Lesson 10
Using Probability to Determine Whether Events Are Independent
10.1: Which One Doesn’t Belong: Events (5 minutes)
Warm-up
This warm-up prompts students to compare four descriptions of two events related to flipping a coin and rolling a standard number cube. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.
Launch
Arrange students in groups of 2–4. Display the descriptions of the events for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning why a particular item does not belong, and together find at least one reason each item doesn't belong.
Student Facing
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
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct.
During the discussion, ask students to explain the meaning of any terminology they use, such as independent events. Also, press students on unsubstantiated claims.
10.2: Overtime Wins (15 minutes)
Activity
The mathematical purpose of this activity is to use a two-way table as a sample space to decide if events are independent and to estimate conditional probabilities. Listen for students mentioning the concept of conditional probability.
Making spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Arrange students in groups of 2. Tell students there are many possible answers for the questions. After quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they are different. Follow with a whole-class discussion.
Design Principles: Maximize Meta-awareness, Support Sense-making
Student Facing
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?
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
Some students may not understand why \(P(\text{A | B}) = P(\text{A})\) is used to determine if two events are independent. Prompt students to think about what \(P(\text{A | B}) = P(\text{A})\) means using words. Emphasize that if the probability of event A under the condition of event B is equal to the probability of event A then it implies that the occurrence of event B does not affect the probability of event A. This means that event A and event B are independent events.
Activity Synthesis
The purpose of this discussion is for students to recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.
Here are some questions for discussion.
- “Where did you use conditional probability in this problem?” (I used it in question 1c and in 2a when I tried to figure out if the events were independent.)
- “How do you figure out how many games did not go into overtime or shootout?” (You take the total number of games played, 410, and subtract the number of games that went into overtime, 95, which mean 315 games did not go into overtime or shootout.)
- “Do you think the hockey team performs better, worse, or about the same in games that go into overtime (or shootout) than usual? Explain your reasoning.” (Overall, the team won about 56% of all their games but only 46% of their overtime games. This suggests that they do worse when games go to overtime. They won about 59% of the games that did not go into overtime.)
- “In 2016, do you think the hockey team performed better, worse, or about the same in games that went into overtime (or shootout) than games that did not go into overtime? Explain your reasoning.” (They performed better in games that went into overtime because they won 18 of the 23 of the overtime games (about 70%) and only about 59% of the 59 games that did not go to overtime.)
- “In 2016, were winning games going into overtime (or shootout) and winning games that did not go into overtime dependent or independent events?” (It suggests that they are dependent events because the probability that they won a game in overtime was higher than winning a game that did not go into overtime.)
Supports accessibility for: Attention; Social-emotional skills
10.3: Genetic Testing (10 minutes)
Activity
The mathematical purpose of this activity is to use a two-way table as a sample space to decide if events are independent and to estimate conditional probabilities. Listen for students mentioning the concept of conditional probability.
Launch
Arrange students in groups of two. Give students quiet time to work the questions, have partners compare answers, and then have a whole-class discussion.
Design Principle(s): Support sense-making
Student Facing
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.
Student Response
For access, consult one of our IM Certified Partners.
Student Facing
Are you ready for more?
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.
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
The purpose of this discussion is for students to describe how to use probabilities to describe dependent and independent events.
Here are some questions for discussion.
- “What is one way to determine whether 'having the disease' and 'having the genetic variation' are dependent or independent events?” (You find the conditional probability of having the disease given the condition of having the genetic variations, and then compare it to the probability of having the disease. If they are equal, they are independent.)
- “What is another way?” (You find the probability of having the disease and having the genetic variations, and compare that to the product of the probability of having the disease and the probability of having the genetic variation.)
- “What other information would it be helpful for the company to report?” (It would be helpful to know the overall risk of getting the disease in addition to knowing the risk of getting it if you have the genetic variation.)
Lesson Synthesis
Lesson Synthesis
The table shows the results of an experiment where participants were given the choice of listening to music or not listening to music while sitting quietly in a room for 5 minutes. Their heart rates were measured at the beginning and the end of the experiment.
Display the table for all to see.
| increase in heart rate | no increase in heart rate | |
|---|---|---|
| music | 30 | 18 |
| no music | 12 | 8 |
Here are some questions for discussion.
- “There are two events described in the table. What are they?” (The two events are “listening to music” and “increase in heart rate.”)
- “How would you figure out if the two events are independent events?” (I would calculate the probability of an increase in heart rate under the condition that music is being listened to. Then I would compare that to the probability of an increase in heart rate.)
- “Is there evidence that suggests the events are independent? Explain your reasoning.” (Yes there is evidence that suggests the events are independent. The events could be independent because the probability of an increase in heart rate under the condition that music is being listened to is \(\frac{30}{48} \approx 0.63\) and the probability that there is an increase in heart rate is \(\frac{42}{68} \approx 0.62\) are very close.
- "What does it mean for two events to be independent?" (Two events are independent if the probability for one event does not change whether the other event occurs or not.)
- "What are two ways to test for independence of events using probability?" (Two events are independent if \(P(\text{A and B}) = P(\text{A}) \boldcdot P(\text{B})\) or \(P(\text{A | B}) = P(\text{A})\).)
10.4: Cool-down - Depends on the Weather (5 minutes)
Cool-Down
For access, consult one of our IM Certified Partners.
Student Lesson Summary
Student Facing
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 \(0.72 \boldcdot 0.72 \approx 0.52\).