Unit 8, Lesson 10
Using Probability to Determine Whether Events Are Independent
Integrated Math 2
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A coin is flipped and a standard number cube is rolled. Which three sets go together? Why do they go together?
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
Does this 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.
Let A represent the event “the hockey team wins a game” and B represent “the game goes to overtime or shootout.”
| 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 |
A group of scientists think that a variation in a certain gene contributes to the likelihood that a person gets a particular disease. 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.”
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:
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:
An example of independence that can be found this way might be the events “gets a hit on the second time at 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
Another way to recognize independence is to look at the probability of both events happening:
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