Two events are **mutually exclusive** if they cannot occur at the same time.

For example, let event A be the event that a dice lands on an even number and let event B be the event that a dice lands on an odd number.

We would define the sample space for the events as follows:

- A = {2, 4, 6}
- B = {1, 3, 5}

Notice that there is no overlap between the two sample spaces. Thus, events A and B are mutually exclusive because they both cannot occur at the same time. The number that a dice lands on can’t be even *and* odd at the same time.

Conversely, two events are **mutually inclusive** if they *can* occur at the same time.

For example, let event C be the event that a dice lands on an even number and let event D be the event that a dice lands on a number greater than 3.

We would define the sample space for the events as follows:

- C = {2, 4, 6}
- D = {4, 5, 6}

Notice that there is overlap between the two sample spaces. Thus, events C and D are mutually inclusive because they can both occur at the same time. It’s possible for the dice to land on a number that is even *and* is greater than 3.

**Probabilities of Events**

If two events are **mutually exclusive**, then the probability that they both occur is zero.

For example, consider the two sample spaces for events A and B from earlier:

- A = {2, 4, 6}
- B = {1, 3, 5}

Since there is no overlap in the sample spaces, we would say P(A and B) = **0**.

But if two events are **mutually inclusive**, then the probability that they both occur will be some number greater than zero.

For example, consider the two sample spaces for events C and D from earlier:

- C = {2, 4, 6}
- D = {4, 5, 6}

Since there are 6 possible numbers that the dice could land on and two of those numbers (4 and 6) belong to both events C and D, we would calculate P(C and D) as 2/6, or **1/3**.

**Visualizing Mutually Inclusive & Mutually Exclusive Events**

We often use Venn diagrams to visualize the probabilities associated with events.

If two events are **mutually exclusive** then they would not overlap at all in a Venn diagram:

Conversely, if two events are **mutually inclusive** then there would be at least some overlap in the Venn diagram:

**Additional Resources**

An Introduction to Theoretical Probability

The General Multiplication Rule

What Are Disjoint Events?