Unit 8, Lesson 10
Using Probability to Determine Whether Events Are Independent
Integrated Math 2
10.1
Warm-up
Standards Alignment
Building On
Addressing
HSS-CP.A.5
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.
Building Toward
Instructional Routines
Which Three Go Together?Materials
NoneActivity Narrative
This Warm-up prompts students to compare four pairs of events. 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.
Activity
NoneA 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
Student Response
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Building on Student Thinking
Activity Synthesis
Invite each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Because there is no single correct answer to the question of which three go together, attend to students’ explanations, and ensure that the reasons given are correct.
During the discussion, prompt students to explain the meaning of any probability terminology they use, such as “independent events,” “dependent events,” or “conditional probability,” and to clarify their reasoning as needed. Consider asking:
- “How do you know _____?”
- “What do you mean by _____?”
- “Can you say that in another way?”
10.2
Activity
Materials
NoneActivity Narrative
In this activity, students use data in a table to estimate conditional probabilities and decide if events are independent or not. They begin by examining the data in the table to construct their own mathematical questions, then persevere in solving the question of whether two events are independent or dependent (MP1).
Making spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5). Students must also pay attention to precision as they decide whether two values are close enough to say that the events are dependent or not (MP6).
This activity uses the Co-Craft Questions math language routine to advance reading and writing as students make sense of a context and practice generating mathematical questions.
Launch
Arrange students in groups of 2. Introduce the context of hockey and overtime. Use Co-Craft Questions to orient students to the context and to elicit possible mathematical questions.
- Display only the table, without revealing the questions. Give students 1–2 minutes to write a list of mathematical questions that could be asked about the situation before comparing questions with a partner.
- Invite several partners to share one question with the class. Record the questions for all to see. Ask the class to make comparisons among the shared questions and their own. Ask, “What do these questions have in common? How are they different?” Listen for and amplify language related to the learning goal, such as “independent events” or “dependent events.”
- Reveal the question “Does this hockey team perform differently in games that go into overtime (or shootout) compared to games that don’t?” and give students 1–2 minutes to compare it to their own question and those of their classmates. Invite students to identify similarities and differences by asking:
- “Which of your questions is most similar to or different from the one provided? Why?”
- “Is there a main mathematical concept that is present in both your questions and the one provided? If so, describe it.”
Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. For example, tell students to complete the first question before checking in with students to provide feedback and encouragement, then proceed with the remaining questions.
Supports accessibility for: Attention, Social-Emotional Functioning
Supports accessibility for: Attention, Social-Emotional Functioning
10.3
Activity
Instructional Routines
MLR8: Discussion SupportsMaterials
NoneActivity Narrative
In this activity, students are given another context and table of data to practice deciding whether events are independent or dependent.
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.
MLR8 Discussion Supports. Display sentence frames to support small-group discussion. Examples:
- “I agree because _____.”
- “I disagree because _____.”
- “Why do you think _____?”
Activity
NoneLesson 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 at 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 and the probability that there is an increase in heart rate is , so they 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 or .)
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 .