# 12.4: Mutually Exclusive Events

**At Grade**Created by: CK-12

**Practice**Mutually Exclusive Events

Remember Tyler and the juggling?

Now that Tyler knows the probability of his successfully juggling four balls for three minutes, he is certain that this is one of the things that he will do for the talent show. After all, he has an 85% probability that he will be successful. Those are very good odds.

Tyler also knows how to balance a pole on his chin. He can do this for 4 minutes without dropping it. Tyler does a similar experiment as he did with the juggling balls and discovers that he has a 75% probability of being able to balance the pole for four minutes. He is a bit concerned about his odds.

“I am not sure if I should do anything,” he tells Liz after figuring out the probability.

“What do you mean?”

“I mean that if my balancing is that off, maybe I shouldn’t do the juggling either.”

“Why not? Juggling and balancing are disjoint events,” Liz said smiling.

Disjoint events? Tyler isn’t sure what she means. He stops to think about this for minute.

Do you know what Liz means?

**This Concept is all about different events and their relationships to each other. When we look at events, knowing how they are related can help us in our work. Take some time to learn about them and then we’ll revisit this problem at the end of the Concept.**

### Guidance

Your Aunt Betsy is having a baby! As a mathematician, you realize that this is an opportunity to explore ** disjoint events**.

*Disjoint events***are two events that have no outcomes in common.**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 a dilemma so that we can better understand disjoint events.

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

**Step 1:** To see if \begin{align*}P (\text{red})\end{align*} and \begin{align*}P (\text{blue})\end{align*} or 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

**Step 2:** Now compare the list. If there are no outcomes in common, the two events are disjoint.

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

Now it's time for you to try a few on your own. Consider each pair of events. Tell whether these are disjoint events meaning that they have no connection to each other.

#### Example A

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

**Solution: Not disjoint events**

#### Example B

Eating an ice cream cone and the sun shining

**Solution: Disjoint events**

#### Example C

Wearing sneakers and running a marathon

**Solution: Not disjoint events**

**Here is the original problem once again. Reread it and then look at Liz’s explanation of disjoint events.**

Now that Tyler knows the probability of his successfully juggling four balls for three minutes, he is certain that this is one of the things that he will do for the talent show. After all, he has an 85% probability that he will be successful. Those are very good odds.

Tyler also knows how to balance a pole on his chin. He can do this for 4 minutes without dropping it. Tyler does a similar experiment as he did with the juggling balls and discovers that he has a 75% probability of being able to balance the pole for four minutes. He is a bit concerned about his odds.

“I am not sure if I should do anything,” he tells Liz after figuring out the probability.

“What do you mean?”

“I mean that if my balancing is that off, maybe I shouldn’t do the juggling either.”

“Why not? Juggling and balancing are disjoint events,” Liz said smiling.

**“What do you mean?” Tyler asks.**

**“Disjoint events are events where the outcome of one event does not affect the results of the other event. How you did juggling does not impact how you do balancing. You can still accomplish one or both of them. How you do in each of the events is separate. Does that make sense?”**

**“Yes. I think I will try both even though my probability of successfully balancing is less than successfully juggling,” Tyler decides.**

### Vocabulary

Here are the vocabulary words in this Concept.

- Disjoint Events
- events that are not connected. One outcome does not affect the other.

- Overlapping Events
- events that have outcomes in common.

- Complementary Events
- One of two events must occur then the two are complementary events. We can subtract one event from 1 to get the other event.

### Guided Practice

Here is one for you to try on your own.

Are these disjoint events? Explain your thinking.

It is going to rain today. I enjoy sailing.

**Answer**

These are disjoint events. There isn't a connection between it raining and the fact that the person "I" likes sailing. These events are not connected, therefore they are disjoint events.

### Video Review

Here is a video for review.

- This is a James Sousa video on probability.

### Practice

Directions: 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?

### Notes/Highlights Having trouble? Report an issue.

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Term | Definition |
---|---|

complement |
A mutually exclusive pair of events are complements to each other. For example: If the desired outcome is heads on a flipped coin, the complement is tails. |

Disjoint Events |
Disjoint or mutually exclusive events cannot both occur in a single trial of a given experiment. |

Event |
An event is a set of one or more possible results of a probability experiment. |

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

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 have no common outcomes. |

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

Overlapping Events |
Overlapping events are events that have outcomes in common. |

Permutation |
A permutation is an arrangement of objects where order is important. |

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

### Image Attributions

Here you'll learn to recognize disjoint events as having no outcomes in common.