# Mutually Exclusive in Statistics: Definition, Formula & Examples

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Lesson Transcript
Instructor: Michael Perekupka
In this lesson, we will discuss mutually exclusive events in statistics. We will talk about the definition, give many real-life examples, and compare mutually exclusive events to independent events.

## What Are Mutually Exclusive Events?

As we grow up, we hear all sorts of sayings and idioms. Some of them seem to make a little sense, and then there are a couple that are so brilliantly simple. Have you ever heard the saying, 'You can't have your cake and eat it, too?' This saying is a perfect way to explain mutually exclusive events. The saying refers to the fact that you cannot both eat your cake and still have it front of you at the same time. Statistically speaking, having your cake, and eating your cake, are mutually exclusive.

Two events are defined to be mutually exclusive if they cannot happen at the same time. In other words, if one event happens, the other event cannot happen. Mutually exclusive events are sometimes referred to as disjointed events.

## Examples and Formula

To help understand this definition, it is best to go through a few real-life examples. For simplicity's sake, let's call two events A and B, and we will see a few examples of events that are mutually exclusive and then a few that are not:

A: Today is Saturday.
B: Today is Tuesday.

A: A coin lands on heads.
B: The same coin lands on tails.

A: You have passed your statistics class.
B: You have failed your statistics class.

Notice that all of the pairs of events cannot happen at the same time. It would be impossible. If today is Saturday, today can't possibly also be Tuesday. If a coin lands on heads, that means it did not land on tails. It is also impossible to both pass and fail the same class.

A: Today is Saturday.
B: It is raining outside.

A: A coin lands on heads.
B: I rolled a die and it landed on four.

A: You passed your statistics class.
B: You passed your accounting class.

These events might sound similar to the previous ones, but they're not. These can happen at the same time. For instance, it is very possible to have a rainy Saturday, to toss a coin to get heads while rolling a die and getting a four, and to pass both your statistics and accounting classes.

In formulaic form, two events, A and B, can be expressed as mutually exclusive as:

P (A and B) = 0

That is, there is no probability of any overlap between the two events. Another way to think of mutually exclusive events is with this picture:

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