MATHEMATICS 10

THIRD QUARTER
WEEK 7

Competencies covered:

Illustrates the probability of a

union of two events.

Find the probability of (A U

Probability of
union of Two
B).

Events

Name of Writer: Renante A. Roldan

Designation: SST - II
School: Poctoy National High School
District: Torrijos

Name of Writer: Renante A. Roldan

Designation: SST - II
School: Poctoy National High School
District: Torrijos

Department of Education • Schools Division of Marinduque

Introductory Message
How useful is probability? Do you want to know your chances in winning a
lottery? Well this topic offers you the simple way to know it. This module
tackles the probability of the union of two events. The probability of the two
events to occur. This will not only be giving you the chance to compute for
the winning combinations in lottery but also giving you the chance to
compute the probability in making right decisions in life.

What I Need to Know

After going through this module, the students should be able to

demonstrate understanding of the key concepts of probability of
compound events, union and intersection of two events, and
appreciate the relationship of the union and intersection of two events
in real-life situation. With these knowledge and skills, they should be
able to use probability in formulating conclusions and in making
decisions

What I Know

Let us see how much do you know about playing a standard deck of
cards. This will guide you in learning the new lesson. If you got all the
answers right or at least 80% of the right answer, then you are now
ready to proceed to the next level. Enjoy!!!

A standard deck of 52 playing cards includes 13 ranks of each of the

four suits: club (♣), spade (♠), diamond (♦) and heart (♥). Each suit
includes an ace, ranks 2 through 10, a jack, a queen and a king. If a
card is drawn from a well-
shuffled deck of cards, find the
probability of drawing:

a. an ace

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 b. a diamond

c. a face card

d. a black card

e. a queen

f. a red ace

Lesson Topic: Illustrating the probability of a union of two

events.

The topic is about illustrating the probability of the union of two events.
This also focus the discussion on how the probability of the union of two
events is illustrated in real-life. You will discover how chances are taking in
two events. This consists of series of activities which made easier for you to
understand and master all the concepts.

What’s In
How much do you know about the new lesson? Did you
encounter the topic in your previous mathematics? Can you do
the activity below? I’m sure you can. Don’t worry because you
will be guided in doing the activity. Relax and breathe deeply.
You can do it.

Activity 1. My Sports
110 grade 10 students from Poctoy National High School are
interviewed if they are willing to join either volleyball or basketball in
the upcoming Intramural Meet.

Shown here is the result of the survey.

Sports Number of Students

Basketball 44
Volleyball 22

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 Basketball and 33
Volleyball

Construct a Venn diagram:

a. What is the probability of the students who are willing to join

volleyball?

b. What is the probability of the students who are willing to join

volleyball only?

c. What is the probability of the students who are willing to join

basketball?

d. What is the probability of the students who are willing to join

basketball only?

e. What is the probability of the students who are willing to join

volleyball and basketball?

f. What is the probability of the students who are willing to join

volleyball or basketball?

What’s New

Are you now ready to learn something new? Well, as you go

through the discussion, set your mind, read and

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 comprehend, and don’t worry because you will be guided by
the sets of examples which truly directs you to the
development of the new concepts. Just take it easy.

Definition:
Compound events – defined as a composition of two or more other
events They can be formed in two ways:

• Union-the union of two events A and B, denoted as A∪B, is

the event that occurs if either A or B or both occur on a single
performance of an experiment.

• Intersection – the intersection of two events A and B, denoted

as A B, is the event that occurs if both A and B occur on a
single performance of the experiment.

Scenario 1. Jack rolled a fair die and wished to find the probability of
“the number that turns up is even or number greater than 3”

Solution:

Sample Space: {1, 2, 3, 4, 5, 6}

From the given statement, A = {2, 4, 6} and

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B = {4, 5, 6} then the number that turns up is even and number
greater than 3, (A ∩ B) = {4, 6}.

So, the probability of “the number that turns up

is even or number greater than 3”

P(A U B) = P(A) + P(B) – P(A ∩ B)

3 3 2 4 2
P(A U B) = + - += or
6 6 6 6 3

Scenario 2. Jack rolled a fair die and wished to find the probability of
“the number that turns up is odd or even”

Solution:
Sample Space: {1, 2, 3, 4, 5, 6}

From the given statement, A = {1, 3, 5}

and B = {2, 4, 6} then the number that

turns up is odd and even (A ∩ B) = {}.

So, the probability of “the number that

turns up is odd or even”

P (A U B) = P(A) + P(B) – P (A ∩ B)

3 3 0 6
P (A U B) = + - += or 1
6 6 6 6

P (A U B) = P(A) + P(B)

3 3 6
P (A U B) = + = or 1
6 6 6

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Scenario 3. Jack rolled a fair die and wished to find the probability of
the number divisible by 5 turns up or the number of odd turns up”.

Sample Space: {1, 2, 3, 4, 5, 6}

From the given statement A = {5} and

B={1, 3, 5} then the number that turns up
is odd and divisible by 5, (A ∩ B) = {5}. So,
the probability of “the number divisible by
5 turns up or the number of odd turns
up”

P (A U B) = P(A) + P(B) – P (A ∩ B)

1 3 1 3 1
P (A U B) = + - += or
6 6 6 6 2

or P (A U B) = P(B)

3 1
P (A U B) = or
6 2

What Is It

You are now on the part where you can do the practice of the
concepts you learned from the discussion. Series of activities were
made for you so that you that you can do the practice differently.
Be calm and brave my friend.

Activity 2. “The Most Probabowl”

Directions: Read and analyse the problems below. Write your

answer on the corresponding bowl at the right.

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A bowl contains 15 chips numbered 1
to 15. If a chip is drawn randomly
from the bowl, what is the probability
that it is;
a. 7 or 15?

b. 5 or a number divisible by 3?

c. even or divisible by 3?

d. a number divisible by 3 or divisible by 4?

Activity 3. Taking Chances

Direction: Consider the situations below and answer the

questions that follow.

Scenario 1

Dario puts 44 marbles in a box in

which 14 are red, 12 are blue, and
18 are yellow. If Dario picks one
marble at random, what is the
probability that he selects a red
marble or a yellow marble?

Scenario 2

Out of 5200 households surveyed,

2107 had a dog, 807 had a cat, and
303 had both a dog and a cat. What is
the probability that a randomly
selected household has a dog or a cat?

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What’s More

Activity 4. Wheel of Fortune

You are chosen to become the contestant of
the Wheel of Fortune at Wowowin and
you are task to spin the wheel below.
The jackpot prize will be one million
pesos and house and lot. Do you think
you can take home the jackpot prize?
Your big chances come only if you get an
orange or a b. How much is your chance of winning the prize if
you spin the will and turn out an orange or a b.

Activity 5. A Pick of Chance

To be able to continue playing a card

game with your playmates, you need
to pay at least 30% of your debt to one
of them, and to do that you need to
draw a heart or 7 from the standard
deck of cards. Do you think you can pay your debt if you draw a
heart or 7? Would you able to play continuously with them?

Activity 6. Puzzle Puzzle Puzzle

A tangram is a puzzle made up of
seven shapes that can be arranged to
form many different designs. But
they're not just any old shapes--a
tangram is made up of two big
triangles, one medium triangle, two small triangles, one square,
and one parallelogram. Retrieveform:https://www.google.com/search?

q=What+is+a+tangram&oq=What+is+a+tangram&aqs=chrome..69i57j0l5.1084

0j0j9&sourceid=chrome&ie=UTF-8. Suppose that the pieces of the

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 tangram puzzle are place in urn and you are asked to pick the
pieces to complete the puzzle, what is the probability that you
pick a big triangle or a small triangle?

What I Have Learned

Basic Concepts:
 The union of two events is the event that occurs if either or both
events occur.
 The Probability of the union of two events A and B (A U B) can
be determine by: P(A U B) = P(A) + P(B) – ( A ∩ B)
 The union of two sets can be illustrated in many aspects of life
especially on decision-making.

What I Can Do
Let us see if you really learn from the concepts and activities
you have encountered previously. The activity below is a
proof of your learning. I’m sure you are familiar with the title
of the activity and I know you are well-versed in playing it.
Think deeply before answering “Fact or Bluff”

Activity 7. FACT or BLUFF

Direction: Underline “FACT” if the statement tells factual

information otherwise “BLUFF”.

1. FACT or BLUFF. The probability of the union of two events

is the outcome of the events?

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2. FACT or BLUFF. If A and B are two events, then the
probability of the union of the events is P(A U B) = P(A) + P(B) –
P(A ∩ B).

3. FACT or BLUFF. A fair die has always a sample space of {1, 2,

3, 4, 5, 6}.

4. FACT or BLUFF. If a fair die is rolled, the probability of even

number and greater than 3 is 1/3.

5. FACT or BLUFF. The sample space in a standard deck of card

is always {club (♣), spade (♠), diamond (♦) and heart (♥)}

6. FACT or BLUFF. The probability of taking a diamond or a 5 in

a standard deck of cards is always 4/13.

7. FACT or BLUFF. The probability of a head or a tail as an

outcome in tossing a coin once is always 1.

8. FACT or BLUFF.
Brian likes to wear colored shirts. He has 10
shirts in the closet. Three of these are blue,
four are in different shades of red, and the
rest are of mixed or different colors. The
probability that he wear a red or a blue is 7/10.

9. FACT or BLUFF. If the spinner on the right is spun, then the

probability of a spin that results is an even number or a less
than 4 is 5/8.

10. FACT or BLUFF. If the probability of heads landing up when

you flip a coin is ½, then probability of getting tails if you flip it
again is also ½.

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Assessment
Are you now ready to take the test? Well I’m sure you are
now very well prepared, ready and excited to take the test.
So, let us start. Just set back and relax. Goodluck.

Directions: Read each question below. Write the letter of the

correct answer on your paper. Use the back portion of
the answer sheet for your solution.

1. A day of the week is chosen at random. What is the

probability of choosing a Monday or Tuesday?
A. 1/7 B. 2/14
C. 2/7 D. none of these
2. In a pet store, there are 6 puppies, 9 kittens,
4 gerbils and 7 parakeets. If a pet is chosen at random, what
is the probability of choosing a puppy or a parakeet?
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A. 1 B. ½
256
C. 11/26 D. None of these
3. The probability of a teenager owning a skateboard is 0.37,
of owning a bicycle is 0.81 and of owning both is 0.36. If a
teenager is chosen at random, what is the probability that
the
teenager owns a skateboard or a bicycle?
A. 1 .18 B. 0.7
C. 0.82 D. none of these
4. A number from 1 to 10 is chosen at random. What is the
probability of choosing a 5 or an even number?
A. 3/5 B. ½
C. 1/5 D. all of these

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5. A single 6-sided die is rolled. What is the probability of
rolling a number greater than 3 or an even number?
A. 1 B. 2/3
C. 5/6 D. none of these
6. A student goes to the library to borrow books. The
probability that she checks out a Mathematics book is .40,
an English book is .30, both Mathematics and English book
is .20. What is the probability that the student checks out
Mathematics, English, or both
A. .40 B. .50
C. .60 D. None of these

7. A card is drawn randomly from a deck of ordinary playing

cards. You win P500.00 if the card is a spade or an ace.
What is the probability that you will win the game?
A. 1/13 B. 13/52
C. 4/13 D. None of these

8. When a fair die is rolled, what is the probability of rolling

on a prime number or a multiple of 3?
A. 1/3 B. 2/3
C. 1 D. None of these

9. From a well shuffled pack of standard cards a card pick

up randomly. What is the probability that the card picked
up is a face card or a diamond?
A. 24/52 B. 25/52
C. 26/52 D. None of these

10. If a coin is tossed 3 times, what is the probability that all

three tosses come up heads given that at least two of the
tosses come up heads?
A.1/6 B.1/4

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 C.1/3 D.3/8

Additional Activities
Where in real life we can use what you learned? Sometimes, one
thing that gives us satisfaction brings to a more complicated
and devastating event. The activity below would tell you which
do you prefer to eat and the chances of destroying your health.

Activity 8. Which is Which

Some street foods were sampled and tested for the presence of

disease-causing bacteria or harmful chemicals. A random

sample of 200 street foods of various types according to how
they are prepared was examined. The table below shows the
results:

Number of
Number of Food with
Number of
Food with Both
Food with
Type of Food Harmful Bacteria Total
Bacteria
Chemicals and
Only
Only Harmful
Chemicals
Fried 35 15 18 68
Boiled 46 14 32 92
Grilled 24 8 8 40
Total 105 37 58 200

1. What is the probability that a street food selected at random

is fried?

2. What is the probability that food selected at random is both

grilled and contained harmful chemicals?

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 3. What is the probability that a randomly selected food has
both bacteria and harmful chemicals?

Answer Key

Recall
a. 1/3 b. 1/4
c. 3/13 d. 1/2
e. 1/13 f. 1/26

Activity 1. My Sports
Venn Diagram
a. 0.7 b. 0.5
c. 0.3 d. 0.9
e. 0.3

Activity 2. The Probabowl

a. 2/5
b. 2/3
c. 2/3
d. 7/15

Activity 3. Taking chances

Scenario 1: 8/11
Scenario 2: 2611/5200

Activity 4. Wheel of Fortune

Answer: 2/3

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Activity 5. A pick of chance

Answer: 4/13

Activity 6. Puzzle puzzle puzzle

Answer: 4/7

Activity 7. Fact or Bluff

1. Fact 6. Fact
2. Fact 7. Bluff
3. Fact 8. Fact
4. Bluff 9. Fact
5. Bluff 10. Fact

Assessment/Evaluation
1. C 6. A
2. C 7. C
3. C 8. B
4. A 9. B
5. B 10. B

Activity 8. Which is which

a. A
b. D
c. B

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References

Callanta, Melvin, Allan Canonigo, Arnaldo Chua, Jerry Cruz, Mirla

Esparrago, Elino Garcia, Aries Marnaye, Fernando Orines, Rowena

Perez and Concepcion Ternida. Mathematics 10 Learner’s Module. 1st

ed. Rex Bookstore, Inc., 2015.

Arciaga, Ronald, and Dan Andrew Magcuyao. Statistics and Probability. 1st

ed. JFS Publishing Services, 2016.

Links:

https://courses.lumenlearning.com/ivytechcollegealgebra/chapter/

computing-the-probability-of-the-union-of-two-events/

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https://www.google.com/search?

q=What+is+a+tangram&oq=What+is+a+tangram&aqs=chrome..69i57j0

l5.10840j0j9&sourceid=chrome&ie=UTF-8.

https://www.spicetheplate.com/product/noodle-soup-bowl/

https://stattrek.com/probability/probability-rules.aspx

https://probabilityformula.org/index.html

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