Session 8:
Homework

Problem H1

Use the table of sums for the sum of two dice to determine the more likely winner of this game of chance:

 • Player A wins when the sum of the two dice is an even number. • Player B wins when the sum of the two dice is an odd number.

Problem H2

Consider another game in which two players roll a pair of dice and look at the magnitude of the difference of the outcomes:

 • Player A wins when the difference is 0, 1, or 2. • Player B wins when the difference is 3, 4, or 5.
 a. Determine whether this game is fair by considering the possible outcomes for two dice. You may want to use the outcomes generated in Part B. b. If the game is fair, come up with a similar game that seems fair but is not. If the game is unfair, change the game in some way to make it fair.

Problem H3

Here is another game that two players can play, which is similar to the game Rock, Paper, Scissors.

Two players each hold between one and three fingers behind their backs, then hold out their hands at the same time:

 • Player A wins if the sum of the number of fingers is even. • Player B wins if the sum of the number of fingers is odd.

Suppose that each player selects randomly among the three choices. Determine whether this game is fair by constructing the possible outcomes (there are a total of nine).

Problem H4

Player B realizes that the game is unfair and changes strategies: Now Player B will always choose two fingers.

 a. If Player A does not change strategies and still picks all three choices with equal probability, who is more likely to win? b. Is this realistic? What is likely to happen if Player B continues with this strategy?

 Problem H5 Is there a strategy that Player B can use to make the game fair, regardless of what Player A tries to do?

Think about the way players win the game: with an "odd" sum or an "even" sum. With each player picking all three choices with equal probability, why aren't "odd" and "even" sums equally likely to occur?   Close Tip

Kader, Gary and Perry, Mike (February, 1998). Push Penny -- What Is Your Expected Score? Mathematics Teaching in the Middle School, 3 (5), 370-377.