Mutually Exclusive Events Examples

Two such events who can never occur together are known as mutually exclusive events. If we choose a card from a given deck of 52 cards, then to get a red and a black card is mutually exclusive. We can get a red card or a black card. For two events A and B, if they are mutually exclusive then,

$P(A\ and\ B) = 0$

$P(A\ or\ B) = P(A) + P(B)$

Example:

From a deck of 52 cards, one card is chosen then getting a king and a black card is mutually exclusive?
Solution:
No, they are not mutually exclusive events as we can get a card which is king and also a black card.

Word Problems

Problem 1:

Find the probability of getting an odd number and a multiple of 2 when a pair of dice is rolled.

Solution:

The event of getting an odd number, A, and a multiple of 2, B, that is, an even number are mutually exclusive. There can come either an odd number or an even number.

$P(A\ and\ B) = 0$
Problem 2:

One card is being chosen from the deck of 52 cards. What is the probability that it is a king and a Jack?

Solution:

Probability of getting a king, $P(A)$ = $\frac{4}{52}$ = $\frac{1}{13}$

Probability of getting a Jack, $P(B)$ = $\frac{4}{52}$ = $\frac{1}{13}$

Either we can can have a king or a jack, hence, these two events are mutually exclusive.

Probability of getting a king or a jack, $P(A\ or\ B)$ = $P(A) + P(B)$ = $\frac{1}{13}$ + $\frac{1}{13}$ = $\frac{2}{13}$
Problem 3:

Asif is a fast bowler playing a inter-college cricket match. What is the probability of him getting 5 wickets or 2 wickets in the match when there are 20 percent chances of him getting 5 wickets and 60 percent chances of him getting 2 wickets?

Solution:

Asif can either get 5 wickets or he can get 2 wickets. Hence, the probability of him getting 5 wickets AND 2 wickets is zero.

Probability of him getting 5 wickets, $P(A) = 0.2$

Probability of him getting 2 wickets, $P(B) = 0.6$

Probability of him getting 5 wickets or 2 wickets, $P(A\ or\ B) = P(A) + P(B) = 0.2 + 0.6 = 0.8.$
Problem 4:

In a box having 4 bulbs, the probability of  having one defected bulb is 0.5 and the probability of having no bulb defected is  0.4. Find the probability of having one defected bulb or zero defected bulb.

Solution:

Probability of 1 bulb being defected, $P(A) = 0.5$

Probability of 0 bulb being defected, $P(B) = 0.4$

As there can be either 1 defected bulb or zero defected bulb as these two events cannot happen together, they are mutually exclusive.

Probability of having one bulb or zero bulb defected, $P(A\ or\ B) = 0.5 + 0.4 = 0.9$
Problem 5:

There are two mobile companies X and Y, and John has to buy a mobile. The probability that he will buy mobile manufactured by company X is 0.3 and the probability that he will buy mobile by company Y is 0.2, then what is the probability that he will buy a mobile made by company X and company Y.

Solution:

Probability of buying a mobile by company $X, P(A) = 0.3$

Probability of buying a mobile by company $Y, P(B) = 0.2$

John can either buy a mobile from company X or company Y. These events are mutually exclusive. Hence,

$P(A\ and\ B) = 0$.