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 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:

Mutually exclusive events

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

Mutually inclusive events

Additional Resources

An Introduction to Theoretical Probability
The General Multiplication Rule

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