# Mutually Exclusive Events

Experiment 1: A single card is chosen at random from a standard deck of 52 playing cards. What is the probability of choosing a 5 or a king?

Possibilities:

1. The card chosen can be a 5.

2. The card chosen can be a king.

Experiment 2: A single card is chosen at random from a standard deck of 52 playing cards. What is the probability of choosing a club or a king?

Possibilities:

1. The card chosen can be a club.

2. The card chosen can be a king.

3. The card chosen can be a king and a club (i.e., the king of clubs).

In Experiment 1, the card chosen can be a five or a king, *but not both at the same time*. These events are mutually exclusive. In Experiment 2, the card chosen can be a club, or a king, or both at the same time. These events are not mutually exclusive.

**Definition: **Two events are **mutually exclusive** if they cannot occur at the same time (i.e., they have no outcomes in common).

Experiment 3: A single 6-sided die is rolled. What is the probability of rolling an odd number or an even number?

Possibilities:

1. The number rolled can be an odd number.

2. The number rolled can be an even number.

Events: These events are mutually exclusive since they cannot occur at the same time.

Experiment 4: A single 6-sided die is rolled. What is the probability of rolling a 5 or an odd number?

Possibilities:

1. The number rolled can be a 5.

2. The number rolled can be an odd number (1, 3 or 5).

3. The number rolled can be a 5 and odd.

Events: These events are not mutually exclusive since they can occur at the same time.

Experiment 5: A single letter is chosen at random from the word SCHOOL. What is the probability of choosing an S or an O?

Possibilities:

1. The letter chosen can be an S

2. The letter chosen can be an O

Events: These events are mutually exclusive since they cannot occur at the same time.

Experiment 6: A single letter is chosen at random from the word SCHOOL. What is the probability of choosing an O or a vowel?

Possibilities:

1. The letter chosen can be an O

2. The letter chosen can be a vowel

3. The letter chosen can be an O and a vowel

Events: These events are not mutually exclusive since they can occur at the same time.

Summary: In this lesson, we have learned the difference between mutually exclusive and non-mutually exclusive events. We can use set theory and Venn Diagrams to illustrate this difference.

**Mutually Exclusive Events**

Two events are mutually exclusive if they cannot occur at the same time (i.e., they have no outcomes in common).

In the Venn Diagram above, the probabilities of events A and B are represented by two disjoint sets (i.e., they have no elements in common).

**Non-Mutually Exclusive Events**

Two events are non-mutually exclusive if they have one or more outcomes in common.

In the Venn Diagram above, the probabilities of events A and B are represented by two intersecting sets (i.e., they have some elements in common).

Note: In each Venn diagram above, the sample space of the experiment is represented by S, with P(S) = 1.

**Exercises**

Directions: Read each question below. Select your answer by clicking on its button. Feedback to your answer is provided in the RESULTS BOX. If you make a mistake, choose a different button.

1. |
Which of the following are mutually exclusive events when a single card is chosen at random from a standard deck of 52 playing cards? |

2. |
All of the following are mutually exclusive events when a single 6-sided die is rolled EXCEPT: |

3. |
Which of the following are mutually exclusive events when a day of the week is chosen at random? |

4. |
A single letter is chosen at random from the word TEACHER. All of the following are mutually exclusive events except: |

5. |
Which of the following are mutually exclusive events when a month of the year is chosen at random? |