Mutually Exclusive Probabilities

You may have heard the phrase "mutually exclusive" before. It's a rather fancy way of saying something very simple: if two events are mutually exclusive, they cannot happen at the same time. It is important in probability mathematics to be able to recognise mutually exclusive events since they have properties that allow us to work out the likelihood of these events happening.

This article will explore the definition, the probability, and examples of mutually exclusive events.

Definition of mutually exclusive events

Take a coin flip for example: you can either flip heads or tails. Since these are obviously the only possible outcomes, and they cannot happen at the same time, we call the two events 'heads' and 'tails' mutually exclusive. The following is a list of some mutually exclusive events:

• The days of the week - you cannot have a scenario where it is both Monday and Friday!

• The outcomes of a dice roll

• Selecting a 'diamond' and a 'black' card from a deck

The following are not mutually exclusive since they could happen simultaneously:

• Selecting a 'club' and an 'ace' from a deck of cards

• Rolling a '4' and rolling an even number

Try and think of your own examples of mutually exclusive events to make sure you understand the concept!

Probability of mutually exclusive events

Now that you understand what mutual exclusivity means, we can go about defining it mathematically.

Take mutually exclusive events A and B. They cannot happen at the same time, so we can say that there is no intersection between the two events. We can show this using either a Venn diagram or using set notation.

The Venn diagram representation of mutual exclusivity

Mutually exclusive events

The Venn diagram shows very clearly that, to be mutually exclusive, events A and B need to be separate. Indeed, you can see visually that there is no overlap between the two events.

The set notation representation of mutual exclusivity

Recall that the "$\cap$" symbol means 'and' or 'intersection'. One way of defining mutual exclusivity is by noting that the intersection does not exist and is therefore equal to the empty set:

$A\cap B=\varnothing$

This means that, since the intersection of A and B doesn't exist, the probability of A and B happening together is equal to zero:

$P(A\cap B)=0$

Rule for mutually exclusive events

Another way to describe mutually exclusive events using set notation is by thinking about the 'union' of the events. The definition of union in probability is as follows:

$P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$.

Since the probability of the intersection of two mutually exclusive events is equal to zero, we have the following definition of mutually exclusive events which is also known as the 'sum rule' or the 'or' rule:

This is a very handy rule to apply. Have a look at the examples below.

Examples of probability of mutually exclusive events

In this section, we will work on a couple of examples of applying the previous concepts.

Independent events and mutually exclusive events

Students sometimes mix up independent events and mutually exclusive events. It's important to be familiar with the differences between them since they mean very different things.