Probability of Mutually Exclusive Events | Solved Examples, FAQs

Class 11 Mathematics | Written by Neeraj Anand
Published by ANAND TECHNICAL PUBLISHERS

Introduction

In probability theory, mutually exclusive events are events that cannot occur simultaneously. If one event happens, the other cannot happen at the same time. This concept is crucial in understanding the basic rules of probability and solving related problems effectively.

What are Mutually Exclusive Events in Probability?

Mutually exclusive events are two or more events that cannot occur at the same time or in the same trial. In mathematical terms, this means that the intersection of these events is an empty set (∅). If one event happens, the other(s) cannot happen simultaneously.

If A and B happen to be mutually exclusive events, then

P(A and B) = 0

or P(A ⋂ B) = 0

Where,

• A and B are Mutually Exclusive Events
• P(A and B) represents Probability of both A and B Occurring Together

Examples of Mutually Exclusive Events

Coin toss:

• Event A: Getting Heads
• Event B: Getting Tails

These are mutually exclusive because a coin can’t be both heads and tails in a single toss.

Rolling a Die:

• Event A: Rolling an even number (2, 4, 6)
• Event B: Rolling an odd number (1, 3, 5)

These are mutually exclusive because a number can’t be both even and odd.

Selecting a Card from a Deck:

• Event A: Drawing a Heart
• Event B: Drawing a Spade

These are mutually exclusive because a single card can’t be both a heart and a spade.

Weather:

• Event A: Sunny Day
• Event B: Rainy Day

These are typically considered mutually exclusive (although light rain on a sunny day is possible in some cases).

Rules for Mutually Exclusive Events

• Addition Rule: For mutually exclusive events A and B: P(A or B) = P(A) + P(B). This is different from non-mutually exclusive events, where: P(A or B) = P(A) + P(B) – P(A and B).
• Probability Sum: The sum of probabilities of all mutually exclusive events that make up the entire sample space is always 1 (or 100%).
• Conditional Probability: For mutually exclusive events A and B: P(A|B) = 0 and P(B|A) = 0 This means the probability of A occurring given that B has occurred (and vice versa) is zero.
• Multiplication Rule: For mutually exclusive events A and B: P(A and B) = 0 This is because mutually exclusive events cannot occur together.
• Complementary Events: If A and B are complementary events (mutually exclusive and exhaustive): P(A) + P(B) = 1

If A and B happen to be mutually exclusive events, then P(A ⋂ B) = 0.

Important Result : The probability of one or other events is equal to the sum of their separate probabilities. For two events, X and Y, we have \(\begin{array}{l}P (X \cup Y) = P(X) + P(Y) – P(X \cap Y)\\\end{array} \) If in case X and Y are mutually exclusive events, then there will be no common event. So, \(\begin{array}{l}P(X \cap Y) = 0\end{array} \) Therefore, \(\begin{array}{l}P (X \cup Y) = P(X) + P(Y)\\\end{array} \)

What are Independent Events?

Independent events are events where the occurrence of one event does not affect the probability of the other event occurring. In other words, the outcome of one event has no influence on the outcome of the other event.

Mathematically, events A and B are independent if:

P(A|B) = P(A) and P(B|A) = P(B)

where,

• P(A|B) is the Probability of A given that B has occurred
• P(A) is the Probability of A occurring on its Own

How to Find Independent Events?

To determine if events are independent follow the steps added below:

Step 1: Calculate the probability of each event occurring separately: P(A) and P(B).

Step 2: Calculate the probability of both events occurring together: P(A and B).

Step 3: If P(A and B) = P(A) × P(B), then the events are independent.

Examples of Independent events

Coin Tosses:

• Event A: Getting heads on the first toss
• Event B: Getting tails on the second toss

These are independent because the outcome of the first toss doesn’t affect the second.

Rolling Dice:

• Event A: Rolling a 6 on the first die
• Event B: Rolling an even number on the second die

These are independent as the roll of one die doesn’t influence the other.

Drawing Cards with Replacement:

• Event A: Drawing a heart on the first draw
• Event B: Drawing a spade on the second draw (after replacing the first card)

These are independent because replacing the card keeps the probabilities constant.

Weather in Different Cities:

• Event A: Rain in New York
• Event B: Sunshine in Los Angeles

Generally independent, as weather in one city usually doesn’t directly affect another distant city.

Rules for Independent events

Multiplication Rule: For independent events A and B: P(A and B) = P(A) × P(B) This extends to multiple events: P(A and B and C) = P(A) × P(B) × P(C)

Conditional Probability: For independent events A and B: P(A|B) = P(A) and P(B|A) = P(B) The occurrence of one event doesn’t change the probability of the other.

Addition Rule: For independent events A and B: P(A or B) = P(A) + P(B) – P(A) × P(B) This accounts for the overlap in probabilities.

Complement Rule: For independent events A and B: P(not A and not B) = P(not A) × P(not B)

Independence of Complements: If A and B are independent, then:

• A and (not B) are Independent
• (not A) and B are Independent
• (not A) and (not B) are Independent

Pairwise Vs Mutual Independence: Events can be pairwise independent but not mutually independent. For true mutual independence, all possible combinations of events must be independent.

Difference Between Mutually Exclusive Events and Independent Events

Major difference between Mutually Exclusive Events and Independent Events are:

Aspect	Mutually Exclusive Events	Independent Events
Definition	Events that cannot occur simultaneously	Events where the occurrence of one does not affect the probability of the other
Probability of Intersection	P(A and B) = 0	P(A and B) = P(A) × P(B)
Addition Rule	P(A or B) = P(A) + P(B)	P(A or B) = P(A) + P(B) – P(A) × P(B)
Conditional Probability	P(A|B) = 0 (if B occurred, A cannot occur)	P(A|B) = P(A) (B’s occurrence doesn’t affect A)
Relationship	Mutually exclusive events are dependent	Independent events are not necessarily mutually exclusive
Sample Space	Occupy distinct parts of the sample space	Can overlap in the sample space
Example	Rolling a 1 OR rolling a 2 on a die	Drawing a heart AND rolling a 6 on a die
Venn Diagram	No overlap between circles	Circles can overlap
Effect on Each Other	Occurrence of one eliminates the possibility of the other	Occurrence of one does not influence the other
Probability Sum	Sum of Probabilities ≤ 1	Sum of Probabilities can be > 1

Solved Examples on Mutually Exclusive Events in Probability

Question 1: What is the probability of a die showing a number 3 or number 5?

Solution: Let,

P(3) is the probability of getting a number 3

P(5) is the probability of getting a number 5

P(3) = 1/6 and P(5) = 1/6

So,

P(3 or 5) = P(3) + P(5)

P(3 or 5) = (1/6) + (1/6) = 2/6

P(3 or 5) = 1/3

Therefore, the probability of a die showing 3 or 5 is 1/3.

Question 2: Three coins are tossed at the same time. We say A as the event of receiving at least 2 heads. Likewise, B denotes the event of getting no heads and C is the event of getting heads on the second coin. Which of these is mutually exclusive?

Solution:  Firstly, let us create a sample space for each event. For the event ‘A’ we have to get at least two head. Therefore, we have to include all the events that have two or more heads.

Or we can write:

A = {HHT, HTH, THH, HHH}.

This set A has 4 elements or events in it i.e. n(A) = 4

In the same way,  for event B, we can write the sample as:

B = {TTT} and n(B) = 1

Again using the same logic, we can write;

C = {THT, HHH, HHT, THH} and n(C) = 4

So B & C and A & B are mutually exclusive since they have nothing in their intersection.

Question 3: The likelihood of the 3 teams a, b, c winning a football match are 1 / 3, 1 / 5 and 1 / 9 respectively. Find the probability that

a] out of the three teams, either team a or team b will win

b] either team a or team b or team c will win

c] none of the teams will win the match

d] neither team a nor team b will win the match

Answer: 

a) P (A or B will win) = 1/3 + 1/5 = 8/15

b) P (A or B or C will win) = 1/3 + 1/5 + 1/9 = 29/45

c) P (none will win) = 1 – P (A or B or C will win) = 1 – 29/45 = 16/45

d) P (neither A nor B will win) = 1 – P(either A or B will win)

= 1 – 8/15

= 7/15

Question 4: If A and B are two independent events, then A and B’ is:

Answer:  A ∩ B’ and A ∩ B are mutually exclusive events such that;

A = (A ∩ B’) ∪ (A ∩ B)

P(A) = P(A ∩ B’) + P(A ∩ B)

P(A ∩ B’) = P(A) – P(A ∩ B)

= P(A) – P(A).P(B)   (Since A and B are independent)

= P(A ∩ B’)

=> P(A) (1 – P(B)) = P(A) P(B’)

Thus, A and B’ are also independent.

Question 5: If P (A) = 2 / 3, P (B) = 1 / 2 and P (A ∪ B) = 5 / 6 then events A and B are mutually exclusive or not:

Answer: 

P (A ∪ B) = P (A) + P (B) − P (A ∩ B)

5 / 6 = (2 / 3) + (1 / 2) − P (A ∩ B)

⇒ P (A ∩ B) = 0

The events A and B are mutually exclusive.

Question 6: A card is drawn at random from a well-shuffled deck of 52 cards. Find the probability that the card drawn is a king or an ace.

Answer: 

As per the definition of mutually exclusive events, selecting an ace and selecting a king from a well-shuffled deck of 52 cards are termed mutually exclusive events.

Assume X to be the event of drawing a king and Y to be the event of drawing an ace.

In a standard deck of 52 cards, there exists 4 kings and 4 aces.

P (an event) = count of favourable outcomes / total count of outcomes

P (selecting a king from a standard deck of 52 cards) = P (X) = 4 / 52 = 1 / 13

P (selecting an ace from a standard deck of 52 cards) = P (Y) = 4 / 52 = 1 / 13

To compute P (king or ace).

By the formula of addition theorem for mutually exclusive events,

P (X U Y) = P (X) + P (Y)

P (X U Y) = (1 / 13) + (1 / 13)

= (1 + 1) / 13

= 2 / 13

The probability of selecting a king or an ace from a well-shuffled deck of 52 cards = 2 / 13.

Question 7: A pair of dice is rolled. Find the probability of

(i) getting either even numbers or odd numbers.

(ii) the sum of the numbers rolled is either 6 or 10.

Solution:

(i) Possible outcomes for even numbers: (2, 2), (4, 4), (6, 6)

⇒ P(even numbers) = 3/36 = 1/12

Possible outcomes for odd numbers:  (1, 1), (3, 3), (5, 5)

⇒ P(Odd numbers) = 1/12

P (even or Odd) = 1/12 + 1/12 = 2/12 = 1/6

(ii)

(1, 5), (2, 4), (3, 3), (4, 2), (5, 1) → (5 outcomes that have sum 6)

(6, 4), (5, 5), (4, 6) → (3 outcomes that have sum 10)

Now,

Probability of 6: P(6) = 5/36

Probability of 10: P(10) = 3/36

Both events are mutually exclusive since the sum of numbers cannot be 6 and 10 at the same time.

P(6 or 10) = P(6) + P(10) = 5/36 + 3/36 = 8/36 = 2/9.

Question 8: Which of the following are mutually exclusive events?

(i) On a throw of a die, “getting 1” and “getting 5.”

(ii) Getting “a head” or “a tail.”

(iii) Choose “a king” or “a queen” from a deck of cards.

(iv) “Having an ace” and “having a spade” from a deck of cards.

Solution:

i) On a throw of a die, the two events “getting 1” and “getting 5” are two mutually-exclusive events because we will never get both 1 and 5  at one time in a throw.

ii) Getting a head or a tail are two mutually-exclusive events.

iii) Drawing a king or a queen are mutually-exclusive events because both cannot be drawn at one time.

(iv) The two events “having an ace” and “having a spade” are not mutually exclusive since we may even draw an “ace of spade”. So, these two events can occur in the same draw.

Question 9:: The probabilities of three mutually exclusive events are 2/3, 1/4 and 1/6, respectively. Verify whether the statement is correct or not.

Solution:

Let the events be A, B, and C.

If the events are mutually exclusive, then A ⋂ B = 0, B ⋂ C = 0 and A ⋂ C = 0.

So, A ⋂ B ⋂ C = 0.

If the above conditions are satisfied, then P(A ⋃ B ⋃ C) = P(A) + P(B) + P(C).

Since P(A ⋃ B ⋃ C) = 13/12 > 1, the probability value lies within 1.

Therefore, the statement is wrong.

Question 10 : If P (A) = 1/3, P (B) = 2/3, then check whether

a] A & B are mutually exclusive.

b] A & B are exhaustive.

Solution:

The events are said to be mutually exclusive if P(A ⋂ B) = 0.

The events are exhaustive if P(A ⋃ B) = 1.

\(\begin{array}{l}P(A)+P(B)=\frac{2}{3}+\frac{1}{3}=1\\ P(A\cup B)=P(A)+P(B)-P(A\cap B)\\1=1-0\end{array} \)

If [A ⋃ B] be the sample space, then the above two conditions are true.

Hence, A and B are mutually exclusive and exhaustive.

Question 11 : Events A, B, and C are mutually exclusive events such that

\(\begin{array}{l}P(A)=\frac{3x+1}{3}, P(B)=\frac{1-x}{4},P(C)=\frac{1-2x}{2},\end{array} \)

, then find the set of all possible values of x are in the interval.

Solution:

Given that the events are mutually exclusive.

\(\begin{array}{l}P(A)\geq 0,P(B)\geq 0,P(C)\geq 0, \text{and}\ P(A\cup B\cup C)\geq 0\\ \Rightarrow P(A)\geq 0,P(B)\geq 0,P(C)\geq 0, \text{and}\ P(A)+P(B)+P(C)\geq 0\\ \frac{3x+1}{3}\geq 0,\frac{1-x}{4}>0, \frac{1-2x}{2}\geq 0\\ \frac{3x+1}{3}+\frac{1-x}{4}+ \frac{1-2x}{2}\geq 0\\ x > \frac{-1}{3}, x \leq 1, x \leq \frac{1}{2}, [1-3x] > 0\\ \frac{-1}{3} \leq x \leq \frac{1}{2} \text{and}\ x\leq \frac{1}{3}\\ \frac{-1}{3} \leq x \leq \frac{1}{2}\Rightarrow x\epsilon [\frac{-1}{3},\frac{1}{2}]\end{array} \)

Question 12 : Two dice are thrown, and the sum of the numbers which come up on the dice is noted. Consider the following events associated with this experiment.

A: The sum is less than or equal to 3.

B: The sum is greater than 11.

Check whether these pairs of events are mutually exclusive.

Solution:

Number of elements in the sample space S = 36

Then, A = {(1,1), (1,2), (2,1)}

B = {(6,6)}

A⋂ B = Ø

Hence, A and B are mutually exclusive events. 

Question 13 : A coin is tossed three times; consider the following events:

A: No tail appears

 B: Exactly one tail appears

Do they form a set of mutually exclusive events?

Solution:

The sample space S = { HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

A = {HHH}

B = { THH, HTH, HHT}

A⋂ B = Ø

Hence, A and B are mutually exclusive events. 

Question 14 : What is the probability of a dice showing a 2 or 5?

Solution:

P(2) = 1/6

P(5) = 1/6

P(2 or 5) = P(2) + P(5)

= (1/6) + (1/6) 

= 1/3

Frequently Asked Questions on Mutually Exclusive Events in Probability

Q1

What do you mean by mutually exclusive events?

The events that cannot happen simultaneously or at the same time are called mutually exclusive events.

Q2

What is the formula of mutually exclusive events?

If A and B are two mutually exclusive events, then probability of A or B is equal to the sum of probability of both the events.
P(A or B) = P(A) + P(B)

Q3

How to find if two events are mutually exclusive?

If two events are mutually exclusive then the probability of both the events occurring at the same time is equal to zero. P(A and B) = 0.

Q4

What is an example of mutually exclusive event?

The examples of mutually exclusive events are tossing a coin, throwing a die, drawing a card from a deck a card, etc. When we toss a coin, we get either a head or a tail. Head and tail cannot happen at the same time. This is an example of mutually exclusive event.

Q5

What is P(A ⋂ B), if A and B are mutually exclusive?

If A and B are mutually exclusive, then P(A ⋂ B) = 0.

Real-Life Examples

• A student can either pass or fail an exam; both outcomes cannot occur together.
• A light switch can either be on or off, not both at the same time.

Importance in Probability

Understanding mutually exclusive events helps simplify complex problems and apply probability formulas correctly. It is particularly important for solving questions related to addition laws of probability and calculating combined probabilities.

Conclusion

Mastering the concept of mutually exclusive events is essential for students preparing for CBSE Board exams and JEE Mains & Advanced. It forms the foundation for more advanced probability topics.

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