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Bill Mandella University of Wisconsin-Milwaukee Wisconsin Mathematics Council

“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.

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Bill Mandella University of Wisconsin-Milwaukee Wisconsin Mathematics Council

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  1. “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.

  2. 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

  3. “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.

  4. A few examplesof possible students’ spinners

  5. 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

  6. 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.

  7. 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.

  8. 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

  9. Simulations • Graphing calculators (TI-84 plus) • Select “MATH” • Scroll to right and select “PRB” • Scroll down and select “5:randInt ( ” • randInt (1, 12, 2) this many chosen at a time min. number max. number In other words, the calculator is set to choose 2 numbers at a time from the numbers 1 to 12 (inclusive).

  10. 1 1 2 4 4 5 5 8 Joan’s Game Mary’s spinner Wins a point Joan Joan’s spinner Mary Mary Joan Joan Joan

  11. 2 1 4 8 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?

  12. “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?

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

  14. “10,000 spins”(EXCEL simulation)

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