“Let's Go for a Spin!”: Understanding Some Important Probability ...
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Transcript of “Let's Go for a Spin!”: Understanding Some Important Probability ...
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“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.
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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 examplesof 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. numbermax. 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
Joan
Joan
Joan
Mary
Mary
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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?
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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
4953 5047 34676 33504