# Mutually exclusive events

In this chapter, we will discuss the concept of mutually exclusive events which is used extensively in probability.

At the end of the chapter, some solved problems are provided for better understanding.

## Mutually exclusive event definition

Two or more events are mutually exclusive if there is no common element present between the events.

Since there is no common elements, these events cannot occur simultaneously in any given experiment.

For example;
Consider the experiment of rolling a dice.

Let A be the event of getting even number and B is the event of getting odd number.
A = {2, 4, 6 }
B = {1, 3, 5 }

Note that there is no common element between event A & B and that’s why events A & B are mutually exclusive.

### Representing mutually exclusive events

Let A & B are mutually exclusive events. Both the events can be expressed as;

A ∩ B = 𝜙

It tells that the intersection of event A & B gives null result. This signifies that there is no common element between the events.

### Formula for mutually exclusive events

Given below is the important formula for mutually exclusive event;

\mathtt{P\ ( A\ \cup \ B\ ) \ =\ P( A) \ +\ P( B)}

If A & B are mutually exclusive events, then probability of union of A & B is equal to sum of probability of A & B.

#### Proof of mutually exclusive formula;

According to general probability addition rule;

P (A U B) = P(A) + P(B) – P(A ∩ B)

Since A & B are mutually exclusive; P(A ∩ B) = 0

So, the formula becomes;
P (A U B) = P(A) + P(B)

### Mutually exclusive events – Solved examples

Question 01
Consider the experiment of rolling a dice. Given below are the events for the experiment;

A = getting even number
B = getting odd number
C = getting multiple of 3

Answer the following questions;
(a) If A & B are mutually exclusive
(b) If B & C are mutually exclusive
(c) If C & A re mutually exclusive

Solution
Let us write the event set for all the given events;

A = {2, 4, 6 }
B = {1, 3, 5 }
C = {3, 6}

(a) A & B are mutually exclusive since there are no common elements between the two.

i.e. A ∩ B = 𝜙

(b) B & C are not mutually exclusive since element 3 is common in both set.

(c) C & A are not mutually exclusive since element 6 is common in both.

Question 02
In a class, one boy and 2 girls are present. In an experiment, two children are selected at random. Given below are events for experiment;

A = both are girls
B = one boy and girl
C = at least one girl

Check which of the given event if mutually exclusive.

Solution
Let’s first write the sample space of given experiment;

S = {BG1, BG2, G1G2}

Writing the elements of the given events;
A = {G1G2}
B = {BG1, BG2}
C = {BG1, BG2, G1G2}

Note that there is no common element between event A and B. Hence, A & B are mutually exclusive events.

Example 03
In a box there are 5 red color, 9 blue color and 6 green color pencils. If a pencil is selected at random, find the probability of getting red or blue color pencil.

Solution
Total pencil = 20

Let A be the event of getting red pencil and B be the event of getting blue pencil.

Calculating the probability of both event A & B;

P (A) = 5 / 20

P (B) = 9 / 20

Note that both events A & B are mutually exclusive. So we can apply following formula;

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

P (A U B ) = 5/20 + 9/20

P (A U B) = 14 / 20 = 7/10

Hence, the probability of getting red or blue pencil is 7/10

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