Mutually Exclusive Events - Formula & Examples

# Mutually Exclusive Events

A set of outcomes is referred to as event. When two events cannot occur at the same time, then we say that the events are mutually exclusive else the events are said to be not mutually exclusive. For example: when we flip a coin then either heads can come or tails. Both heads and tails cannot be outcomes simultaneously.
Example of not mutually exclusive events is the event of getting a total of 5 or getting prime numbers on top in the experiment of throwing a pair of dice. In probability we studied about various terms like events, sure events, favourable events, outcomes and mutually exclusive events etc. The probability of an event lies between zero and one. In this section we will learn about mutually exclusive events with some solved examples.

## Formula

There are two things that are clear when we say two events say $M\ and\ N$ that are associated with an experiment are mutually exclusive:

1) $M\ and\ N$:

The intersection set is equal to {NULL} and hence $P (M\ and\ N)$ = 0.
This is because when the events cannot occur simultaneously there can be nothing in common in them.

2) $M\ or\ N$
P $(M\ or\ N)$ = $P (M) + P (N)$.

The probability of union of two mutually exclusive events is given by the sum of the probabilities of the events individually.
If the events are not mutually exclusive then it is not necessary that P (M @ N) = 0 and hence we have

$P (M\ or\ N) = P (M) + P (N) - P (M\ and\ N)$

But does this mean that mutually exclusive events are independent as well? NO.

Mutually exclusive events cannot be independent. As in case of independent events P (M @ N) = P (M) * P (N). But when two events are mutually exclusive then $P (M and N)$ = 0 always which is not the case in case of independent events. Independent events does not affect outcome of each other but they are not mutually exclusive as both of the independent events will have some individual non zero probability of their own and two non zero numbers can never product to zero which is a major condition of mutually exclusive events.

## Examples

Example 1: From the following events judge the pair of mutually exclusive or not mutually exclusive events and justify your answers.

a) Drawing a king or an ace from a deck of cards

b) Getting a white ball or a red ball from an urn of white and red balls.

c) Getting a number multiple of 3 and divisible by 2 when a dice is thrown.

d) Drawing a red card or a jack from a given 52 cards deck.

e) Getting three heads or three tails when three coins are flipped.

Solution:

a) This set of events is mutually exclusive and we can either have an ace or king but both cannot be draw simultaneously.

b) When we draw a ball from the urn of white and red balls it can be only of one color. So again the events here are mutually exclusive.

c) In this case 6 is the number which is a multiple of 3 and is also divisible by 2. Hence we have a case that proves that the events here are not mutually exclusive.

d) Here, a jack can be red in color too along with black. So the events are not mutually exclusive.

e) Three heads or three tails are mutually exclusive events as it is clear that if one occurred the second cannot occur at all.