Determine whether or not the two events are mutually exclusive

Possibilities:

1. The card chosen can be a 5.

2. The card chosen can be 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.

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.

Related Pages
More Lessons On Probability
Probability Tree Diagrams
Independent Events
Dependent Events

The following diagrams show the formulas for the probability of mutually exclusive events and non-mutually exclusive events. Scroll down the page for examples and solutions.

Probability Of Mutually Exclusive Events

Two events are said to be mutually exclusive if they cannot happen at the same time.

For example, if we toss a coin, either heads or tails might turn up, but not heads and tails at the same time. Similarly, in a single throw of a die, we can only have one number shown at the top face. The numbers on the face are mutually exclusive events.

If A and B are mutually exclusive events then the probability of A happening OR the probability of B happening is P(A) + P(B).

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

Example:
The probabilities of three teams A, B and C winning a badminton competition are 1/3, 1/5 and 1/9 respectively.

Calculate the probability that a) either A or B will win b) either A or B or C will win c) none of these teams will win

d) neither A nor B will win

Solution:

c) P(none will win) = 1 – P(A or B or C will win)

d) P(neither A nor B will win) = 1 – P(either A or B will win)

Mutually Exclusive Events

Probabilities of Mutually Exclusive Events
If two events are ‘mutually exclusive’ they cannot occur at the same time.

Learn all about mutually exclusive events in this video.
For mutually exclusive events the total probabilities must add up to 1.

Probability - P(A ∪ B) and Mutually Exclusive Events P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

For mutually exclusive events, P(A ∩ B) = 0.

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Mutually Exclusive Events And Non-Mutually Exclusive Events

The following video shows how to calculate the probability of mutually exclusive events and non-mutually exclusive events.

Examples:

1. Find the probability of drawing a yellow ball or drawing a three.
2. Find the probability of drawing a red ball or drawing an odd number ball.

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Mutually Exclusive Events - Introduction

Examples:

1. What is the probability of drawing a heart and a black card?
2. What is the probability of drawing a heart or a black card?
3. What is the probability of drawing a heart and a face card?
4. What is the probability of drawing a heart or a face card?

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Mutually Exclusive Events Vs Independent Events

Example: The figure shows how 25 people travelled to work: B for bicycle, T for Train and W for Walk. a) Write down two of these events that are mutually exclusive. Give a reason for your answer.

b) Determine whether or not B and T are independent events.

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Independence And Mutually Exclusive

Mutually exclusive events cannot be independent events.
Independent events cannot be mutually exclusive events.

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Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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