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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…

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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?

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Answer

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

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Question

Are the following events mutually exclusive?

Rolling a 6 and rolling an even number

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Answer

Yes

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Question

Are the following events mutually exclusive?

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

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Answer

Yes

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Question

Are the days of the week mutually exclusive?

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Answer

Yes

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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?

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Answer

They are mutually exclusive events.

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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?

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Answer

Going to the cinema and going to the theatre

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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?

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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.

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