Probability

Lesson 7

Math

Unit 8

10th Grade

Lesson 7 of 10

Objective


Calculate and analyze relative frequencies in two-way tables to make statements about the data and determine independence. 

Common Core Standards


Core Standards

  • S.CP.A.4 — 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.
  • S.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.
  • S.ID.B.5 — Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

Foundational Standards

  • 8.SP.A.4

Criteria for Success


  1. Calculate relative frequencies in a two-way table.
  2. Assess whether particular variables are independent or dependent from a two-way table.
  3. Calculate conditional probabilities in a two-way table.
  4. Use multiple representations (two-way frequency tables, tree diagrams, Venn diagrams, etc.) to show the sample space of a set of variables.
  5. Determine an acceptable range of values that would be deemed “equal” and thus determine variables independent.

Tips for Teachers


Students have analyzed two-way tables in both Algebra I and the eighth grade. This lesson introduces the idea of calculating relative frequencies and determining independence from two-way tables. 

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Anchor Problems


Problem 1

Each student in a random sample of seniors at a local high school participated in a survey. These students were asked to indicate their gender and their eye color. The following table summarizes the results of the survey.

Eye Color

  Brown Blue Green Total
Male 50 40 20 110
Female 40 40 10 90
Total 90 80 30 200


Based on this data, are “blue eyes” and “male” independent variables? Explain your reasoning.

Guiding Questions

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References

Illustrative Mathematics Two-Way Tables and Probability

Two-Way Tables and Probability, accessed on June 15, 2017, 9:04 a.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Modified by Fishtank Learning, Inc.

Problem 2

On April 15, 1912, the Titanic struck an iceberg and rapidly sank with only 710 of her 2,204 passengers and crew surviving. Data on the survival of passengers are summarized in the table below.

  Survived Did not survive Total
First class passengers 201 123 324
Second class passengers 118 166 284
Third class passengers 181 528 709
Total passengers 500 817 1317
  1. Choose two variables that you think are NOT independent. Test your assumption.
  2. Choose two variables that you think ARE independent. Test your assumption.
  3. What do you notice in part (a) and part (b)?

Guiding Questions

Create a free account or sign in to access the Guiding Questions for this Anchor Problem.

References

Illustrative Mathematics The Titanic 2

The Titanic 2, accessed on June 15, 2017, 9:10 a.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Modified by Fishtank Learning, Inc.

Target Task


Problem 1

A state nonprofit organization wanted to encourage its members to consider New York as a vacation destination. They are investigating whether their online ad campaign influenced its members to plan a vacation in New York within the next year. Out of 1,000 people, they found the following.

  Plan to vacation in New York within the next year Do not plan to vacation in New York within the next year Total
Watched the online ad 300 450 750
Did not watch the online ad 100 150 250
Total 400 600 1,000

Are the events “a randomly selected person watched the online ad” and “a randomly selected person plans to vacation in New York within the next year” independent or not independent? Justify your answer using probabilities calculated from information in the table.

References

EngageNY Mathematics Algebra II > Module 4 > Topic A > Lesson 3Exit Ticket, Question #1

Algebra II > Module 4 > Topic A > Lesson 3 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.
EngageNY Mathematics Algebra II > Module 4 > Topic A > Lesson 4Exit Ticket, Question #2

Algebra II > Module 4 > Topic A > Lesson 4 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

Problem 2

 A survey conducted at a local high school indicated that 30% of students have a job during the school year. If having a job and being in the eleventh grade are not independent, what do you know about the probability that a randomly selected student who is in the eleventh grade would have a job? Justify your answer. 

References

EngageNY Mathematics Algebra II > Module 4 > Topic A > Lesson 4Exit Ticket, Question #2

Algebra II > Module 4 > Topic A > Lesson 4 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Additional Practice


The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

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Lesson 6

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Lesson 8

Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Conditional Probability and the Rules of Probability

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