Grade

10

MATHEMATICS
QUARTER 3 – MODULE 6
MELC 8

The Probability of
PART I
MELC 8: Finds the probability of

A. Introduction and Discussion

In this module, you will learn about the concept
of probability of . The union of two or more
sets is the set that contains all the elements of each of
the sets; an element is in the union if it belongs to at
least one of the sets. The symbol for union is , and is
associated with the word “or”, because is the
set of all elements that are in or (or both.) To find
the union of two sets, list the elements that are in
either (or both) sets. In terms of a Venn Diagram, the 𝑨 𝑩
union of sets and can be shown as two
completely shaded interlocking circles.

Union of Two Sets: The shaded Venn Diagram shows the union of set (the circle on left)
with set (the circle on the right). It can be written shorthand as

In symbols, since the union of A and B contains all the points that are in A or B or
both, the definition of the union is:

For example, if and

then Notice that the element 1
is not listed twice in the union, even though it appears
in both sets and This leads us to the general
addition rule for the union of two events:

P (A or B)

where is the intersection of the two sets. We must subtract this out to avoid double
counting of the inclusion of an element.

If sets and are disjoint, however, the event has no outcomes in it, and is an
empty set denoted as or { } which has a probability
of zero. So, the above rule can be shortened for
disjoint sets only:

This can even be extended to more sets if

they are all disjoint:

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B. Illustrative Examples

Consider the following examples.

Example 1:
When drawing one card out of a deck of playing cards, what is the probability of
the following?
a.) Getting heart or a face card (king, queen, or jack)?
Let denote drawing a heart and denote drawing a face card. Since there
are hearts and a total of face cards ( of each suit: spades, hearts, diamonds, and
clubs), but only face cards of hearts, we obtain:

Using the addition rule, we get:

The reason for subtracting the last term is that otherwise we would be counting the
middle section twice (since and overlap).

b.) A card that is a Queen or a Club is selected.

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c.) A card that is a Queen or a King is selected.

Example 2:
The table summarizes results from 976 pedestrian deaths that were caused by
automobile accidents.

If one of the pedestrian deaths is randomly selected, find the probability that the
pedestrian was not intoxicated, or the driver was intoxicated?

Driver Pedestrian Intoxicated?

Intoxicated? Yes No Total
Yes 62 82 144
No 226 606 832
Total 288 688 976

Example 3:
A tutoring service specializes in preparing adults for high school equivalence tests.
Among all the students seeking tutoring service, seek tutoring service in Mathematics,
seek tutoring service in English, and seek tutoring service in both Mathematics
and English. What is the percentage of students who seek tutoring service in either
Mathematics or English?

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Let M denote the event “the student needs help in Mathematics”.
Let E denote the event “the student needs help in English.”

Example 4:
Given A number is taken at random. Find the probability of
getting an odd number or a number less than 8.

Step 1: Step 4:

Step 2:
5 7 4

8
Step 3:

4
5

Example 5:
Suppose that in your class of students,
students are in band, students play a sport, and
students are both in band and play a sport. Let be the
event that a student is in band and let be the event
that a student plays a sport. Create a Venn diagram that
models this situation.
To fill in the Venn diagram, remember that the
total of the numbers in circle must be and the total of
the numbers in circle must be . The intersection of the two circles must contain a .
is the probability that a student is in band or plays a sport or both. With the help of
the Venn diagram, this is not too difficult to calculate:

You could also compute this probability using the Addition Rule:

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A GENERAL NOTE: PROBABILITY OF THE UNION OF TWO EVENTS
The probability of the union of two events and (written equals the sum of the
probability of A and the probability of B minus the probability of A and B occurring together
(which is called the intersection of A and B and is written as ).

If sets 𝐴 and 𝐵 are disjoint, however, the event 𝐴 𝐵 has no outcomes in it, and is an
empty set denoted as or { } which has a probability of zero. So, the addition rule can be
shortened for disjoint sets only:
𝑃 𝐴 𝐵 𝑃 𝐴 𝑃 𝐵

PART II: ACTIVITIES

𝑃 𝐴 𝐵 𝑃 𝐴 𝑃 𝐵
A. Directions: Solve each problem using the concept of probability of a union of two
events.

1. State the Addition Rule for probability and explain when it is used.
2. A card is drawn from a standard deck. Find the probability of drawing a heart or a .
3. A card is drawn from a standard deck. Find the probability of drawing a heart or a spade.
4. In a hospital on one evening, there were 8 nurses and 5 doctors. 7 of the nurses and 3 of
the doctors were female. What is the probability that a person chosen at random will be
male or a nurse?
5. Your dad only ever makes one meal for dinner. The probability that he makes pizza
tonight is 30%. The probability that he makes pasta tonight is 60%. What is the probability
that he makes pizza or pasta?

B. Directions: Consider the situation below and answer the questions that follow.

1. After your little sister has gone trick-or-treating for Halloween, your mom says she can
have 2 pieces of candy. The probability of her having a Snickers is . The probability
of her having a peanut butter cup is . The probability of her having a Snickers or a
peanut butter cup is . What is the probability of her having a Snickers and a peanut
butter cup?
2. A survey of college students finds that like country-music, like gospel music,
and like both. What is the probability that a student likes either country music or
gospel music or both?
3. Sarah tells her mom that there is a 40% chance she will clean her room, a 70% she will
do her homework, and a 24% chance she will clean her room and do her homework.
What is the probability of Sarah cleaning her room or doing her homework?

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4. Of the customers who bought items at a store on a particular day, of them bought
clocks and of them bought watering cans. What is the largest possible number of
people that bought either a clock or a watering can on that day?
4.
𝑜𝑟 3.
𝑜𝑟 2.
1.
B.

5. The probability that he makes pizza or pasta is 0.90.

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4. The probability that a person chosen at random will be male or a nurse is
3. The probability of drawing a heart or a spade is 2
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2. The probability of drawing a heart or a is
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probability of event A or event B occurring.
It is used to find the 𝐵 𝑃 𝐴 1. The Addition Rule is 𝑃 𝐴 𝐵 𝑃 𝐴 𝑃 𝐵
A.
ANSWER KEY

PART III: ASSESSMENT

Directions: Choose the correct answer and write only the letter on the space
provided.

1. The General Addition Rule is used to determine the probability of either event A or event
B occurring and states – Why is
?
A. You do not have to subtract P(A and B) if A and B are statistically independent.
B. This rule determines the probability of event A or B occurring, not A and B, so this
factor is subtracted.
C. There is no explanation other than it’s just a part of the theory.
D. P(A and B) has been counted twice through P(A) and P(B).

2. The table below provides information about the students in an accounting class:

If a student is selected at random from the class, what is the probability that the student is
a male or a senior?
A. 0.882 B. 0.588 C. 0.471 D. 0.745

3. The table below provides information about the students in an accounting class:

A student is randomly selected from the class. Given that the student is female,

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 determine the probability that the student is also a senior.
A. 0.118 B. 0.412 C. 0.222 D. 0.647

4. The table below provides information about the students in an accounting class:

A student is randomly selected from the class. Determine the probability that the student
is a female if it is known that the student is a senior.
A. 0.529 B. 0.118 C. 0.222 D. 0.43

5. A card is drawn from a standard deck. Find the probability of drawing a heart or a
spade.
A. B. C. D.
2 4 8 6

6. A group of students took a test. The grades and gender are summarized below. If one
student is chosen at random, find the probability that the student was female or got a .
A B C Total
Male 18 19 3 40
Female 12 4 9 25
Total 30 23 12 65

A. B. C. D.

7. Given A number is taken at random. Find the probability of getting an

even number or a number less than 5.
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A. B. C. D.
4 4 2 8

8. Given A number is taken at random. Find the probability of getting the

number or a prime number.
5 5 5 8
A. 8 B. 4 C. 2 D. 5

9. 50 students in a class were divided based on gender and type of sport they love to
play.
Type of sport
Gender
Indoor games Outdoor games
Male 2 12
Female 6 30
A student is selected at random. Find the probability that the student is female or love to
play outdoor games.
A. B. C. D.

10. Among a group of boys, like chocolate ice cream, like strawberry ice cream,
and like both. If a boy is randomly selected from the group, what is the probability
that he likes either chocolate or strawberry ice cream, but not both?
A. B. C. D.

11. Suppose that today there is a 0.90 chance of rain, a 0.20 chance of a strong winds, and
a 0.15 chance of both rain and strong winds. What is the probability of rain or strong

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 winds?
A. 0.90 B. 0.15 C. 0.20 D. 0.95

12. Now suppose that today there is a chance of rain, an chance of rain or strong
winds, and a chance of rain and strong winds. What is the chance of strong winds?
A. 0.25 B. 0.75 C. 0.50 D. 0.85

13. On any given night, the probability that Nick has a cookie for dessert is . The
probability that Nick has ice cream for dessert is . The probability that Nick has a
cookie or ice cream is . What is the probability that Nick has a cookie and ice cream
for dessert?
A. B. C. D.

14. Yvonne tells her mom that there is a 0.40 chance she will clean her room, a 0.70 she will
do her homework, and a 0.24 chance she will clean her room and do her homework.
What is the probability of Sarah cleaning her room or doing her homework?
A. 0.86 B. 0.96 C. 0.85 D. 0.95

15. In a survey of 65 people, 28 consider themselves republicans, 27 consider themselves

democrats. The rest are considered independent. What is the probability that a person
chosen at random will be a democrat or independent?
38 65 28 65
A. B. C. D.
65 38 65 28

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REFERENCES:

Ex: Determine the Probability of the Union of Two Events (OR) - YouTube
Probability of Union Event - YouTube
Probability of a Union of Two Events - YouTube
Probability of Unions | CK-12 Foundation
CCSS Geometry - Lesson 11.6 Solutions ( Read ) | User Generated Content | CK-12
Foundation
What Are the Chances? | Boundless Statistics (lumenlearning.com)
Computing the Probability of the Union of Two Events | College Algebra
(lumenlearning.com)
Computing the Probability of Mutually Exclusive Events | College Algebra
(lumenlearning.com)