# Mutually Inclusive Events

Probability is a prediction that an event will occur. Probability of an event lies between zero and one. There are two types of events commonly when we talk about probability: mutually exclusive and mutually inclusive.

Mutually exclusive events are the ones when they cannot happen at same time, that is, there is no outcomes common in these events. For example: getting a head or a tail when a coin is tossed. On tossing a coin, we can never get head or tail together, so these two events are mutually exclusive.

In this case the combined probability of two events can be obtained by simply adding up the individual properties of the events:

P $(X \cup Y)$ = P (X) + P (Y), where X and Y are mutually exclusive events.

Mutually inclusive events are the ones in which there are some common outcomes in between the given events. Like getting an odd number or getting a prime number when we throw a dice. In these two events there are common outcomes {3, 5} repeating in both the events. So these two events are mutually inclusive events.

## Formula

In case of mutually inclusive events, we can evaluate the combined probability of the two events with the help of following formula:

P $(M \cup N)$ = P (M) + P (N) - P $(M \cap N)$

Where, P $(M \cup N)$ is the combined probability of occurrence of either M or N,
P (M) is the probability of occurrence of event M, P (N) is the probability of occurrence of event N, and P $(M \cap N)$ is the probability of joint occurrence of both M & N.
When there is no common outcome between M and N then P $(M \cap N)$ = 0

## Problems

Let us see some problems on mutually inclusive events for better knowledge.

Example 1:

Which of the following events are mutually inclusive and which are not? Give reasons.

a) On throwing two dices together getting a sum of 4 or getting exactly two 2’s.

b) On tossing two coins together, getting exactly two head or exactly two tails.

c) On drawing a card from a deck getting a 2 or getting a heart.

Solution:

a) In this case the events of getting a sum of 4 and exactly two 2’s are mutually inclusive as the outcome (2, 2) is common in both.

b) When we toss two coins we can get either two heads or two tails or other combinations but both two head and two coins cannot be occurring together. Thus these two events are not mutually inclusive.

c) When we draw a card of number 2 from a deck it can also be a 2 of hearts. This implies that these two events are mutually inclusive.

Example 2: Find the probability of getting a number greater than 3 or a multiple of 2 when we throw a dice.

Solution:

Clearly n (S) = 6

Let A = number greater than 3 = {4, 5, 6}
=> P (A) =  $\frac{3}{6}$

Let B = multiple of 2 = {2, 4, 6}

=> P (B) =  $\frac{3}{6}$

$A \cap B$ = {4, 6}

=> P (A $\cap$ B) =  $\frac{4}{6}$

Hence the two events are mutually inclusive.
Hence P (A $\cup$ B) = P (A) + P (B) - P (A $\cap$ B) = $\frac{3}{6}$ + $\frac{3}{6}$ - $\frac{2}{6}$ = $\frac{4}{6}$ = $\frac{2}{3}$