Topics
12. Probability
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United States of America
Grades 9-12

12.05 Analyzing probability strategies (M)

Lesson
Practice

Introduction

This lesson utilizes the modeling cycle with probability contexts to analyze fairness and create strategies for fair and unfair outcomes. We have used the modeling process with other topics in Geometry, including in lessons  5.06 Modeling with congruent triangles  ,  8.08 Modeling with triangles  , and  9.06 Geometric modeling  . When creating a geometric model, we will need to:

A modeling cycle. Starting with the phrase Identify the problem inside a circle. An arrow pointing to the right where the phrase Create a model is inside a rectangle. Next is an arrow pointing downward where the phrase Apply and analyze is inside a rectangle. Then, an arrow pointing to the right where the phrase Interpret results is inside a rectangle. Next is an arrow pointing upward where the phrase Verify the model is inside a triangle. Then, an arrow pointing to the right where the phrase Report findings is inside a circle. There is an arrow pointing to the left from the phrase Verify the model to Create a model.

  1. Identify the essential features of the problem

  2. Create a model using a diagram, graph, table, equation or expression, or statistical representation

  3. Analyze and use the model to find solutions

  4. Interpret the results in the context of the problem

  5. Verify that the model works as intended and improve the model as needed

  6. Report on our findings and the reasoning behind them

Analyzing probability strategies

The goal of analyzing probability strategies includes using probabilities to make fair decisions and analyzing decisions through probability. This involves creating a model for probability scenarios and analyzing the model to determine fairness.

Throughout the unit, the various types of models we've worked with to visualize probability outcomes are as follows:

  • Venn diagrams

  • Two-way frequency tables

  • Tree diagrams

  • Lists

  • Organizational tables

Examples

Example 1

Consider the three games that follow:

  1. Game A: Two players each toss one coin. If two heads turn up, the first player wins. If a head and a tail turn up, the second player wins. If two tails turn up, you play again.

  2. Game B: Two players each roll a six-sided die. If the sum of the numbers is odd, the first player gets 1 point. If the sum is even, the second player gets 1 point.

  3. Game C: Two players each roll a six-sided die. If the product of the numbers is odd, the first player gets 1 point. If the product is even, the second player gets 1 point.

a

For each game that is described above, list the sample space and determine the probability of the listed outcomes.

Worked Solution
Apply the idea

Game A: S_A= \{HH, HT, TH, TT \}

  • First player wins: P \left(HH \right)=\frac{1}{4}

  • Second player wins: P \left(HT, TH \right)=\frac{2}{4}= \frac{1}{2}

  • Play again: P \left(TT \right)=\frac{1}{4}

Game B: S_B:

123456
1234567
2345678
3456789
45678910
567891011
6789101112

  • First player gets 1 point: P \left( \text{odd sum} \right) = \frac{18}{36} = \frac{1}{2}

  • Second player gets 1 point:

    P \left( \text{even sum} \right) = \frac{18}{36} = \frac{1}{2}

Game C: S_C:

123456
1123456
224681012
3369121518
44812162024
551015202530
661218243036

  • First player gets 1 point: P \left( \text{odd product} \right) = \frac{9}{36} = \frac{1}{4}

  • Second player gets 1 point:

    P \left( \text{even product} \right) = \frac{27}{36} = \frac{3}{4}
Reflect and check

For this modeling problem, listing the sample spaces and determining probabilities of the listed outcomes is a part of identifying the essential features of the problem. Listing sets or using a table models the possible outcomes.

b

They say that a game is a fair game if both players are equally likely to win. For each game in part (a), determine whether it is a fair game and then explain why or why not.

Worked Solution
Apply the idea

For game A, the second player has a higher probability of winning the game than the first player. Since both players are not equally likely to win, the game is not fair.

For game B, both players have the same probability of earning the same number of points in the game. Since both players are equally likely to win, the game is fair.

For game C, the second player has a higher probability of earning a point. Since both players are not equally likely to win, the game is not fair.

Reflect and check

Using the model to find probabilities of specific events, we can then interpret the fairness of the game based on the probabilities calculated in part (a).

c

If you determine that any of the games described is not a fair game, how could you change the rules to make it a fair game? Describe your new game and explain how you know the game is fair.

Worked Solution
Apply the idea

For game A, the second player had a higher likelihood of winning the game. If we instead change the game rules to the following, the players will have an equal chance of winning the game:

  • Two players each toss one coin. If two heads turn up, the first player wins. If two tails turn up, the second player wins. If a head and a tail turn up, you play again.

With these new rules, each player has a 25 \% probability of winning.

For game C, the second player had a higher likelihood of winning the game. If we instead change the game rules to the following, the players will have an equal chance of winning the game:

  • Two players each roll a six-sided die. If the product of the numbers is odd, the first player gets 1 point. If the product is a multiple of 5 that is greater than 5, the second player gets 1 point.

With these new rules, each player has a 25 \% probability of winning.

Reflect and check

We're changing the assumptions of the problem situation first. Then, we may need to revise our model, or our interpretation of the model. In this case, the sample space of outcomes remained the same, so our list and tables did not need to change. But, we interpreted the model differently because we were analyzing different probability outcomes to match the new rules.

We may need to try out various assumptions and revise our model until we find a set of rules that makes the game fair. Then we're reporting on our solution, which includes justifying our reasoning.

Idea summary

Models help us organize our ideas and mathematical calculations when working with probability. We can use models to make decisions about the fairness of certain strategies.

Outcomes

S.MD.B.6 (+)

Use probabilities to make fair decisions.

S.MD.B.7 (+)

Analyze decisions and strategies using probability concepts.

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