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# Mutually Inclusive vs. Mutually Exclusive Events

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 canâ€™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. Itâ€™s possible for the dice to land on a number that is even and is greater than 3.

### Probabilities of Events

If two events areÂ 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: