Math & History: Mutually Exclusive & Mutually Inclusive Events

Mutually Exclusive

Events A and B are mutually exclusive events if they cannot occur at the same time. This means that A and B do not share any outcomes and P(A and B) = 0 aka P(A∩B) = 0.

Notice in the diagram below that there is no intersection between the possible outcomes of event A and the possible outcomes of event B. For example, if you were asked to pick a number between 1 and 10, you cannot pick a number that is both even and odd. These events are mutually exclusive.

Mutually Inclusive

"Mutually inclusive" (often just called inclusive events) is a term used to describe two events that can happen at the same time. If Event A and Event B are mutually inclusive, their "intersection" is not zero (P(A∩B) ≠ 0.

Notice in the diagram below that there is some overlap with events A and B indicating they can occur at the same time. For example, if a random number was chosen from 1 to 10 and event A is picking a number less than 4 and event B is picking an even number then the two events are inclusive since 2 is a number which both events share.

Example 1

Suppose the sample space S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Let A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, and C = {7, 9}.

A and B = {4, 5}. P(A and B) = 2/10 and is not equal to zero. Therefore, A and B are not mutually exclusive.

A and C do not have any numbers in common so P(A and C) = 0. Therefore, A and C are mutually exclusive.

Calculating Probability

Probability of Mutually Exclusive Events

Example 1

Imagine rolling a standard six-sided die. What is the probability of rolling a 5 or a 6?

S = {1, 2, 3, 4, 5, 6}

A = {5}

B = {6}

P(A) = 1/6

P(B) = 1/6

P(A or B) = P(A∪B) = 1/6 + 1/6 = 2/6 = 1/3

Here, the favorable outcomes are a 5 or a 6. Since we are trying to find the probability of either of them happening, we simply add their individual probabilities. We can do this since they are mutually exclusive events. That is, neither of them can happen at the same time.

Probability of Mutually Inclusive Events

Example 2

What is the probability of choosing a number from 1 to 10 that is less than 5 or odd?

S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A = {1, 2, 3, 4}
B = {1, 3, 5, 7, 9}

P(A) = 4/10 = 2/5

P(B) = 5/10 = 1/2

P(A and B ) = 2/10 = 1/5

P(A or B) = P(A) + P(B) - P(A and B)

P(A or B) = 2/5 + 1/2 - 1/5

P(A or B) = 4/10 + 5/10 - 2/10

P(A or B) = 7/10

When we calculated the probability of the mutually exclusive events in example 1, you see that we simply added their probabilities of event A and event B together. With example 2, we have to do a little more work and also subtract out the probability of A and B. Why? Because with mutually inclusive events, events A and B share some overlap (See A and B Venn diagram above). Namely, 1 and 3 appear in both sets. If we simply added the probabilities of A and B together (4/10 + 5/10), we would be counting the numbers 1 and 3 twice—once because they are less than 5, and again because they are odd. This is called double counting.

To fix this, we must subtract the probability of the events that they share (the intersection) exactly once. This removes the extra count and gives us the correct probability.

Addition Rule

P(A or B) = P(A) + P(B) - P(A and B)

stated using set notation

P(A∪B) = P(A) + P(B) - P(A∩B)

Example 3

What is the probability of choosing a card from a deck of cards that is a club or a ten? Let A be the event of selecting a club and B be the event of selecting a ten.

P(A) = 13/52

P(B) = 4/52

P(A and B) = 1/52

P(A or B) = P(A) + P(B) - P(A and B)

P(A or B) = 13/52 + 4/52 - 1/52

P(A or B ) = 16/52 = 4/13

Example 4

Two dice are rolled, and the events F and T are as follows:
F = {The sum of the dice is four} and T = {At least one die shows a three}

Find P(F∪T).

The sample space for this problem is 6 x 6 = 36

P(F) = 3/36

P(T) = 11/36