“Let’s Go for a Spin!”:Understanding Some Important Probability

Concepts through Fair Game Analysis

Bill MandellaUniversity of Wisconsin-Milwaukee

Wisconsin Mathematics Council41st Annual Conference

Green Lake, WIMay 6-8, 2009

The Milwaukee Mathematics Partnership (MMP) is supported by the National Science Foundation under Grant No. 0314898.

• In this presentation, we will explore several probability topics such as:

– “Fair Game” Analysis– Simulations

• Using physical objects and graphing calculators

– Tree Diagrams– Experimental Probability vs. Theoretical

Probability– Equally Likely Outcomes– “Law of Large Numbers”– Expected Value

“Two Spinners Game”• To play the games, you would like your students to

create spinners using these guidelines:

– The spinner must be divided into 2, 3, or 4 regions.

– The spinner can be divided into equal regions, but it doesn’t have to be.

– Each region will be numbered using the numbers 0 through 9, with no number used more than once per spinner.

– The relative size of each number on the spinner must be inversely related to the size of its region.

– The sum of the regions must be 10.

• Students will be paired together and asked to create a “fair game” which uses the spinners they made.

A few examplesof possible students’

spinners

Joan and Mary• Two students, Joan and Mary, are paired up to play.

However, Joan and Mary each have their own idea about what a fair game would be using their two spinners.

Joan’s spinner Mary’s spinner

Joan’s idea for a fair game:

• Each person spins their own spinner.

• Whoever’s spin results in the larger number wins 1 point.

• The player with the most points after 20 spins wins the game.

Mary’s idea for a fair game:

• Each person spins their own spinner.

• Each player gets as many points as the result of his/her spin.

• The player with the most points after 20 spins wins the game.

Analyzing the fairness of Joan and Mary’s games

• Compare EXPERIMENTAL probabilities– Generate data from playing each game

– Simulations• Spinners• Graphing Calculators

ProbSim—“Spin Spinners” “randInt” —generate random numbers

• Compare THEORETICAL probabilities – Build TREE diagrams of outcomes for each game

Simulations• Graphing calculators (TI-84 plus)

– Select “MATH”

– Scroll to right and select “PRB”

– Scroll down and select “5:randInt ( ”– randInt (1, 12, 2)

min. number

max. number

this many chosen at a time

In other words, the calculator is set to choose 2 numbers at a time from the numbers 1 to 12 (inclusive).

2

8

1

4

5

1

4

5

Joan’s Game

Joan’s spinner

Mary’s spinner

Wins a point

Joan

Joan

Joan

Joan

Mary

Mary

2

8

1

4

5

Mary’s GameJoan’s spinner:

ProbabilitiesMary’s spinner:

Probabilities

What would the “average” spin be for:

• Joan’s spinner?

•Mary’s spinner?

“Challenge round”• Is it possible to change the numbers on Joan

and Mary’s spinners so that Mary’s game is fair?

• Can you create two new spinners such that both Joan and Mary’s games would be fair?

Conclusion• Could you use this in your own classroom?

• What changes might you make?

“10,000 spins”(EXCEL simulation)

Number of spins Joan

won

(Joan’s game)

Number of spins Mary

won

(Joan’s game)

JOAN’s total points

MARY’s total points

4953 5047 34676 33504