Lesson 10

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.

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.

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.

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).

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.

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. 

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.)

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.

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.

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.)