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Mutually inclusive events

Mutually inclusive events are events that can occur at the same time as opposed to mutually exclusive events.


Example of inclusive events

Let event A be that you pick an even number between 11 and 20 and event B that you pick a number larger than 10 and less than 15. The Venn diagram would show an overlap, or a joint section: 

Mutually inclusive events


 12 and 14 are mutually inclusive as they can occur at the same time. We can denote:

Mutually inclusive events

The probability of A and B is the joint section which consists of two numbers: 12 and 14, So there is a 2 out of 10 probability (= 20%) event A and B occurring.


The addition rule for mutually inclusive events

Let event A be 1,2,3,4 and event B be 3,4,5,6:

A = {1,2,3,4}

B = {3,4,5,6}

As described in Sample space, events and probabilities we would double count the joint area if we just add up A and B. In this case, we would count the elements 3 and 4 twice. That is one time too many, so we need to subtract one of these counts. Therefore, we say A+B – (A+B).

In our example, it would be the objects of 1, 2, 3, 4, 3, 4, 5, 6 – (the objects 3 & 4) and the general formula for calculating the probability of A or B for mutually inclusive events becomes:

Mutually inclusive events


Card example with mutually inclusive events

What is the probability of drawing a card from a standard deck of cards that is either an ace or a spade?

Let A be the event of drawing a spade and B be the event of drawing an ace:

Mutually inclusive events

The probability of selecting a spade or an ace would be the two events added up, only that we then count the ace of spades twice, so we need to subtract that one card:

Mutually inclusive events

P(A and B) is P(A) x P(B) = 13/52 x 4/52 = 1/52 which can be seen as the fact that A and B is ace of spades. It is one card out of the 52, so:

Mutually inclusive events


The probability of drawing an ace or a spade is 4/13 31%.



CK-12 Foundation video: Mutually Inclusive Events


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