# 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:

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

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:

## 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:

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:

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:

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

## Learnings

CK-12 Foundation video: Mutually Inclusive Events

#### Carsten Grube

Freelance Data Analyst

##### Normal distribution

##### Two-sample inference

##### Confidence intervals

##### Simple linear regression, fundamentals

##### ANOVA & the F-distribution

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