Mutually Exclusive Probabilities

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.

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Are the following events mutually exclusive?Rolling a 6 and rolling an even number

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Are the following events mutually exclusive?Drawing a 4 from a deck of cards, and drawing a diamond.

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Are the days of the week mutually exclusive?

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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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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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Are the following events mutually exclusive?Rolling a 6 and rolling an even number

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Immunology
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Mo

Are the following events mutually exclusive?Drawing a 4 from a deck of cards, and drawing a diamond.

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Mo

Are the days of the week mutually exclusive?

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Last Updated: 07.08.2022
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This article will explore the definition, the probability, and examples of mutually exclusive events.

Definition 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!

Probability of mutually exclusive events

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.

The Venn diagram representation of mutual exclusivity

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.

The set notation representation of mutual exclusivity

Recall that the " $\cap$ " 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:

$A \cap B = \emptyset$

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:

$P (A \cap B) = 0$

Rule for mutually exclusive events

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:

$P (A \cup B) = P (A) + P (B) - P (A \cap B)$ .

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.

$P (A \cup B) = P (A) + P (B)$

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

Examples of probability of mutually exclusive events

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.

$P (" r o l l i n g a n e v e n n u m b e r ") = P (" r o l l i n g a 2, 4, o r 6 ") = P (" r o l l i n g 2 ") + P (" r o l l i n g 4 ") + P (" r o l l i n g 6 ") = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}$

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.

$P (' a t l e a s t o n e c h i l d i s a b o y') = P (G B o r B B o r B G) = \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = \frac{3}{4}$

Independent events and mutually exclusive events

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	$P (A \cap B) = P (A) \times P (B)$	$P (A \cup B) = P (A) + P (B)$ $P (A \cap B) = 0$
Venn diagram	Venn diagram of independent events	Venn diagram of mutually exclusive events
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.

Mutually Exclusive Probabilities - Key takeaways

Two events are mutually exclusive if they cannot happen at the same time
There are two mathematical definitions of mutual exclusivity:
- $P (A \cup B) = P (A) + P (B)$
- $P (A \cap B) = 0$
The 'sum' or 'or' rule: the union of two mutually exclusive events equals the sum of the probabilities of the events

Flashcards in Mutually Exclusive Probabilities 5

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Are the following events mutually exclusive?

Rolling a 6 and rolling an even number

Yes

Are the following events mutually exclusive?

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

Yes

Are the days of the week mutually exclusive?

Yes

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?

They are mutually exclusive events.

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

Going to the cinema and going to the theatre

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Frequently Asked Questions about Mutually Exclusive Probabilities

What is mutually exclusive in probability?

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

How do you know if two probabilities are of mutually exclusive events?

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

What is the formula for solving Mutually Exclusive Probabilities?

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

What is an example of Mutually Exclusive Probabilities?

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

What is the method for solving Mutually Exclusive Probabilities?

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

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