What are Complementary Events in Probability?

The probability of an event is a measure of the chance of occurrence of an event when an experiment is done. Complementary events occur when there are only two outcomes, for example, clearing an exam or not clearing an exam. The complement means the exact opposite of an event. In this article, we will learn what are complementary events in probability.

Definition

For any event A, there exists another event A‘ which shows the remaining elements of the sample space S. A’ denotes the complementary event of A.

A’ = S – A.

Event A and A’ are mutually exclusive and exhaustive.

Consider the example of tossing a coin. Let P(E) denote the probability of getting a tail when a coin is tossed. Then,

Solved Examples

Example 1:

If P(A₁ ⋃ A₂) = 1 – P(A₁^c) P(A₂^c) where c stands for complement, then events A₁ and A₂ are

A) Mutually exclusive

B) Independent

C) Equally likely

D) None of these

Solution:

Given P(A₁ ⋃ A₂) = 1 – P(A₁^c)P(A₂^c)

= 1 – (1 – P(A₁))(1 – P(A₂))

= P(A₁) + P(A₂) – P(A₁) P(A₂)

Hence, the events are independent.

So option (B) is the answer.

Example 2:

If P(A) + P(B) = 1; then which of the following options explains the event A and B correctly?

(A) Event A and B are mutually exclusive, exhaustive and complementary events

(B) Event A and B are mutually exclusive and exhaustive events

(C) Event A and B are mutually exclusive and complementary events

(D) Event A and B are exhaustive and complementary events

Solution:

Mutually exclusive events are events that cannot happen at the same time. Mutually exhaustive events are events in which at least one of the events should occur. P(A) and P(B) cannot be zero at the same time since it is given the sum of both is 1. Complementary events are those two events which are the only possible events. Since P(A) + P(B) = 1, A and B are possible events. Hence, A and B are mutually exclusive, exhaustive and complementary events.

Hence option (A) is the answer.

Example 3: A number is chosen at random from a set of whole numbers from 1 to 50. Calculate the probability that the chosen number is not a perfect square.

(A) 1/25

(B) 43/50

(C) 7/50

(D) 7/25

Solution:

Let B denotes the event of choosing a perfect square.

Let B’ denotes the event that the number chosen is not a perfect square.

B = { 1, 4, 9, 16, 25, 36, 49}

Number of elements in B = n(B) = 7

P(B) = n(B)/n(S) = 7/50

P(B’) = 1-P(A) 

= 1-(7/50)

= 43/50

So the required probability is 43/50.

Hence option (B) is the answer.

Example 4: Two dice are thrown. What is the probability that the sum of the numbers appearing on the two dice is 11, if 5 appears on the first.

(A) 1/36

(B) 1/6

(C) 5/6

(D) None of these

Solution:

5 appears on the first. Hence to get sum 11, the number on the second die must be 6.

So required probability = 1/36

Hence option (A) is the answer.

Stay tuned with BYJU’S and practice more Probability Problems asked in JEE Main exam.