“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.
• 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 examples
of 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
this many chosen at a time
max. number
In other words, the calculator is set to choose 2 numbers
at a time from the numbers 1 to 12 (inclusive).
Joan’s Game
Mary’s
spinner
Wins a point
Joan’s
spinner
1
Joan
2
4
Mary
5
Mary
1
Joan
4
Joan
5
Joan
8
Mary’s Game
Joan’s spinner:
Mary’s spinner:
Probabilities
Probabilities
2
1
4
8
5
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
MARY’s total
total points
points
4953
5047
34676 33504