options: Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.

Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).

Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.

Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.
problem_activity: 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.
Speaking, Representing: MLR8 Discussion Supports.
Use this routine to support small group discussion. After students have completed a large portion of the task prompts, provide students with sentence frames to support their conversation comparing their answers and determining if they are correct. For example, "I agree because . . .”, "I disagree because . . .”, “Why do you think . . . ?” This will help students understand how to use probabilities to describe dependent and independent events.
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      |                 |                           |
| does not have the genetic variation |                 |                           |
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
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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
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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.)


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