Using Probability to Determine Whether Events Are Independent
Integrated Math 2
10.1
Warm-up
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
10.2
Activity
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
Use the data to estimate the probabilities. Explain or show your reasoning.
We have seen two ways to check for independence using probability. Use your estimates to check whether each might be true.
Based on these results, do you think the events are independent?
10.3
Activity
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.”
Use the table to find the probabilities. Show your reasoning.
Based on these probabilities, are the events independent? Explain your reasoning.
From your analysis, do you think there is evidence to support the scientists’ theory that this genetic variation contributes to the likelihood that a person gets the disease?
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.
Student Lesson 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
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 and by looking only at the second at bat after a strikeout, we can estimate that . Although the probabilities are not exactly the same, we might decide that being within 0.7% is close enough to equality, depending on the number of at bats in the data set, to say that these 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
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 , and . The events are independent since .
