Unit 8, Lesson 10
Using Probability to Determine Whether Events Are Independent
Geometry
10.1
Warm-up
5 mins
Which Three Go Together: Events
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
None
Activity 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
NoneStudent Task Statement
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
Student Response
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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
15 mins
Overtime Wins
10.3
Activity
10 mins
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 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
- "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
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
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
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