.
Consider a point C not on-line AB. Locate Point D on line AB such that DC is perpendicular to AB. Now,
consider any point E on AB except point D. Triangle CDE is right triangle. By Theorem 9, CD < CE.
Theorem 10: If two angles of a triangle are not congruent, then the longer side is opposite the larger angle.
Given: ABC, m⦟ABC > m⦟BAC Prove: AC > BC
The proof is left as an exercise.
Theorem 11: Triangle Inequality Theorem
The sum of the lengths of two sides of a triangle is greater than the length of the third side.
Given: ABC
Prove: AB + AC > BC
Let’s dig
deeper

8 ǁ Math 8
L
H
A
I
STATEMENT REASON
1. ABC Given
2. Locate point D on CB such that AD is
perpendicular to C
Construction of perpendicular lines
3. BD < AB; DC < AC Corollary to Theorem 9
4. AB + AC > BD + DC Addition Property of Inequality
5. AB + AC > BC Postulate 10, Substitution
Theorem 12: Hinge Theorem
If two sides of one triangle are congruent to two sides of another triangle and the included angle of the first
triangle is larger than the included angle of the second, then third side of the first is longer than the third side of
the second triangle.
Consider the given isosceles trapezoid such that LA = 14, LH = 7, HI = 6. Write True if the
statement is correct, otherwise, write False.
___________ 1. m⦟HAI > m⦟AHI
___________ 2. m⦟LAH > m⦟HLA
___________ 3. m⦟LHA > m⦟HLA
___________ 4. m⦟LAH > m⦟LAI
___________ 5. m⦟HAI > m⦟AHI
Keep this in Mind
If two sides of a triangle are not congruent, then the larger angle is opposite the
longer side.
The sum of the lengths of the two sides of a triangle is greater than the length of
the third side.
Think about this
Activity 1

9 ǁ Math 8
P
A G
A
Y
L
D
E N
Indicate the correct answer in the box provided.
FIGURE Smallest Angle Largest Angle
PAG
LAY
DEN
Answer the following.
1. If two sides of a triangle measures 12 and 20 units, determine the only possible integer values of the
triangle.
2. If two sides of a triangle both measure 10 units, determine the only possible integer values of the third
side of the triangle.
Activity 2
Activity 3

10 ǁ Math 8
A
B
C
D
WORD SEARCH: Find each of the five words given below by encircling the letters.
FIND THESE WORDS:
1. INEQUALITY
2. LONGER
3. LARGER
4. SIDE
5. ANGLE
6. HINGE
Parallel lines are straight lines which are coplanar and do not intersect however far they are extended.
The symbol ││is used to denote those two lines are parallel. For example, “AB ││CD” is read as “lines AB and
CD are parallel or AB is parallel to CD.
E R L Y T A L I T E
G N A N G L E O Y G
U A S L O Q L E S N
Y T I L A U Q E N I
E N D I R R Q D I H
Q N E G R E G N O L
R O G R I D E E N I
A L L S E L G N R A
Let’s dig
deeper
Engage
yourself
Lesson 10.1
1. identify and illustrate parallel lines and transversal line
2. identify pairs of angles formed by a pair a line and a transversal line
G
O
A
L
S
Chapter 10

11 ǁ Math 8
A
E
B
C D
1 2
3 4
5
6
7 8
A transversal of two or more lines is a line is a line that cuts across these lines. The transversal intersects
these two coplanar lines at two different points. In the figure below, EF is a transversal of AB and CD.
The angles between the two lines are called interior angles while the angles outside are called exterior angles.
Thus, in the figure above
INTERIOR ANGLE EXTERIOR ANGLE
⦟3, ⦟4, ⦟5, ⦟6 ⦟1, ⦟2, ⦟7, ⦟8
Same side interior angles are two interior angles on the same side of the transversal. The following are pairs of
same side interior angles in the figure above: ⦟3 and ⦟5; ⦟4 and ⦟6.
Given two lines that are cut by a transversal, the angles on the same side of the transversal and on the same
side of the lines are called corresponding angles.
The following are pairs of corresponding angles based on the figure above: ⦟4 and ⦟8, ⦟3 and ⦟7, ⦟1and ⦟5,
⦟2 and ⦟6.
Alternate interior angles of two lines cut by a transversal are non-adjacent angles between the two lines and on
opposite sides of transversal.
Alternate exterior angles of two lines cut by a transversal are non-adjacent angles the two lines and opposite
sides of the transversal.
ALTERNATE INTERIOR ANGLE ALTERNATE EXTERIOR ANGLE
⦟4 and ⦟5, ⦟3 and ⦟6 ⦟2 and ⦟7, ⦟1 and ⦟8
Keep this in Mind
Parallel lines are non-intersecting coplanar lines
Transversal line of two lines is a line that intersects the two lines at different points.

12 ǁ Math 8
Draw an illustration of the following.
A pair of parallel lines Transversal line
A pair of interior angles A pair of alternate interior angles
A pair of corresponding angles A pair of alternate exterior angles
Given the figure, prove whether the given pairs of angles are corresponding, same side
interior angles, alternate interior angle, or alternate exterior angles.
Angle Pairs Answer
1.) ⦟2 ; ⦟6
2.) ⦟4 ; ⦟5
3.) ⦟11 ; ⦟13
4.) ⦟1 ; ⦟8
5.) ⦟10 ; ⦟14
6.) ⦟6 ; ⦟15
7.) ⦟3 ; ⦟10
8.) ⦟2 ; ⦟9
9.) ⦟5 ; ⦟16
10.) ⦟7 ; ⦟15
Activity 1
Think about this
Activity 2

13 ǁ Math 8
D
Postulate 22: Corresponding Angles Postulate: If two parallel lines are cut by a transversal, then
corresponding angles are congruent. (Corresponding angles of parallel are congruent.
Theorem 13: If two lines are parallel, each pair of alternate interior angles are congruent (alternate interior
angles of parallel lines are congruent)
Given: AB, CD are cut by a transversal line; and AB ││CD
Prove: m⦟2 = m⦟3
STATEMENT REASON
1. AB and CD are cut by transversal line Given
2. AB ││ CD Given
3. m⦟1 = m⦟2 Postulate 22: Corresponding Angles Postulate
4. m⦟3 = m⦟1 Vertical angle theorem
5. m⦟2 = m⦟3 3, 4, Transitive property of equality
Lesson 10.2
1. state conditions when are parallel
2. state and illustrate properties of parallel
G
O
A
L
S
Let’s dig
deeper
3

14 ǁ Math 8
Theorem 14: If two lines are parallel, each pair of interior angles on the same side of the transversal are
supplementary.
The proof of Theorem 14 is left as an exercise.
In each of the following, find the measure of x, y, and z. Assume the pairs of lines are
parallel.
GIVEN ANSWER
Keep this in Mind
1. Corresponding angles of parallel lines are congruent.
2. Alternate interior angles of parallel lines are congruent.
3. Same side interior angles of parallel lines are supplementary.
Activity 1
Think about this

15 ǁ Math 8
In each of the following, refer to the given figure below. Let l1 ││l2, l3 ││l4
1. if m⦟16 = 40°, find the measure of all the other angles.
2. if m⦟11 = 129°, find the measure of all the other angles.
Find the value of x.
1. ⦟A and ⦟B are corresponding angles of parallel lines l1 and l2, m⦟A = (3x + 4)°, and m⦟B = 69°
2. ⦟A and ⦟B are corresponding angles of parallel lines l1 and l2, m⦟A = (3x - 4)°, and m⦟B = 128°
Activity 3
Activity 2

16 ǁ Math 8
List all the elements or the results when you perform the following activities:
1. Tossing a Coin
Possible results:
2. Rolling a Die
Possible Results:
3. Birth month of your classmates
Possible Outcomes:
Chapter 10
Lesson 10.1
1. Illustrate an experiment, outcome, sample space, and event; and
2. Count the number of occurrences of an outcome in an experiment
G
O
A
L
S
Engage
yourself

17 ǁ Math 8
Before we start, since it is your first time to study probability in you Mathematics, let us define it first.
Probability – simply how likely something is to happen. Whenever we're unsure about the outcome of
an event, we can talk about the probabilities of certain outcomes—how likely they are.
It was created by a mathematician who was either a friend of gamblers or a gambler itself, to help
gamblers win more often. The probability theory is generally considered to have originated in the 16th century
when the Italian physician, mathematician, and gambler Gerolamo Cardano wrote the first theoretical study of
probabilities in gambling. However, it was not viewed as a serious branch of mathematics because of its
association with gambling.
If we will distinguish in the activity the three concepts in the definition (experiment, outcome, and
sample space):
Experiment Outcome Sample Space
Tossing a Coin Head or Tails {H, T)
Rolling a Die 1, 2, 3, 4, 5, or 6 {1, 2, 3, 4, 5, 6}
Asking for a Birth Month January, February, March, {January, February, March,
April, May, June, July, April, May, June, July,
August, September, October, August, September, October,
November, or December November, December}
Here are some of more complex examples.
Examples:
1. List all the possible outcomes for the experiment of tossing two coins (a 5-peso coin and a 10-peso
coin).
Solutions:
The possible outcomes of a 5-peso coin are ‘heads’ or ‘tails’.
The possible outcomes of a 10-peso coin are also ‘heads’ or ‘tails’.
Experiment – a process that has a number of distinct possible outcomes in which
the result cannot be predicted with certainty.
Outcome – any possible result of the experiment.
Sample Space – the set of all possible outcomes of an experiment.
Definition
Let’s dig
deeper

18 ǁ Math 8
Using a table, we can list all the possible outcomes for the experiment.
5-peso coin 10-peso coin Outcome
Hence, the sample space is the set {HH, HT, TH, TT}.
2. List the sample space when a pair of dice is tossed.
Solutions:
To list the sample space, we will use a grid.
Representing the dots in the face of the dice by numbers, the sample space is:
{ (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1,6), (2, 1), (2, 2), (2, 3), (2, 4) (2, 5), (2, 6),
(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),
(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) }
Moreover, there are 36 possible outcomes.
1. List the sample space for a two-digit number formed from the digits 1, 3, and 5 with no digits being
repeated.
Solutions:
To list the sample space, we will use a tree diagram.
First Digit Second DigitOutcome (Two-digit numbers)
3 13
1
5 15
1 31
3
5 35
1 51
5
3 53
(H)
(H)
(T)
(T)
(H)
(H)
(T)
(T)
(Both Heads)
(Heads for P5,
Tails for P10)
(Tails for P5,
Heads for P10)
(Both Tails)
The sample space is {13, 15, 31, 35, 51, 53}, and there are 6 elements.

19 ǁ Math 8
Events
Sometimes we are interested in the occurrence of each outcome or in the occurrence of the several
outcomes of the sample space. We consider the experiment of tossing two dice. The sample space is:
{ (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1,6),
(2, 1), (2, 2), (2, 3), (2, 4) (2, 5), (2, 6),
(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),
(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),
(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6),
(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) }
We may be interested in the event ‘a sum of 7’. In this case, we are satisfied if (1, 6), (2, 5), (3, 4), (4,
3), (5, 2), and (6, 1) occur as shown below.
{ (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1,6),
(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),
(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),
(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),
(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6),
(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) }
Or we may be interested in the event ‘a sum of 4’. In this case, we are satisfied if either (1, 3), (2, 2), or
(3, 1) occurs.
{ (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1,6),
(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),
(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),
(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),
(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6),
(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) }
Examples:
The club moderator wishes to choose two students from a set of 6 equally qualified students, four
boys and 2 girls, to represent the club in the next math competition.
1. What is the sample space of the club in the next competition?
Solution:
We will let A, B, C, and D represent the four boys
and E and F represent the two girls.
The sample space for this experiment is:
{ AB, AC, AD, AE, AF, BC, BD, BE, BF, CD, CE, CF, DE, DF, EF }
2. What is the subset of the sample space defining the event ‘a boy and a girl’?
Solution:
We are looking for the subset of the sample space where each outcome is a boy and a girl. To
easily answer that, we will look find the answers from the subset we had at number 1.
Now, look for a pair of a boy and a girl.
Thus, the event ‘a boy and a girl’ is the subset:
{AE, AF, BE, BF, CE, CF, DE, DF}
Event – a subset of the sample space.
Definition

20 ǁ Math 8
Match each term in column A with its description in column B.
A B
________1. Experiment a. subset of the sample space.
________2. Outcome b. a process that has a number of possible outcomes
________3. Sample Space c. how likely something is to happen.
________4. Event d. the set of all possible outcomes of an experiment.
________5. Probability e. any possible result of the experiment
List the sample space for each of the following experiments.
1. Three coins are tossed. (one-peso coin, five-peso coin, and ten-peso coin)
2. A merienda and a drink from Tatay Apen’s Menu.
3. Three dice tossed.
Keep this in Mind
 Probability – simply how likely something is to happen.
 Experiment – a process that has a number of distinct possible outcomes in which
the result cannot be predicted with certainty.
 Outcome – any possible result of the experiment.
Sample Space – the set of all possible outcomes of an experiment
Think about this
Activity 1
Merienda
Turon
Pancit
Sopas
Drinks
Iced-Tea
Juice
Water
Activity 2

21 ǁ Math 8
When you toss a coin, the possible outcomes are heads (H) or tails (T). Together, all possible
outcomes of an experiment make up a sample space.
Experiment Outcomes Sample Space
Tossing a Coin H
T
If you toss a coin twice, here are the possible outcomes.
Experiment Outcomes
Tossing a Coin
In the first experiment, there are only two possible outcomes: Heads (H) or Tails (T). In the second
experiment, there are a total of 4 possible outcomes if you are going to toss a coin twice: HH, HT, TH, and TT.
When you roll a die, the possible outcomes are:
Hence, there are six possible outcomes.
When a die is rolled, then you toss a coin, these are the possible outcomes:
Lesson 10.2
Engage
yourself
1.) determine the possible number of outcomes
2.) count the number of occurrences of an outcome using the
fundamental counting principle.
G
O
A
L
S
Let’s dig
deeper
H
C
V
T
C
V
H
C
V
T
C
V
H
C
V
T
C
V
H
C
V
T
C
V

22 ǁ Math 8
There are 12 possible outcomes.
The Fundamental Principle of Counting
If one thing can occur in m ways and a second thing can occur in n ways, and the third thing
can occur in p ways, and so on, then the sequence of things can occur m x n x p x … ways.
Examples:
1. Cody’s Café serves two desserts: a cake and a pie. They also serve three beverages: coffee, tea,
and juice. Suppose you choose one dessert and one beverage. How many possible outcomes are
there?
Solution:
To answer the question, we will use two ways.
First, a tree diagram.
Coffee CC
cake tea CT
juice CJ
coffee PC
pie tea PT
juice PJ
Counting the final outcomes, there are six possible outcomes.
The second way is by the use of the Fundamental Principle of Counting.
(Number of Desserts) x (Number of Beverages) = (Number of Possible Outcomes)
2 x 3 = 6
Aside from the combination of dishes or food, another famous real-life application of Fundamental
Counting Principle is combination of passcodes/passwords or other things that require characters. It can be
applied either the characters in a code can or cannot be repeated.
Examples:
A plate number is made up of three letters from the English alphabet followed by a three-digit number.
How many plate numbers are possible to form if:
H
C
V
T
H
C
V
H
C
V
T
H
C
V
H
C
V
T
H
C
V
H
C
V
T
H
C
V
H
C
V
T
H
C
V
H
C
V
T
H
C
V
Have you notice a pattern to count the
possible number of outcomes aside from
counting the final outcomes?
Was it easier and more convenient?
Is that applicable in all cases?

23 ǁ Math 8
a. the letters and digits can be repeated in the same plate number?
Solution:
Since there are 26 letters in an English alphabet and 10 digits, it will take us a long time to answer it
using a tree diagram. We will now use the Fundamental Counting Principle, instead.
The plate number is made up of 6 character; hence, we will make six boxes: 3 for the letters and 3 for the
digits.
There are 26 choices for the letters and 10 choices for the numbers. Since the letters and numbers can
be repeated, the number of choices does not decrease by 1 each time. Thus, the total number of
possible numbers to be formed is:
There are 17, 576, 000 possible plate numbers to be formed when the letters and numbers can be
repeated.
b. the letters and digits cannot be repeated.
There are 26 choices for the first box. Since the letters cannot be repeated, there are only 25 choices
for the second box and 24 choices for the third box. There are 10 choices for the fourth box, 9
choices for the fifth box, and 8 choices from the sixth box.
There are 11, 232, 000 possible plate numbers to be formed when the letters and numbers cannot be
repeated.
State the number of possible outcomes for each of the following.
________1. A pair of dice is tossed
________2. Three coins are tossed
________3. A coin and a die are tossed
________4. Five coins are tossed
________5. Three dice are tossed.
Think about this
Activity 1
letters digits
26 26 26 10 10 10
X X X X X = 17, 576, 000
26 25 24 10 9 8
X X X X X = 11, 232, 000

24 ǁ Math 8
Answer each question using the Fundamental Counting Principle. Show all your solutions.
1. A men’s department store sells 3 different suit jackets, 6 different shirts, 8 different ties, and 4 different
pairs of pants. How many different suits consisting of a jacket, shirt, tie, and pants are possible?
2. In the main canteen of CSRLI, there are 5 kinds of viand available, available drinks are water, iced tea,
soft drinks, and juice. Also, there are desserts such as leche flan and sweetened banana. How many
possible combinations of a meal including a viand, drink, and a dessert are there?
3. You forgot your password to unlock your cellphone. Suppose that your password consists of 4 letters in
English alphabet followed by 2 digits. How many possible combinations of password may you try?
Activity 2
Keep this in Mind
The Fundamental Principle of Counting
If one thing can occur in m ways and a second thing can occur in n ways, and
the third thing can occur in p ways, and so on, then the sequence of things can
occur
m x n x p x … ways.
 Solutions of a linear equation are ordered pairs that make the equation true.

25 ǁ Math 8
Get a coin and perform an experiment of tossing it 20 times.
Record your data on the tables below. Use H for heads and T for Tails.
Can you see any trend in the outcomes?
Find the percentage of getting heads in your experiment.
Using the result in the coin tossing experiment, what conjecture can you make about the
chance of getting heads?
Trial 1st
2nd
3rd
4th
5th
6th
7th
8th 9th
10th
Outcome
Trial 11th
12th
13th
14th
15th
16th
17th
18th 19th
20th
Outcome
Lesson 10.3
1. find the probability of an event;
2. illustrate an experimental probability and a theoretical probability; and
3. solve problems involving probabilities of simple events.
G
O
A
L
S
Engage
yourself

26 ǁ Math 8
Probability of an event (Theoretical)
The probability of an event E, denoted by P(E) is
𝑃(𝐸) =
𝑛(𝐸)
𝑛(𝑆)
if the experiment’s outcomes are equally likely to occur.
Examples
In picking a card in regular deck of playing cards, what is the probability of the event:
a. picking a king?
b. picking a black card?
Solutions:
We all know that there are 52 cards in a regular deck of playing cards. There are 13 symbols (Ace,
2-10, Jack, Queen, and King) and each symbol has 4 logos.
a. To use the formula, we need to know the total number of outcomes in an event n(E), and the total
number of elements in a sample space n(S).
Since there are 4 kings in a regular deck of playing cards, n(E) = 4. There are total of 52 elements
in a sample space.
Hence,
b. Since there are only 2 colors in a regular deck of playing cards, therefore there are 26 black cards
and 26 red cards.
Probability of an event
not happening
Let’s dig
deeper
𝑃(𝐾) =
𝑛(𝐸)
𝑛(𝑆)
𝑃(𝐾) =
4
52
𝑃(𝐾) =
1
13
𝑃(𝐵) =
𝑛(𝐸)
𝑛(𝑆)
𝑃(𝐵) =
26
52
𝑃(𝐵) =
1
2
= 1 – Probability of an event happening

27 ǁ Math 8
Example:
In picking a card in a regular deck of playing cards, what is the probability of not picking a king?
Solution:
On our previous example, we solved that the probability of picking a king in a regular deck of playing
cards is 1/13. Now, using the formula
Match the situation A with its probability in B.
A (Situation) B (Probability)
_______1.) A sure event a. 1/2
_______2.) a chance of 75 out of 100 b. 0
_______3.) a chance of 1 out of 100 c. 0.001
_______4.) A 50% chance d. 1
_______5.) An impossible event e. 0.75
 The probability of an event E, denoted by P(E) is
𝑃(𝐸) =
𝑛(𝐸)
𝑛(𝑆)
 Probability of an event = 1 – Probability of an event happening
not happening
Activity 1
Think about this
Keep this in Mind
𝑃(𝐾′) = 1 −
1
13
𝑃(𝐾′) =
13−1
13
𝑃(𝐾′) =
12
13

28 ǁ Math 8
I. Give the sample space, and outcome of an event for each experiment.
1. Experiment: Tossing a coin, then rolling a die.
Sample Space:
Subset of an event ‘head and even number’:
2. Experiment: Choosing 2 students out of 5 Grade 7 students and 3 Grade 8 students.
Sample Space:
Subset of an event ‘one grade 7 and one grade 8’:
Subset of an event ‘two grade 8 students’:
II. Using the Fundamental Counting Principle, answer the following questions. Show all your
solutions.
1. You go to the snack bar to buy a bagel and a drink for lunch. You can choose from a plain bagel, a
blueberry bagel, or a raisin bagel. The choices for a drink include water or a sports drink. How many
different lunches could be made with these choices?
2. When you get ready to get dressed for an online class you open your closet to find that you have
the following choices: a red, blue, or white shirt; jeans or sweatpants; tennis shoes or sandals. How
many different outfits could be made with these choices?
3. A website requires users to set up an account that is password protected. If the password format is
four letters followed by a single digit number, how many different passwords are possible?
III. Using the concept of Probability of an Event, answer the following.
1. In rolling a die, what is the probability of getting an even number?
2. In picking a card in a regular deck of playing cards, what is the probability of:
2.1. picking an odd number?
2.2. picking a jack of hearts?
2.3. picking a red number?
Extend your
understanding

29 ǁ Math 8
Choose and encircle the letter of the correct answer.
1.) There are two kinds of merienda available at the canteen and there are also three kinds of beverages.
How many possible combinations are there?
a.) 5
b.) 6
c.) 7
d.) 8
2.) What is the probability of a sure event?
a.) 1
b.) 1/2
c.) 1/3
d.) 1/4
3.) What is the probability of an impossible event?
a.) 1
b.) 1/2
c.) 1/3
d.) 0
4.) Which of the following is the correct sample space in tossing a coin?
a.) {H, T}
b.) {H, H}
c.) {T, T}
d.) {H, T, T}
5.) How many possible combinations can be made from a 4-digit pin code given that the digits can be
repeated?
a.) 1,000
b.) 10,000
c.) 720
d.) 5,040
6.) He is an Italian physician, mathematician, and gambler who wrote the first theoretical study of
probabilities in gambling.
a.) Gerolamo Cardano
b.) Rene Descartes
c.) Albert Einstein
d.) Leonardo Fibonacci
7.) In probability, _____ is a possible result of the experiment.
a.) sample space
b.) outcome
c.) experiment
d.) event
8.) What is the probability of getting heads in tossing a coin?
a.) 1 b.) 1/2 c.) 1/4 d.) 1/4
9.) It is a process that has a number of distinct possible outcomes in which the result cannot be predicted
with certainty.
a.) outcome
b.) sample space
c.) event
d.) experiment
10.) It is a subset of the sample space.
a.) Sample space
b.) Event
c.) Experiment
d.) Outcome
11.) How many possible outcomes are there in rolling two dice?
a.) 6
b.) 12
c.) 24
d.) 36
12.) It is the set of all possible outcomes of an experiment.
a.) Sample space
b.) Experiment
c.) Event
d.) Outcome
Check your understanding

30 ǁ Math 8
13.) You asked your friend his/her birth month. How many elements of sample space are there?
a.) 6
b.) 8
c.) 10
d.) 12
14.) If one thing can occur in m ways and a second thing can occur in n ways, and the third thing
can occur in p ways, and so on, then the sequence of things can occur m x n x p x … ways. This
principle explains the ______.
a.) Fundamental Counting Principle c.) Principle of Multiplication
b.) Fundamental Operations d.) Basic Principles in Mathematics
15.) What is the probability of getting a jack card in a regular deck playing cards?
a.) 4/52
b.) 2/26
c.) 1/13
d.) 1

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