Grade

10

MATHEMATICS
QUARTER 3 – MODULE 4
MELC 6

Exploring Events

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PART I.
MELC 6: Illustrates events, and union and intersection of events.

Lesson 1 - EVENTS

A. Introduction and Discussion

Predicting events requires accuracy and precision, qualities vital to our day-to-day
living and to the society as a whole. Many problems that could not be quantified because of
uncertainties can now be analyzed mathematically. The probability theory is very helpful in
this respect.
In simple probability, activities like tossing of coins, rolling of dice, picking a card from
a deck of cards or randomly choosing a ball from a box which could be repeated over and
over again and which have well-defined results are called experiments. The possible results
of the experiment are called outcomes. The set of all possible outcomes of an experiment is
called a sample space. Any subset of possible outcomes for an experiment is known as an
event.
When an event involves a single element of the sample space, it is called simple
event. A combination of simple events is a compound event.
In this module, you will learn how to illustrate events, recognize simple from
compound events, illustrate union and intersection of events, find the complement of an
event and identify mutually exclusive events.

B. Illustrative Examples:
To understand better the difference between simple and compound event, let us
study the examples below:

Simple Events
Experiment Sample Space (S) Events (E)
The event that 2 appears.
Rolling a die
E = {2}
S = {1, 2, 3, 4, 5, 6}

Tossing a coin The event that a head will

occur.
Head (H) Tail (T) E = {H}
S = {H, T}

Compound Events
Experiment Sample Space (S) Events (E)
The event that even number
Rolling a die appears.
S = {1, 2, 3, 4, 5, 6} E = {2, 4, 6}

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 The event that an even
Rolling a die number will come out.
E = {2H, 4H, 6H, 2T, 4T, 6T}
and
The event that an odd
tossing a coin number and a tail will come
S = {1H, 2H, 3H, 4H, 5H, 6H, out.
1T, 2T, 3T, 4T, 5T, 6T} E = {1T, 3T, 5T}
Identical number cards (1, 2,
The event that the number
…, 10) are placed in a box S = {1, 2, 3, 4, 5, 6, 7, 8, 9,
drawn is prime.
and a card is drawn at 10}
E = {2, 3, 5, 7}
random
Two blue balls and five red
The event that a blue ball will
balls of the same size are
S = { B, B, R, R, R, R, R} be drawn.
placed in a jar and a ball is
E = {B, B}
drawn at random

Since the events of an experiment are subsets of the sample space of the
experiment, union and intersection of events will also be considered. Complement of an
event and mutually exclusive events will also be discussed.

Lesson 2 – UNION OF TWO EVENTS

A. Introduction and Discussion

The union of two events (𝑨 ∪ 𝑩) is one of the most-commonly performed operations
on events. As the name suggests the word "union" means to join two things irrespective of
how they behave individually. The union of two events (𝑨 ∪ 𝑩) refers to the set of
outcomes which belong to either event.
There are two ways to represent the union of two or more events: we make use of either
‘∪’ or a “u-shaped symbol” representing union of two or more events (𝑨 ∪ 𝑩) and another
way is logical “OR”. The OR is written in between the events (A OR B).

B. Illustrative Examples:
Suppose that you roll a die once. Consider two events that may happen. It may turn
up a face less than 4 or it may turn up an even number.

Solution:
Let A be the event “less than 4” and B be the event “even number.” The union
of the events A and B is 𝐴 ∪ 𝐵 and contains the outcomes of both events A and B.
A = {1, 2, 3}
B = {2, 4, 6}
𝑨 ∪ 𝑩 = {1, 2, 3, 4, 6}

𝐴 ∪ 𝐵 is the event “either A occurs OR B occurs.” It occurs if the outcome is

either in A, or in B, or in both A and B.

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 To illustrate the union 𝐴 ∪ 𝐵 using Venn diagram, we have
U
A B 5

1 4
2
3 6

Be reminded that: 1. The universal set U is the sample space S = {1, 2, 3, 4, 5, 6}.
2. The shaded region is the union of A and B (𝑨 ∪ 𝑩).

Lesson 3 – INTERSECTION OF TWO EVENTS

A. Introduction and Discussion

The intersection of two events (𝑨 ∩ 𝑩) is the event whose outcomes belong to both
events A and B. This means that the two events are occurring together.
We represent the intersection with symbol ‘∩’ or at times by writing logical ‘AND’ in
between the events. If A and B are two events then we represent A intersection B as A ∩ B
or A AND B.

B. Illustrative Examples:
Using the similar experiment which requires you to roll a die once, consider two
events: that it may turn up a face less than 4 AND it may turn up a face which is an even
number.
Solution:
Let A be the event “less than 4” and B be the event “even number.” The
intersection of the events A and B is 𝐴 ∩ 𝐵 and contains the outcomes that occur in
both events A and B.
A = {1, 2, 3}
B = {2, 4, 6}
𝑨 ∩ 𝑩 = {2}

To illustrate the intersection of the events A and B using Venn diagram, see the
figure below.
U
A B 5

1 4
2
3 6

Be reminded that: 1. The universal set U is the sample space S = {1, 2, 3, 4, 5, 6}.
2. The shaded region is the intersection of A and B (𝐀 ∩ 𝑩).

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Lesson 4 – COMPLEMENT OF AN EVENT (A’)

A. Introduction and Discussion

The complement of an event is an important concept. There will be times when you
need to consider the outcomes not found in a particular event. This is when you need the
complement of an event.
The complement of an event (A’) is read as “A complement” meaning event A does
not occur. So the event A’ is the set consisting all the outcomes in the sample space S that
are not in A.

B. Illustrative Examples:
In our example above, A = {1, 2, 3} and S = {1, 2, 3, 4, 5, 6}, so A’ = {4, 5, 6}.

The shaded region in the Venn diagram below shows the A’.
U
A 5
4
1
2

3
6

Furthermore, in our example, B = {2, 4, 6}. Since A = {1, 2, 3} and B = {2, 4, 6}, there
is an outcome that is common to A and B. That is, 𝑨 ∩ 𝑩 = {𝟐}.

Therefore, we could also illustrate the complement of A and the complement of B

using the Venn diagrams below.

The shaded region in the Venn diagram The shaded region in the Venn diagram
below shows the A’. below shows the B’.
U U
A B A B
4 4
1 1
2 5 2 5

3 6 3 6

A’ = {4, 5, 6} B’ = {1, 3, 5}

Lesson 5 – MUTUALLY EXCLUSIVE EVENTS

A. Introduction and Discussion

Events may be either mutually exclusive or not mutually exclusive. Events that
cannot occur at the same time are called mutually exclusive events.
On the other hand, if two events intersect or there are outcomes that are common to
them, then, they are called not mutually exclusive events.

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B. Illustrative Examples:
Suppose you roll a die. Let us define one event as the set of outcomes where the
face shows a number less than 4 and the other event as the set of outcomes where the face
shows a number more than 3. Are the events mutually exclusive?
Solution:
A and B are mutually exclusive events if and only if they cannot occur
simultaneously. That is, 𝑨 ∩ 𝑩 = ∅.
Let event A = {1, 2, 3} so B = {4, 5, 6}. Event A does not contain any of the
outcomes of event B. 𝑨 ∩ 𝑩 = ∅.
There are no common outcomes for the events A and B.
The Venn diagram below shows the mutual exclusivity of events.
U
A B
1 2 4 5

3 6

Be reminded that: 1. The universal set U is the sample space S = {1, 2, 3, 4, 5, 6}.
2. The two disjoint circles illustrate the mutual exclusivity of events.
3. ∅ is the symbol for null/empty set.

PART II. Activities

Activity 1
Directions: Write the sample space (S) for each experiment.
___________________ 1. Tossing of two coins
___________________ 2. Rolling a pair of dice
___________________ 3. Drawing a ball randomly from a box containing 3 red balls, 4
white balls and 2 blue balls
___________________ 4. Tossing of three coins
___________________ 5. Tossing of two coins and rolling a die

ACTIVITY 2
Directions: For each of the given experiment, write the outcomes of the events
corresponding to each description.

1. Three coins are tossed 3. Tossing a coin and rolling a die

a. Event A: all come up heads a. Event A: a head and a prime number
b. Event B: all come up tails b. Event B: a head and an even number
c. Event C: two heads and one tail c. Event C: a tail and a number divisible
d. Event D: at least one head by 5
d. Event D: no heads

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2. Two dice are rolled 4. One card drawn from a standard deck
a. Event A: double (same no. of dots) of 52 cards
b. Event B: not a double a. Event A: a red card
c. Event C: the sum of numbers is even b. Event B: a spade card
d. Event D: the sum of the numbers is 6. c. Event C: an ace
e. Event E: the sum of the number is d. Event D: a face card
greater than 8 e. Event E: a king of hearts

ACTIVITY 3
Directions: A spinning wheel is divided into 12 equal sectors and numbered 1-12. The
following events were listed. Find the following:

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

A = {1, 3, 5, 7, 9, 11} C = {2, 3, 5, 11} E = {1, 4, 9}
B = { 2, 4, 6, 8, 10, 12} D = {1, 2, 3, 4, 5, 6} F = {1, 8}

1. 𝑨 ∪ 𝑩 3. 𝑬 ∩ 𝑨 5. A’
2. 𝑪 ∪ 𝑫 4. 𝑪 ∩ 𝑭 6. 𝑨 ∪ 𝑩′

ACTIVITY 4
Direction: The following events were listed, find the following.

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A = {1, 3, 5, 7, 9} B = {3, 4, 5, 6, 8, 10} C = {2, 4, 6, 8, 10}

1. B’ 3𝑨∩𝑩 5. 𝑨 ∩ (𝑩 ∪ 𝑪)
2. 𝑨 ∪ 𝑩 4. 𝑨′ ∩ 𝑩 6. 𝑨 ∪ (𝑩 ∩ 𝑪)

ANSWER KEY
HH6, HT6, TH6, TT6} 3. S = {R, R, R, W, W, W, W, B, B}
HH5, HT5, TH5, TT5, (1, 6), (2, 6), (3, 6), (4, 6), (5, 6), (6, 6)}
HH4, HT4, TH4, TT4, (1, 5), (2, 5), (3, 5), (4, 5), (5, 5), (6, 5),
HH3, HT3, TH3, TT3, (1, 4), (2, 4), (3, 4), (4, 4), (5, 4), (6, 4),
HH2, HT2, TH2, TT2, (1, 3), (2, 3), (3, 3), (4, 3), (5, 3), (6, 3),
5. S = {HH1, HT1, TH1, TT1, (1, 2), (2, 2), (3, 2), (4, 2), (5, 2), (6, 2),
THH, THT, TTH, TTT} 2. S = {(1, 1), (2, 1), (3, 1), (4, 1), (5, 1), (6, 1),
4. S = {HHH, HHT, HTH, HTT, 1. S= {HH, HT, TH, TT}
Activity 1

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a. event b. outcome c. experiment d. sample space
3. What do you call the set of all possible outcomes of an experiment?
b. tossing a coin twice d. rolling a die
a. choosing a marble from a bag c. getting a head
2. Which of the following is an outcome of an experiment?
b. rolling a die d. all of the above
a. tossing a coin c. choosing a card from a deck of cards
1. Which of the following is an experiment?
A. Directions: Encircle the letter of the correct answer.
Assessment PART III.
Activity 4
1. B’ = {1, 2, 7, 9} 6. 𝑨 ∪ (𝑩 ∩ 𝑪) = {1, 3, 4, 5, 6, 7, 8, 9, 10}
2. 𝑨 ∪ 𝑩 = {1, 3, 4, 5, 6, 7, 8, 9, 10} Since (𝑩 ∩ 𝑪) = {4, 6, 8, 10} and
3. 𝑨 ∩ 𝑩 = {3, 5} A = {1, 3, 5, 7, 9}, therefore,
4. 𝑨′ ∩ 𝑩 = {4, 6, 8, 10} 𝑨 ∪ (𝑩 ∩ 𝑪) = {1, 3, 4, 5, 6, 7, 8, 9, 10}.
5. 𝑨 ∩ (𝑩 ∪ 𝑪) = {3, 5}
Since 𝑩 ∪ 𝑪 = {2, 3, 4, 5, 6, 8, 10}
and A = {1, 3, 5, 7, 9},
therefore, 𝑨 ∩ (𝑩 ∪ 𝑪) = {3, 5}.
Activity 3
1. 𝑨 ∪ 𝑩 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} = U 4. 𝑪 ∩ 𝑭 = ∅ or { }
2. 𝑪 ∪ 𝑫 = {1, 2, 3, 4, 5, 6, 11} 5. A’ = {2, 4, 6, 8, 10, 12} = B
3. 𝑬 ∩ 𝑨 = {1, 9} 6. 𝑨 ∪ 𝑩′ = {1, 3, 5, 7, 9, 11} = A
Activity 2
1. a. 1 outcome; E = {HHH} c. 3 outcomes; E = {(HHT), (HTH), (THH)}
b. 1 outcome; E = {TTT} d. 7 outcomes; E = {(HHH),(HHT),(HTH),(THH),(HTT),(THT),(TTH)}
2. a. 6 outcomes; E = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}
b. 30 outcomes; E = {(2, 1), (3, 1), (4, 1), (5, 1), (6, 1), (1, 2), (3, 2), (4, 2), (5, 2), (6, 2),
(1, 3), (2, 3), (4, 3), (5, 3), (6, 3), (1, 4), (2, 4), (3, 4), (5, 4), (6, 4),
(1, 5), (2, 5), (3, 5), (4, 5), (6, 5), (1, 6), (2, 6), (3, 6), (4, 6), (5, 6)}
c. 18 outcomes; E = {(1, 1), (3, 1), (5, 1), (2, 2), (4, 2), (6, 2), (1, 3), (3, 3), (5, 3),
(2, 4), (4, 4), (6, 4), (1, 5), (3, 5), (5, 5), (2, 6), (4, 6), (6, 6)}
d. 5 outcomes; E = {(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)}
e. 10 outcomes; E = {(6, 3), (5, 4), (6, 4), (4, 5), (5, 5), (6, 5), (3, 6), (4, 6), (5, 6), (6, 6)}
3. a. 3 outcomes; E = {H2, H3, H5} c. 1 outcomes; E = {T5}
b. 3 outcomes; E = {H2, H4, H6} d. 6 outcomes; E = {1T, 2T, 3T, 4T, 5T, 6T}
4. a. 26 outcomes (13 Heart and 13 Diamond) b. 13 outcomes (13 Spade)
c. 4 outcomes (4 Aces) d. 12 outcomes (4 Jacks, 4 Queens and 4 Kings)
e. 1 outcome (King of Hearts)
 4. Which of the following is the subset of all possible outcomes of an experiment?
a. outcome b. event c. sample space d. experiment
5. What do you call an event with a single outcome?
a. simple c. complementary
b. compound d. mutually exclusive
6. Which of the following is a combination of simple events?
a. simple c. compound
b. complementary d. mutually exclusive
7. What is the symbol of the intersection of A and B?
a. 𝑨 ∪ 𝑩 b. A ≥ B c. A’ or Ā d. 𝑨 ∩ 𝑩
8. Which of the following describes the union of two events?
a. A’ or Ā b. A ≤ B c. 𝐴 ∪ 𝐵 d. 𝐴 ∩ 𝐵
9. Which of the following is a sample space when 2 coins are tossed?
a. {H, T} c. {HH, HT, TH, TT}
b. {H, T, H, T} d. none of the above
10. Which of the following is a sample space when 3 coins are tossed?
a. {H, T} c. {HH, HT, TH, TT}
b. {H, T, H, T} d. none of the above
11. What is the sample space for choosing a prime number less than 12?
a. {1, 2, 3, 4, 5, 7, 9, 11} c. {2, 3, 5, 7, 9, 11}
b. {1, 2, 3, 4, 5, 7, 11} d. {2, 3, 5, 7, 11}
12. What is the sample space for choosing one ball at random from a box with 3 white, 4
green, and 6 yellow balls?
a. {3, 4, 6} c. {white, green, yellow}
b. {3 white, 4 green, 6 yellow} d. { W, W, W, G, G, G, G, Y, Y, Y, Y, Y, Y}
13. What does the shaded region represent?
a. A or B c. A U
b. A and B d. B A B

14. What does the shaded region represent? U

a. A or B c. A and B
b. not A d. not B A B

15. What does the shaded region represent? U

a. B’ c. A’
b. A and B d. A or B A B
16. When a pair of dice is rolled, in how many ways can the sum of the two numbers be
7? a. 4 b. 5 c. 6 d. 7
B. Directions: Use a Venn Diagram to illustrate the following.
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A = {all odd numbers} C = {2, 5}
B = {all even numbers} D = {numbers greater than 5}

1. 𝑨 ∪ 𝑩 2. 𝑨 ∩ 𝑪

U U

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3. A’ 4. 𝑪 ∩ 𝑫

U U

REFERENCES:
Jose Maria L. Escaner IV, PhD. et al. 2014. Spiral Math 10. Philippines. Trinitas
Publishing, Inc.

Government of the Philippines. Department of Education. 2015. Mathematics

Learner’s Module Grade 10. Pasig City. Rex Book Store, Inc.

Sr. Iluminada C. Coronel,F.M.M. et al. 2015. Growing Up with Math. Philippines.

FNB Educational, Inc.

Abelardo A. Villareal and Gemmalyn S. Gestoso. 2017. Mathematics for Grade 10 A

Spiral Approach Explanations, Examples, Exercises. Quezon City. Educational
Resources Corporation

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