Explainer: Mutually and Nonmutually Exclusive Events

In this explainer, we will learn how to find the probability of mutually and non-mutually exclusive events.

You may recall that if events are mutually exclusive or disjoint, they cannot occur at the same time. For example, an animal cannot be both a cat and a dog: “being a cat” and “being a dog” are mutually exclusive or disjoint events. However, a person may like both cats and dogs, so “likes cats” and “likes dogs” are not mutually exclusive or disjoint events.

To calculate probabilities for mutually exclusive and nonmutually exclusive events, we will need to use some of the rules of probability. Let us remind ourselves of these.

Some Probability Rules and Definitions

For any event 𝐴, if 𝑃(𝐴) is the probability of event 𝐴 occurring, we have the following:

Rule 1

0𝑃(𝐴)1

Rule 2

The sum of the probabilities of all possible outcomes is equal to 1 (or 100%).

The complement of event 𝐴, written as 𝐴, refers to everything that is not 𝐴.

Rule 3

𝑃𝐴=1𝑃(𝐴)

Note: The complement of event 𝐴 is sometimes also written as 𝐴. Also note that 𝑃(𝐴𝐵) refers to the probability of the occurrence of 𝐴 and the nonoccurrence of 𝐵.

If two events 𝐴 and 𝐵 cannot occur at the same time, we say that they are mutually exclusive or disjoint events. In this case, the joint probability of 𝐴 and 𝐵 occurring at the same time is zero. We write this as 𝑃(𝐴𝐵)=0. If events 𝐴 and 𝐵 are mutually exclusive, the probability that 𝐴 or 𝐵 occurs is the sum of their probabilities.

That is,

Rule 4

𝑃(𝐴𝐵)=𝑃(𝐴)+𝑃(𝐵)

If events 𝐴 and 𝐵 are not mutually exclusive, the probability that either 𝐴 or 𝐵 or both occur is

Rule 5

𝑃(𝐴𝐵)=𝑃(𝐴)+𝑃(𝐵)𝑃(𝐴𝐵).

We are now armed with all the tools we need to tackle some examples. This first example uses the “total probability” rule for mutually exclusive events.

Example 1: Total Probability and Mutually Exclusive Events

In an animal rescue shelter, 39% of the current inhabitants are cats (C) and 41% are dogs (D).

  1. Find the probability that an animal chosen at random is either a cat or a dog.
  2. Find the probability that an animal chosen at random is neither a cat nor a dog.

Answer

Part 1

To find the probability that an animal chosen at random is either a cat or a dog, we note that the events “cat” and “dog” are mutually exclusive. That is, an animal can be either a cat or a dog but not both. This means that 𝑃()=0CD. We can illustrate this in a Venn diagram as below.

If we let 𝑃()C be the probability that an animal chosen at random is a cat and let 𝑃()D be the probability that an animal chosen at random is a dog, converting our percentages into probabilities by dividing by 100, we have 𝑃()=0.39C and 𝑃()=0.41D. We can now use the probability rule which states that, for mutually exclusive events, 𝑃()=𝑃()+𝑃()CDCD This gives us 𝑃()=𝑃()+𝑃()=0.39+0.41=0.8.CDCD

Hence, the probability that an animal chosen at random is either a cat or a dog is 0.8. Multiplying by 100 to get back to a percentage, we have an 80% chance of selecting either a cat or a dog.

Part 2

To find the probability that an animal chosen at random is neither a cat nor a dog, we use the rule that the sum of the probabilities of all possible outcomes is equal to 1. We know that the probability of either a cat or a dog is 0.8, so the probability of selecting neither a cat nor a dog is 10.8=0.2. As a percentage (×100), this means that 20% must be neither cats nor dogs.

The next example tests our understanding of mutual exclusivity.

Example 2: Are the Events Mutually Exclusive?

Isabella has a deck of 52 cards.