# Lesson 10

Using Probability to Determine Whether Events Are Independent

- Let’s take a closer look at dependent and independent events.

### 10.1: Which One Doesn’t Belong: Events

A coin is flipped and a standard number cube is rolled. Which one doesn’t belong?

Set 1

Event A1: the coin landing heads up

Event B1: rolling a 3 or 5

Set 2

Event A2: rolling a 3 or 5

Event B2: rolling an odd number

Set 3

Event A3: rolling a prime number

Event B3: rolling an even number

Set 4

Event A4: the coin landing heads up

Event B4: the coin landing tails up

### 10.2: Overtime Wins

Does a hockey team perform differently in games that go into overtime (or shootout) compared to games that don't? The table shows data about the team over 5 years.

year | games played | total wins | overtime or shootout games played | wins in overtime or shootout games |
---|---|---|---|---|

2018 | 82 | 46 | 19 | 6 |

2017 | 82 | 46 | 18 | 7 |

2016 | 82 | 51 | 23 | 16 |

2015 | 82 | 54 | 18 | 10 |

2014 | 82 | 34 | 17 | 5 |

total | 410 | 231 | 95 | 44 |

Let A represent the event “the hockey team wins a game” and B represent “the game goes to overtime or shootout.”

- Use the data to estimate the probabilities. Explain or show your reasoning.
- \(P(\text{A})\)
- \(P(\text{B})\)
- \(P(\text{A and B})\)
- \(P(\text{A | B})\)

- We have seen two ways to check for independence using probability. Use your estimates to check whether each might be true.
- \(P(\text{A | B}) = P(\text{A})\)
- \(P(\text{A and B}) = P(\text{A}) \boldcdot P(\text{B})\)

- Based on these results, do you think the events are independent?

### 10.3: Genetic Testing

A suspected cause of a disease is a variation in a certain gene. A study gathers at-risk people at random and tests them for the disease as well as for the genetic variation.

has the disease | does not have the disease | |
---|---|---|

has the genetic variation | 80 | 12 |

does not have the genetic variation | 1,055 | 1,160 |

A person from the study is selected at random. Let A represent the event “has the disease” and B represent “has the genetic variation.”

- Use the table to find the probabilities. Show your reasoning.
- \(P(\text{A})\)
- \(P(\text{B})\)
- \(P(\text{A and B})\)
- \(P(\text{A | B})\)

- Based on these probabilities, are the events independent? Explain your reasoning.
- A company that tests for this genetic variation has determined that someone has the variation and wants to inform the person that they may be at risk of developing this disease when they get older. Based on this study, what percentage chance of getting the disease should the company report as an estimate to the person? Explain your reasoning.

Find or collect your own data that can be represented using a two-way table. Some ideas include finding data about a sports team (shooting percentage greater than 50% or less than 50% vs. winning or losing) or clinical trials for a medication (placebo or medication vs. symptom or no symptom).

- Create a two-way table to represent the sample space.
- Use probabilities to determine whether events summarized in the table are dependent or independent events.
- When data is collected in an experiment or a survey it is often difficult to show evidence that events are dependent. Why do you think this occurs? Explain your reasoning.

### Summary

Although it may not always be easy to determine whether events are dependent or independent based on their descriptions alone, there are several ways to check for independence using probabilities.

One way to recognize independence is by understanding the experiment well enough to see if it fits the definition:

- Events A and B are independent if the probability of Event A occurring does not change whether Event B occurs or not.

An example of independence that can be found this way might be the events “a coin landing heads up” and “rolling a 4 on a number cube” when flipping a coin and rolling a standard number cube. Whether the coin lands heads up or not does not change the probability of rolling a 4 on a number cube.

A second way to recognize independence is to use conditional probability:

- Events A and B are independent if \(P(\text{A | B}) = P(\text{A})\)

An example of independence that can be found this way might be the events “gets a hit on the second time to bat in a game” and “struck out in the first at bat in a game” for a baseball player in games for a season. By looking at what happens when the player has his second at bat, we can estimate that \(P(\text{hit on second at bat}) = 0.324\) and by looking only at the second at bat after a strikeout, we can estimate that \(P(\text{hit on second at bat | strike out on first at bat}) = 0.324\) so the events are independent.

Another way to recognize independence is to look at the probability of both events happening:

- Events A and B are independent if \(P(\text{A and B}) = P(\text{A}) \boldcdot P(\text{B})\)

An example of independence that can be found this way might be the events “making the first free throw shot” and “making the second free throw shot” for a basketball player shooting two free throws after a foul. By looking at the outcomes of the two shots for the player throughout the year, we can estimate that \(P(\text{make the first shot}) = 0.72\), \(P(\text{make the second shot}) = 0.72\) and \(P(\text{make the first shot and make the second shot}) = 0.52\). The events are independent since \(0.72 \boldcdot 0.72 \approx 0.52\).

### Glossary Entries

**conditional probability**The probability that one event occurs under the condition that another event occurs.

**dependent events**Dependent events are two events from the same experiment for which the probability of one event depends on whether the other event happens.

**independent events**Independent events are two events from the same experiment for which the probability of one event is not affected by whether the other event occurs or not.