# The Difference between Mutually Exclusive and Independent Events

An *event* is one possible outcome of a random experiment. Events may sometimes be related to each other. Two key ways in which events may be related are known as mutually exclusive and independent.

## How to Identify Mutually Exclusive Events

Two events are said to be *mutually exclusive* if they can’t both happen at the same time. Here are two events that are mutually exclusive:

*A* = The roll of a die is odd.

*B* = The roll of a die is even.

Clearly, the roll of a die must result in a number that is either odd or even; it can’t be both. Therefore, events *A* and *B* are mutually exclusive.

As another example, in a coin-flipping experiment, suppose that two events are defined:

*G* = Two heads turn up.

*H *= Two tails turn up.

It’s impossible for *both* two heads to turn up *and* two tails to turn up. This means that *G* and *H* are mutually exclusive. This result can be demonstrated using sets as follows:

*G* = {HH} and *H* = {TT}. These events have no elements in common; their intersection is the *empty set*

The probability of the empty set is zero; therefore, the event that both *G* and *H* occur is *impossible*. This means that *G* and *H* are mutually exclusive.

## How to Identify Independent Events

Two events *A* and *B* are said to be *independent* if the outcome of event *A* doesn’t affect the outcome of event *B* and vice versa. For example, suppose that based on the coin-flipping experiment, event *A* is defined as the event that the first flip is a head, and event *B* is defined as the event that the second flip is a head. In other words:

*A* = {HH, HT}

*B* = {HH, TH}

Because the outcome of the first flip has no influence over the outcome of the second flip, events *A* and *B* are *independent events*.

Note that *A* and *B* are *not* mutually exclusive; both *A* and *B* can occur.