StudySmarter – The all-in-one study app.

4.8 • +11k Ratings

More than 3 Million Downloads

Free

Americas

Europe

You may have heard the phrase “mutually exclusive” before. It’s a rather fancy way of saying something very simple: if two events are mutually exclusive, they cannot happen at the same time. It is important in probability mathematics to be able to recognise mutually exclusive events since they have properties that allow us to work out the likelihood of these…

Explore our app and discover over 50 million learning materials for free.

Save the explanation now and read when you’ve got time to spare.

Save

Lerne mit deinen Freunden und bleibe auf dem richtigen Kurs mit deinen persönlichen Lernstatistiken

Jetzt kostenlos anmelden

Nie wieder prokastinieren mit unseren Lernerinnerungen.

Jetzt kostenlos anmelden

You may have heard the phrase “mutually exclusive” before. It’s a rather fancy way of saying something very simple: if two events are mutually exclusive, they cannot happen at the same time. It is important in probability mathematics to be able to recognise mutually exclusive events since they have properties that allow us to work out the likelihood of these events happening.

This article will explore the definition, the probability, and examples of mutually exclusive events.

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

Take a coin flip for example: you can either flip heads or tails. Since these are obviously the only possible outcomes, and they cannot happen at the same time, we call the two events ‘heads’ and ‘tails’ mutually exclusive. The following is a list of some mutually exclusive events:

The days of the week – you cannot have a scenario where it is both Monday and Friday!

The outcomes of a dice roll

Selecting a ‘diamond’ and a ‘black’ card from a deck

The following are not mutually exclusive since they could happen simultaneously:

Selecting a ‘club’ and an ‘ace’ from a deck of cards

Rolling a ‘4’ and rolling an even number

Try and think of your own examples of mutually exclusive events to make sure you understand the concept!

Now that you understand what mutual exclusivity means, we can go about defining it mathematically.

Take mutually exclusive events A and B. They cannot happen at the same time, so we can say that there is no intersection between the two events. We can show this using either a Venn diagram or using set notation.

Mutually exclusive events

The Venn diagram shows very clearly that, to be mutually exclusive, events A and B need to be separate. Indeed, you can see visually that there is no overlap between the two events.

Recall that the “” symbol means ‘and’ or ‘intersection’. One way of defining mutual exclusivity is by noting that the intersection does not exist and is therefore equal to the empty set:

This means that, since the intersection of A and B doesn’t exist, the probability of A and B happening together is equal to zero:

Another way to describe mutually exclusive events using set notation is by thinking about the ‘union’ of the events. The definition of union in probability is as follows:

.

Since the probability of the intersection of two mutually exclusive events is equal to zero, we have the following definition of mutually exclusive events which is also known as the ‘sum rule’ or the ‘or’ rule:

The union of two mutually exclusive events equals the sum of the events.

This is a very handy rule to apply. Have a look at the examples below.

In this section, we will work on a couple of examples of applying the previous concepts.

You roll a regular 6-sided dice. What is the probability of rolling an even number?

Solution

The sample space is the possible outcomes from rolling the dice: 1, 2, 3, 4, 5, 6. The even numbers on the dice are 2, 4, and 6. Since these results are mutually exclusive, we can apply the sum rule to find the probability of rolling either 2, 4 or 6.

A couple has two children. What is the probability that at least one child is a boy?

Solution

Our sample space consists of the different possible combinations that the couple can have. Let B denote a boy and G denote a girl.

Our sample space is therefore S = {GG, GB, BB, BG}. Since none of these options can occur simultaneously, they are all mutually exclusive. We can therefore apply the ‘sum’ rule.

Students sometimes mix up independent events and mutually exclusive events. It’s important to be familiar with the differences between them since they mean very different things.

Independent Events Mutually Exclusive Events
Explanation One event occurring does not change the probability of the other event. Two events are mutually exclusive if they cannot happen at the same time.
Mathematical definition
Venn diagram
Example Drawing a card from a deck, replacing the card, shuffling the deck, then drawing another card.Explanation: since you are replacing the first card, this does not affect the likelihood of drawing any card the second time. Flipping a coin.Explanation: the outcome of a coin flip is either heads or tails. Since these two events cannot occur simultaneously, they are mutually exclusive events.

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

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

The union of two mutually exclusive events equals the sum of the probabilities of the events.

The two events “heads” or “tails” when flipping a coin are mutually exclusive events.

The union of two mutually exclusive events equals the sum of the probabilities of the events.

Question

What does ‘mutually exclusive’ mean?

Show answer

Answer

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

Show question

Question

Are the following events mutually exclusive?

Rolling a 6 and rolling an even number

Show answer

Answer

Yes

Show question

Question

Are the following events mutually exclusive?

Drawing a 4 from a deck of cards, and drawing a diamond.

Show answer

Answer

Yes

Show question

Question

Are the days of the week mutually exclusive?

Show answer

Answer

Yes

Show question

Question

Fashions change, but at the time of writing, everyone agrees that neckties should be worn with shirts, not t-shirts. A necktie would look ridiculous with a t-shirt.

Based on the above, which one of the following is true about wearing a t-shirt and wearing a necktie at the time of writing?

Show answer

Answer

They are mutually exclusive events.

Show question

Question

I’m planning what I will do this evening. I could go to a restaurant, I could cook a meal at home, I could go to the cinema and I could go to the theatre. I won’t eat two meals. I can’t go to both the cinema and the theatre, because their shows are at the same time.

Which of the following pairs of events must be mutually exclusive?

Show answer

Answer

Going to the cinema and going to the theatre

Show question

Question

According to a travel guide, the country of Bhutan has no traffic lights, but there are traffic wardens everywhere.

In Bhutan, are traffic wardens and traffic lights mutually exclusive?

Show answer

Answer

Yes. Since there are no traffic lights, the probability of traffic lights is 0. Since traffic wardens are everywhere, the probability of traffic wardens is 1. Since these two cannot happen at the same time, they are mutually exclusive events.

Show question

of the users don’t pass the Mutually Exclusive Probabilities quiz! Will you pass the quiz?

Start Quiz

How would you like to learn this content?

94% of StudySmarter users achieve better grades.

Sign up for free!

94% of StudySmarter users achieve better grades.

Sign up for free!

How would you like to learn this content?

Free math cheat sheet!

Everything you need to know on . A perfect summary so you can easily remember everything.

Be perfectly prepared on time with an individual plan.

Test your knowledge with gamified quizzes.

Create and find flashcards in record time.

Create beautiful notes faster than ever before.

Have all your study materials in one place.

Upload unlimited documents and save them online.

Identify your study strength and weaknesses.

Set individual study goals and earn points reaching them.

Stop procrastinating with our study reminders.

Earn points, unlock badges and level up while studying.

Create flashcards in notes completely automatically.

Create the most beautiful study materials using our templates.

Sign up to highlight and take notes. It’s 100% free.

Save explanations to your personalised space and access them anytime, anywhere!

Sign up with Email Sign up with Apple

By signing up, you agree to the Terms and Conditions and the Privacy Policy of StudySmarter.

Already have an account? Log in

You are watching: Mutually Exclusive Probabilities: Explanation. Info created by GBee English Center selection and synthesis along with other related topics.