Probability of a Union
Mathematics
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10
NOT
Mathematics
Quarter 3 - Module 7
Probability of a Union of Two Events
Department of Education ● Republic of the Philippines
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Mathematics- Grade 10
Alternative Delivery Mode
Quarter 3 - Module 7: Probability of a Union of Two Events
First Edition, 2020
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Published by the Department of Education
Secretary: Leonor Magtolis-Briones, PhD
Undersecretary: Diosdado M. San Antonio, PhD
Development Team of the Module
Author/s:
Reviewers:
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Rhodel A. Lamban, PhD
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10
Mathematics
Quarter 3 - Module 7
Probability of a Union of Two Events
This instructional material was collaboratively developed and reviewed
by educators from public and private schools, colleges, and or/universities.
We encourage teachers and other education stakeholders to email their
feedback, comments, and recommendations to the Department of Education
at bukidnon@deped.gov.ph.
We value your feedback and recommendations.
Department of Education-Division of Bukidnon ● Republic of the Philippines
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Table of Contents
PAGE
COVER PAGE
COPYRIGHT PAGE
TITLE PAGE
TABLE OF CONTENTS
WHAT THIS MODULE IS ABOUT
Note to the Teacher/Facilitator
Note to the Learner
Note to the Parents/Guardian
Module Icons
WHAT I NEED TO KNOW
1
WHAT I KNOW(PRETEST)
2
LESSON 1: PROBABILITY OF A UNION OF TWO EVENTS
4
What I Need to Know
What I know
What’s in
What’s New
What Is It
What’s More
What I Have Learned
What I Can Do
Assessment
Additional Activities
4
4
5
5
7
10
11
12
12
13
SUMMARY
14
POSTTEST
15
ANSWER KEY
17
REFERENCES
18
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WHAT THIS MODULE IS ABOUT
This module will enable the learners to independently provide visual structure of
certain given events together with its operations on the union and intersections of these
events. The attainment of the main goal of this module was made possible through the
appropriate and currently applicable activities formulated by the author to cater the
needs of the learners. The necessary details and sufficient discussions needed for the
learners to successfully answer all the conditioning exercises were logically provided in
this paper so that the relevant concept for this module will be totally mastered.
Notes to the Teacher/Facilitator
To the teachers, this module is designed to let the learners learn by themselves
at times like this crisis that face to face instruction at school cannot be possible. We are
responsible to take precautionary measures in ensuring quality education for our
learners. It is the ultimate dream in the Department of Education that no learners must
be left behind and so realizing this in this unstable situation due to COVID-19 pandemic
become even more challenging. But it our oath made that at all times, quality education
must be accessible to all no matter how demanding this could be.
In this module, you will be teaching the learners the probability of a union of two
events. Used the different activities provided here to motivate and challenge the
learners to continue learning and be able to adopt the new normal in the field of
education nowadays.
Notes to the Parents/Guardians
To the parents/guardians, this module is for our learners to independently learn
the probability of a union of two events. Various tasks are provided here to allow them
to fully develop the needed knowledge and skills. Let’s be reminded that education for
our learners has never been just the work of the school or teachers but it’s always a
combine effort of the school and society. Thus, we must actively participate the mission
and vision of the Department of Education especially at this time where every single
serious attempt to nurture the learning capacity of our learners is vital.
Notes to the Learners
To the learners, this module talks about the probability of a union of two events.
To be able to understand the topic, this module is made easy for your convenience but
the learning you’ll get is sure and compact. The demand of the world to quality
education must not be sacrificed at this crisis for if it will happen, it is your future and
the future of our nation that will be renounced. The Department of Education has been
doing all possible means to deal with the current situation the country has been facing
of. Making sure that your education won’t be left behind amidst this pandemic.
Administrators, teachers, stakeholders and parents had been working hand in hand to
make leaning feasible to the majority. Now, it is in your hand to take part of these
efforts and campaign by simply conscientiously reading and answering this module. Do
it with honest convection that education is not solely about how teachers provide
instructions but it also rely on the commitment of every learner to expand what they
have learned, understood what confused them, and acquire information and
knowledge new to them.
Good luck and God bless learners!
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Icons of this Module
What I Need to Know
This will give you an idea of the skills or
competencies you are expected to learn in
the module.
This part includes an activity that aims to
check what you already know about the
lesson to take. If you get all the answers
correct (100%), you may decide to skip this
module.
What I Know
What’s In
This is a brief drill or review to help you link
the current lesson with the previous one.
What’s New
In this portion, the new lesson will be
introduced to you in various ways such as a
story, a song, a poem, a problem opener, an
activity or a situation.
This section provides a brief discussion of the
lesson. This aims to help you discover and
understand new concepts and skills.
What is It
This comprises activities for independent
practice to solidify your understanding and
skills of the topic. You may check the
answers to the exercises using the Answer
Key at the end of the module.
What’s More
What I Have Learned
This
includes
questions
or
blank
sentence/paragraph to be filled in to process
what you learned from the lesson.
What I Can Do
This section provides an activity which will
help you transfer your new knowledge or skill
into real life situations or concerns.
Assessment
This is a task which aims to evaluate your
level of mastery in achieving the learning
competency.
Additional Activities
In this portion, another activity will be given to
you to enrich your knowledge or skill of the
lesson learned. This also tends retention of
learned concepts.
Answer Key
This contains answers to all activities in the
module.
References
This contains all the sources which helped in
the formulation of the module.
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What I Need to Know
This Quarter 3 - Module 7 is all about the probability of a union of two events.
Certain competency and subtask are provided below.
Competency:
Illustrates the probability of a union of two events.
(Code: M10SP-IIIg-1)
Subtasks/Objectives:
At the end of the week, the students are expected to:
1. define probability;
2. define union of two events; and
3. illustrates the probability of a union of two events.
How to Learn from this Module
To achieve the objectives cited above, you are to do the following:
•
Take your time reading the lessons carefully.
•
Follow the directions and/or instructions in the activities and exercises
diligently.
•
Answer all the given tests and exercises
1
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What I Know
Direction: Read each item carefully and choose the letter of the correct answer.
1. It is defined as a ratio of how many times an event may occur as compared to
the total possible outcomes.
A. Chances
B. Probability
C. Union of events D. Statistics
2. When choosing an “Ace” from a standard deck of cards, the probability is ___.
B.
C.
D.
A.
3. The sum of the probability of getting a head and the probability of getting a tail
when tossing a coin once is equal to ____.
A. 1
B.
C.
D. 0
4. The school principal decided to repaint the school administration building. The
school custodian present four available paint colors to choose from (green,
neon pink, yellow and royal blue). What is the probability that the school
principal will choose green?
B.
C.
D.
A.
5. In an urn, there are 20 balls in which 4 of these are in color blue, 5 are on
color orange, 8 are in color violet and 3 are in color red. What is the probability
of getting orange when ask to draw one ball from the urn?
A.
B.
C.
D.
6. (from # 5) If A is an event of red balls and B is an event of violet balls, find
A.
B.
C.
D.
7. What is the probability of picking two spade cards or two face cards?
A.
B.
C.
D.
8. In tossing a coin twice, what is the probability of getting one or two heads?
C.
D.
A. 1
B.
For numbers 9 to 11, please refer your answers on the figures provided below.
Figure 1.
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9. What is the probability of event A to happen (
A.
B.
C.
)?
10. What is the probability of event B to happen (
A.
B.
C.
)?
11. Find
A.
D.
D.
.
B.
C.
12. Given two events Y and Z. If the
D.
and the
, then this
means that _____.
A. the events are not mutually exclusives.
.
B.
C. both events satisfied the condition that
D. the events are intersecting at some point.
.
13. Given to events A and B. If the
and
that ______.
A. the events are mutually exclusives.
B. both events have probability of less than 1.
C. the events are intersecting.
D. both events satisfied the condition which states that
, then this implies
.
14. In a Teachers’ Division Seminar for the new normal set-up in school opening
via zoom, 90 school heads joined. 53 of the school heads voted for modular
class as an alternative way of delivering lessons to the learners (event A) and
70 voted for online class (event B). Find the probability that school heads
voted online class only?
A.
B.
C.
D.
Figure 2.
15. A spinner in figure 2 is equally divided into 8 parts and numbered from 1 to 8.
Let
and
. What is
?
A.
B.
C.
D. 0
3
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Lesson
1
Probability of a Union of Two
Events
What I Need to Know
This Quarter 3 - Module 7 is all about Probability of a Union of Two Events.
In this module you expected to develop competency on the ability to illustrate the
probability of a union of two events. To complete this goal, you just have to
accomplish certain subtasks. First, you need to define what is meant by the word
“probability”. Second, you need to determine union of two events. And lastly, you
need to demonstrate the probability of a union of two events.
All these three subtasks are provided with appropriate and current activities to
entice your interest so you will be able to successfully go through this module with
full understanding on the main concept on Probability of a Union of Two Events.
What I Know
Direction: Identify the correct answer of the following questions.
1. When choosing an “Ace” and a “King” from a standard deck of cards, the
probability is __________.
Answer: _________________________________________
2. The sum of the probability obtained in all possible outcomes when throwing a
die is equal to ____.
Answer: _________________________________________
3. In an urn, there are 10 balls in which 3 of these are in color blue, 2 are on
color orange, and 5 are in color violet. What is the probability of getting orange
when ask to draw one ball from the urn?
Answer: _________________________________________
4. (from # 3) If A is an event of blue balls and B is an event of violet balls, find
Answer: _________________________________________
5. What is the probability of picking one heart cards or one face cards?
Answer: _________________________________________
6. In tossing a coin twice, what is the probability of getting two tails?
Answer: _________________________________________
4
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7. In an English alphabet, what is the probability of getting a vowel letter when
you’ll be asked to pick one?
Answer: _________________________________________
)? When
8. In a word MERCY, what is the probability of event B to happen (
B is an event of obtaining a non-vowel letter.
Answer: _________________________________________
9. Find
, when A is the event of the word PRAYERS and B is an event
of the word WONDROUS.
Answer: _________________________________________
10. In a set of natural numbers less than 10, when you’ll be asked to pick one,
what is the probability of getting a prime number?
Answer: _________________________________________
What’s In
In the previous module, you have been taught of illustrating events, together
with its types and its basic operation on union and intersection. You were able to
define event and were also able to identify what type of event applied in a scenario
given. Items on finding the union and intersection of events were also given which
surely you had successfully answered making you able to proceed in this next
module. Those items will be very useful to easily understand what this module all
about.
After being able to know the union of say event A and B, your next task would
be to know the probability that such union would happen.
Be ready to learn and enjoy as you go along with this module.
What’s New
Let’s explore
For the 40th day of enhanced community quarantine at Barangay Santa Rita
due to the COVID-19 pandemic, Athena tried to prepare breakfast. But she is having
hard time in choosing what flavor of can goods is she going to open for meal.
If you can still perfectly remember, you helped her choose her viand of the
day.
You now know that what Athena picked and cooked as viand of the day is an
example of an event.
Now, lets’ do this.
• What is the probability that
each can will be chosen?
• How will the answer be
obtained?
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• How about the probability
of Athena’s viand of the
day be any can goods or
a hotdog?
• Is there a chance that
Athena would choose an
egg or a canton noodles?
Let’s try this
Identify the correct answer in each of the questions base on the figures
provided.
For numbers 1 to 5.
1. Describe event A.
……………………………………………
2. Describe event B.
……………………………………………
? Explain your answer
3. Is
shortly.
……………………………………………
? How about
?
4. What is
……………………………………………
5. Find
.
……………………………………………
For numbers 6 to 10.
6. Describe event X.
……………………………………………
7. Describe event Y.
……………………………………………
8. Is
? Explain your answer
shortly.
……………………………………………
? How about
?
9. What is
……………………………………………
10. Find
.
……………………………………………
6
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For numbers 11 to 15.
11. Describe U.
……………………………………………
12. Describe event M and N.
……………………………………………
13. Is
? Explain your answer
shortly.
……………………………………………
? How about
?
14. What is
……………………………………………
15. Find
.
……………………………………………
What Is It
Probability
It is defined as a ratio of Example 1:
how many times an event
In an experiment of
may occur as compared to tossing a coin twice, what is
the total possible outcomes.
the probability that a tail
Probability of an event would come out?
P(M) is equal to the total Solution:
number of desired event (M)
Let P(M) be the
divided by the total number of probability of getting a tail,
possible outcomes (L).
M be total number of
desired event, and L be the
total number of possible
outcomes.
Note:
M = 1 (in tossing a coin
✓
once, only 1 tail is
the outcome)
✓ The
sum
of
the
probabilities
of
all
L=2
(head and tail)
outcomes is equal to 1.
(substitute the
(Example: In flipping a die once,
the possible outcomes are 6. Thus,
each outcome has a probability of .
Adding all these probabilities
) would result
(
to 1.)
obtain values)
or 0.5
If the answer has to be
converted into percentage
form, then 0.5 has to be
multiplied by 100.
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Thus,
or 50% is the
probability of getting a tail in
an experiment of tossing a
coin.
Example 2:
What is the probability of getting a 5 when rolling a regular die once?
Solution:
Let P(X) - be the probability of getting a 5 when rolling a die once.
X - be the total number of desired event.
Z - be the total number of possible outcomes.
X=1
Z=6
(since in a regular die, only one face has a 5 dots mark)
(since in a regular die has 6 faces)
Thus, or 16.67% is the probability of getting
5 when rolling a regular die once.
4
Union of two events
Given two events M and Example:
N, the union of these two
If given that events
events is the elements which
&
,
are found in M, in N or in
then
Both M and N.
.
In symbol,
.
To illustrate. . .
Note:
The word or here means
one or the other, or both.
Take note that in listing
the elements in the union of
two events, the common
elements have to be listed
once only.
Probability of a
Union of two
Events
Given two events M and Example 1:
N, the probability of a union
In
the
previous
of these two events (which example we have
can be written as
)
&
is equal to the sum of the
.
probability of event M and the
Find
.
8
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probability of event N minus
the probability of
(the
elements occurring to both
events). In this case it is
assume that M and N are not
mutually exclusive events, in
.
other words
Solution:
Event
M
has
6
elements; event N has 5
elements: and the total
number of elements is 11.
In symbol,
.
Note:
If these two events (M
and
N)
are
mutually
exclusive, then
.
an event of getting head and M is
an event of getting tail. Here, there
is no chance that the outcome will
be of both event making
. And so,
empty, thus
.
, one head out of 2
possible outcome
one tail out of 2
possible outcome
)
,
So,
(Example: Tossing a coin once, L is
.
At this point, we have
to
take
note
that
, meaning there
are elements common to
both events (as illustrated
in the figure above).
Now,
how
many
elements found in
?
and so
.
Substituting the values
obtained in the equation
below, then
Thus, the probability of
a union of events M and N
is . In symbol,
.
Example 2:
In an experiment of drawing a card from a regular deck of cards, what is
the probability of drawing a heart or a face card.
Solution:
Let M be an event with heart cards and N be an event with face cards.
Note that the total number of cards in a regular deck of cards is 52.
M = 13 cards (since there are 13 heart cards out of 52) and so,
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N = 12 cards (since there are 12 face cards out of 52) and so,
, meaning the two events are not
Note that in this case,
mutually exclusives. Why?
Elements common to
both events.
1
Thus, or 50% is the probability of getting a
2
tail in an experiment of tossing a coin.
, making
It can be seen from the figure that,
Substituting
the
values
obtained
, then
above
Therefore,
.
in
the
equation
.
What’s More
Let’s do this
Define shortly but concisely the terms listed below and provide simple
illustration to help define the terms. Make use of your own words in
constructing your answers.
1. Probability
2. Union of two Event
3. Probability of a Union of two Events
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What I Have Learned
Provide the illustration needed in each items to answer the all sub-questions
which follow.
1. Draw a Venn diagram. Let Z be an event of natural numbers, let A be an
event of natural numbers greater than or equal to 5 but less than 25 and let B
be an event of natural numbers greater than or equal to 15 but less than 35.
1.1. What does Z in the Venn diagram represent?
_________________________________________________________________
1.2. Find
.
_________________________________________________________________
1.3. Find
.
_________________________________________________________________
1.4. Find
.
_________________________________________________________________
2. Draw a Venn diagram. Let X be the event of red cards in a standard deck of
cards and let Y be the event of face cards.
2.1. What is the universal set?
_________________________________________________________________
2.2. Find
.
_________________________________________________________________
2.3. Find
.
_________________________________________________________________
2.4. Find
.
_________________________________________________________________
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What I Can Do
At this point, you will be challenged again to formulate your own real life
problem in which the application of the topic on the probability of a union of two
events can be seen. After formulating your problem, identify the universal set, the
events and the union of events.
Assessment
Given the following scenario, provide the Venn diagram to illustrate the probability
of a union of two events.
1. U = {x: x is a natural number less than 20}
A = {y: y is an odd number less than 20}
B = {z: z is a prime number}
2. U = {x: x is a letter in an English alphabet}
A = {y: y is a consonant letter}
B = {z: z is a letter in the word “FOREVER”}
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3. A = {x: x is even number less than 11}
B = {y: y is an odd number less than or equal to 11}
4. U = The Milky Way.
A = The Solar System
B = The Stars in the Milky Way
Additional Activity
In this section, you are challenge to formulate your own story based on
the figure provided below. The application on the probability of a union of two
events has to be applied.
13
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Summary
Probability
It is defined as a ratio of how many times an event may occur as
compared to the total possible outcomes.
Probability of an event P(M) is equal to the total number of desired
event (M) divided by the total number of possible outcomes (L).
Note:
o
o The sum of the probabilities of all outcomes is equal to 1.
Union of Two Events
Given two events M and N, the union of these two events is the
elements which are found in M, in N or in Both M and N
In symbol,
.
Note:
o The word or here means one or the other, or both.
Probability of a Union of Two Events
Given two events M and N, the probability of a union of these two
) is equal to the sum of the probability
events (which can be written us
of event M and the probability of event N minus the probability of
(the
elements occurring to both events). In this case it is assume that M and N are
.
not mutually exclusive events, in other words
In symbol,
.
Note:
If these two events (M and N) are mutually exclusive, then
.
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Assessment: (Post-Test)
Direction: Read each item carefully and choose the letter of the correct answer.
1. When choosing an “Red Ace” from a standard deck of cards, the probability is
_____.
B.
C.
D.
A.
2. The sum of the probability of getting two heads and the probability of getting
two tails when tossing a coin twice is equal to ____.
A.
B.
C.
D.
3. A ratio of how many times an event may occur as compared to the total
possible outcomes.
A. Chances
B. Statistics C. Union of events D. Probability
4. In an urn, there are 15 balls in which 4 of these are in color blue, 3 are on
color orange, 3 are in color violet and 5 are in color red. What is the probability
of getting blue ball when ask to draw one ball from the urn?
A.
B.
C.
D.
5. (from # 5) If A is an event of red balls and B is an event of blue balls, find
A.
B.
C.
D.
6. The school principal decided to repaint the school administration building. The
school custodian present four available paint colors to choose from (green,
neon pink, yellow, neon red and royal blue). What is the probability that the
school principal will choose green?
B.
C.
D.
A.
7. What is the probability of picking three spade face cards or two red cards?
A.
B.
C.
D.
8. Given two events Y and Z. If the
and the
, then this
means that _____.
A. the events are not mutually exclusives.
B.
.
C. both events satisfied the condition that
D. the events are intersecting at some point.
For numbers 9 to 10, please
refer your answers on the figures
provided below.
Figure 1.
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.
9. What is the probability of the complement of event A to happen?
B.
C.
D.
A.
10. What is the probability of
A.
B.
?
C.
D.
11. In tossing a coin thrice, what is the probability of getting one or two heads?
C.
D.
A. 1
B.
12. Given to events A and B. If the
and
that ______.
A. the events are mutually exclusives.
B. both events have probability of less than 1.
C. the events are intersecting.
D. both events satisfied the condition which states that
, then this implies
.
13. In a Teachers’ Division Seminar for the new normal set-up in school opening
via zoom, 90 school heads joined. 53 of the school heads voted for modular
class as an alternative way of delivering lessons to the learners (event A) and
70 voted for online class (event B). Find the probability that school heads
voted modular class only?
A.
B.
C.
D.
Figure 2.
14. A spinner in figure 2 is equally divided into 8 parts and numbered from 1 to 8.
and
. What is the probability of hitting prime
Let
number?
A. 0
B.
C.
D.
15. In a wedding, there are a total of 500 attendees. 380 of the attendees are
friends of the bride. 330 of the attendees are friends of the groom. What is the
probability that an attendee is both bride and groom’s friends?
B.
C.
D.
A.
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Key to Answers
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References
Calanta, Melvin, Canonigo, Allan, Chua, Arnaldo. Cruz, Jerry, Esparrago,
Mirla, Garcia, Elino, Magnaye, Aries, et. al., mathematics Learner’s Module.
First edution.DepEd-IMCS: Meralco Avenvue, Pasig City, 2015.
https://saylordotorg.github.io/text_introductory-statistics/s07-01-samplespaces-events-and-their.html
LumenLeraning, “Computing the Probability of the Union of Two Events”.
https://courses.lumenlearning.com/ivytech-collegealgebra/chapter/cpmputingthe-probability-of-the-union-of-two-evnts/
Math is Fun, “Probability: Types of Events” accessed May 7, 2020.
https://www.mathisfun.com/data/probability-events-types.html
Math
is
Fun.
“Probability”
accessed
https://www.mathisfun.com/data/probability.html
May
7,
MattePlanet, “Probability of Events” accessed May 7,
https://www.mathplanet.com/education/pre-algebra/probability-andstatistic/probability-of-event
2020.
2020.
“Union of Events” accessed May 7, 2020. https://probabilityformula.org/unionof-events-examples.html
Statistics Libretexts, “Sample Spaces, Events, and Their Probabilities”
accessed
May
7,
2020.
https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Book%3A_Intro
ductory_Statistics_(Shafer_and_Zhang)/03%3A_Basic_Concepts_of_Probabil
ity/3.01%3A_Sample_Spaces%2C_and_Their_Probabilities
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