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Mutually Exclusive Events

Probability of two events that cannot occur at the same time P(A or B) = P(A) + P(B)

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Mutually Exclusive Events

Let's Think About It

Elizabeth will be the lead dancer in a ballet performance tonight.  She practiced all afternoon and successfully accomplished 25 pirouettes (spins) out of 30 attempts.  She feels as though if she walks around the block twice before she leaves for the performance, she will successfully complete her pirouettes during the performance.  Are the two events, pirouettes and walking around the block, disjoint events?  In other words, are they mutually exclusive?

In this concept, you will learn about probability and mutually exclusive events.

Guidance

Disjoint events are two events that have no outcomes in common. They are mutually exclusive of each other.  For example, for a pregnant woman, here are two events that have no outcomes in common.

  • Event B: Having a baby boy.
  • Event G: Having a baby girl.

Disjoint events are either-or events.  For example, consider flipping a coin. The two events–flipping heads or flipping tails–have no outcomes in common. You either flip heads or you flip tails.

The probability of one of the two disjoint events occurring is just the sum of the probabilities of the events. Since the probability of flipping heads is \begin{align*}\frac{1}{2}\end{align*} and the probability of flipping tails is \begin{align*}\frac{1}{2}\end{align*}:

\begin{align*}P (\text{heads or tails}) &= P (\text{heads}) + P (\text{tails})\\ &= \frac{1}{2} + \frac{1}{2}\\ &= 1\end{align*}

In other words, all possibilities are covered. The probability of either heads or tails is 1–it will be either heads or tails 100 percent of the time.

Let’s look at an example.

For a single spin, are events \begin{align*}R (\text{red})\end{align*} and \begin{align*}B (\text{blue})\end{align*} disjoint events?

First, to see if \begin{align*}P (\text{red})\end{align*} and \begin{align*}P (\text{blue})\end{align*} are disjoint events, make a list of the outcomes of \begin{align*}P (\text{red})\end{align*} and \begin{align*}P (\text{blue})\end{align*}:

\begin{align*}R\end{align*} outcomes: red

\begin{align*}B\end{align*} outcomes: blue

Next, compare the list to decide if the events have common outcomes. If there are no outcomes in common, the two events are disjoint.

Then, if there are no outcomes in common, the two events are disjoint. 

The answer is \begin{align*}R\end{align*} and \begin{align*}B\end{align*} are disjoint events because they have nothing in common.

Guided Practice

Are these disjoint events? Explain your thinking.

It is going to rain today. I enjoy sailing.

First, to see if \begin{align*}P (\text{rain})\end{align*} and \begin{align*}P (\text{sailing})\end{align*} are disjoint events, make a list of the outcomes:

\begin{align*}rain\end{align*} outcomes: rain

\begin{align*}sailing\end{align*} outcomes: sailing

Next, compare the list to decide if the events have common outcomes.

Then, if there are no outcomes in common, the two events are disjoint.

The answer is these are disjoint events.  There isn't a connection between it raining and the fact that the person, "I", likes sailing. 

Examples

Tell whether the following events are disjoint events.

Example 1

At a baseball game, hitting a ball and scoring a home run.

First, to see if \begin{align*}P (\text{hitting a ball})\end{align*} and \begin{align*}P (\text{scoring a home run})\end{align*} are disjoint events, make a list of the outcomes:

\begin{align*}hitting\ a\ ball\end{align*} outcomes: foul, out, home run 

\begin{align*}home\ run\end{align*} outcomes: home run

Next, compare the list to decide if the events have common outcomes.

Then, if there are no outcomes in common, the two events are disjoint.  If there are outcomes in common, the two events are not disjoint.

The answer is these are not disjoint events.  The event of hitting a ball can result in a home run.  Therefore, the events have common possible outcomes.

Example 2

Eating an ice cream cone and the sun shining

First, to see if \begin{align*}P (\text{eating ice cream cone})\end{align*} and \begin{align*}P (\text{sun shining})\end{align*} are disjoint events, make a list of the outcomes:

\begin{align*}eating\ ice\ cream\ cone\end{align*} outcomes:  eating ice cream cone

\begin{align*}sun\ shining\end{align*} outcomes: sun shining

Next, compare the list to decide if the events have common outcomes.

Then, if there are no outcomes in common, the two events are disjoint.

The answer is these are disjoint events.  There isn't a direct connection between eating ice cream and the sun shining.  Either can occur without the other. 

Example 3

Wearing sneakers and running a marathon

First, to see if \begin{align*}P (\text{sneakers})\end{align*} and \begin{align*}P (\text{running})\end{align*} are disjoint events, make a list of the outcomes:   

\begin{align*}sneakers\end{align*} outcomes: performing athetics such as tennis or running  

\begin{align*}running\end{align*} outcomes: running, as in a marathon

Next, compare the list to decide if the events have common outcomes.

Then, if there are no outcomes in common, the two events are disjoint.  If there is a commonality, the two events are not disjoint.

The answer is these are not disjoint events.  There is a connection between wearing sneakers and running a marathon.

Follow Up

Remember Elizabeth and her pirouettes?  She had accomplished 25 pirouette attempts out of 30 during practice.  Are the two events, performing a pirouette and walking around the block, disjoint events?  Are they mutually exclusive events?

First, to see if \begin{align*}P (\text{pirouettes})\end{align*} and \begin{align*}P (\text{walking around the block})\end{align*} are disjoint events, make a list of the outcomes:

\begin{align*}pirouettes\end{align*} outcomes:  pirouettes

\begin{align*}walking\ around\ the\ block\end{align*} outcomes: walking around the block

Next, compare the list to decide if the events have common outcomes. If there are no outcomes in common, the two events are disjoint.

Then, if there are no outcomes in common, the two events are disjoint. 

The answer is performing pirouettes and walking around the block are disjoint events because they have nothing in common.

Video Review

Explore More

Use what you have learned to solve each problem.

1. For a single toss of a number cube, are events \begin{align*}F (\text{four})\end{align*} and \begin{align*}E (\text{even})\end{align*} disjoint events or overlapping events?

2. For a single toss of a number cube, are \begin{align*}T (\text{three})\end{align*} and \begin{align*}E (\text{even})\end{align*} disjoint events or overlapping events?

3. For a single toss of a number cube, are \begin{align*}O (\text{odd})\end{align*} and \begin{align*}E (\text{even})\end{align*} disjoint events or overlapping events?

4. For a single toss of a number cube, are \begin{align*}T (\text{two})\end{align*} and \begin{align*}L5\end{align*} (less than 5) disjoint events or overlapping events?

5. For a single toss of a number cube, are \begin{align*}S (\text{six})\end{align*} and \begin{align*}O (\text{one})\end{align*} disjoint events or overlapping events?

6. For a single toss of a number cube, are \begin{align*}F (\text{five})\end{align*} and \begin{align*}L5\end{align*} (less than 5) disjoint events or overlapping events?

7. For a single spin, are \begin{align*}R (\text{red})\end{align*} and \begin{align*}Y (\text{yellow})\end{align*} disjoint events or overlapping events?

8. For a single spin, are \begin{align*}R (\text{red})\end{align*} and \begin{align*}L (\text{left})\end{align*} disjoint events or overlapping events?

9. For a single spin, are \begin{align*}P (\text{yellow})\end{align*} and \begin{align*}P (\text{right})\end{align*} disjoint events or overlapping events?

10. For a single spin, are \begin{align*}R (\text{right})\end{align*} and \begin{align*}L (\text{left})\end{align*} disjoint events or overlapping events?

11. For a single spin, are \begin{align*}L (\text{left})\end{align*} and \begin{align*}G (\text{green})\end{align*} disjoint events or overlapping events?

12. For a baby, are \begin{align*}B (\text{boy})\end{align*} and \begin{align*}R (\text{right-handed})\end{align*} disjoint events or overlapping events?

13. For a baby, are \begin{align*}L (\text{left-handed})\end{align*} and \begin{align*}R (\text{right-handed})\end{align*} disjoint events or overlapping events?

14. For a baby, are \begin{align*}G (\text{girl})\end{align*} and \begin{align*}B (\text{brown hair})\end{align*} disjoint events or overlapping events?

15. For a baby, are \begin{align*}G (\text{girl})\end{align*} and \begin{align*}H (\text{heavier than 8 pounds})\end{align*} disjoint events or overlapping events?

Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 12.4. 

Vocabulary

Event

Event

An event is a set of one or more possible results of a probability experiment.
Independent Events

Independent Events

Two events are independent if the occurrence of one event does not impact the probability of the other event.
Intersection

Intersection

Intersection is the probability of both or all of the events you are calculating happening at the same time (less likely).
Mutually Exclusive Events

Mutually Exclusive Events

Mutually exclusive events have no common outcomes.
Outcome

Outcome

An outcome of a probability experiment is one possible end result.
Permutation

Permutation

A permutation is an arrangement of objects where order is important.
statistical probability

statistical probability

A statistical probability is the calculated probability that a given outcome will occur if the same experiment were completed an infinite number of times.
union

union

\cup is a symbol that stands for union and is used to connect two groups together. It is associated with the logical term OR.

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