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“Let’s Go for a Spin!” : Understanding Some Important Probability Concepts through Fair Game Analysis Bill Mandella University of Wisconsin-Milwaukee Wisconsin Mathematics Council 41st Annual Conference Green Lake, WI May 6-8, 2009 The Milwaukee Mathematics Partnership (MMP) is supported by the National Science Foundation under Grant No. 0314898.

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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

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“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.

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A few examples of possible students’ spinners

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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

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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.

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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.

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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

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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).

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2 8 1 4 5 1 4 5 Joan’s Game Joan’s spinner Mary’s spinner Wins a point Joan Mary

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2 8 1 4 5 Mary’s Game Joan’s spinner: Probabilities Mary’s spinner: Probabilities What would the “average” spin be for: Joan’s spinner? Mary’s spinner?

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“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?

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Conclusion Could you use this in your own classroom? What changes might you make?

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“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 495350473467633504

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