### Chris and Jerel’s Initial Idea of Fairness in Ordinary Dice Games (Part 1 of 3)

PurposesLesson activity; Student collaboration; Reasoning
DescriptionThis analytic is the first of three analytics that showcase the formulation of two students’ ideas of mathematical fairness. The analytics follow the argumentation of two students, Chris and Jerel, throughout two after-school sessions and one interview session as they investigate what makes a game involving rolling one or two dice fair. The first after-school session occurred on April 29, 2004. The second after-school session occurred on May 5, 2004, with the interview happening directly after the second session on the same day. The first analytic begins with Chris and Jerel discussing the fairness of a game involving a single die during the first session. By the end of the first session, Chris has designed his own game involving rolling two dice. The second analytic looks at Chris and Jerel’s work during the second after-school session and interview session where the pair explore a task investigating the fairness of a game involving rolling two dice. This game is different from the one Chris creates in the first analytic. The third analytic includes an investigation on the probability of rolling a sum of 7 versus a sum of 6 when rolling two dice that occurs during the interview.
In this analytic (the first of three analytics) during the first after-school session Chris and Jerel, both sixth-grade students, are solving a problem about the fairness of a game using a single fair die. Gameplay involves rolling a single fair die: if a 1, 2, 3, or 4 is rolled Player A gets one point (and Player B gets 0) and if a 5 or 6 is rolled Player B gets one point (and Player A gets 0). The first player to accumulate ten points is the winner. The questions that students are asking about this game are: 1) Is this a fair game? Why or why not? 2) Play the game with a partner. Do the results of playing the game support your answer? 3) If you think the game is unfair, how could you change it so that it would be fair?
Chris and Jerel begin gameplay before making a prediction. They claim that the given game is unfair using the results of their playing the game as support for their conclusion. In this analytic the students provide two indicators of fairness. The first indicator is the number of possible outcomes for each player (in this case Player A has four outcomes and Player B has two outcomes). The second indicator is the score difference of the game. In their case, Player A won ten points to Player B’s two points. When asked to create their own fair game Chris and Jerel create a game where each player has three outcomes: Player A gets a point for a roll of 1, 2, or 3 and Player B gets a point for a roll of 4, 5, or 6.
After playing their fair game, Chris and Jerel play a new game invented by Chris involving two dice. In this game Player A gets a point for rolling a sum that is even and Player B gets a point for rolling a sum that is odd. They assert that this game is fair by the same two indicators, that there is an equal number of outcomes (Player A: 2, 4, 6, 8, 10 and 12, Player B: 1, 3, 5, 7, 9, 11) and that the score of the game is very close (Player B wins 10 points to Player A’s 9 points). In this game, Chris and Jerel exhibit the equiprobability bias, the belief that all outcomes have the same probability of occurring (Shay, 2008). At the end of the session, a researcher suggests to the students that it is not possible to roll a sum of 1 with two dice. The session concludes before the two students were able to explore this further. Chris and Jerel explore further the sample space of rolling two dice in Part 2 of this series.

One dice problem:
Roll one die. If the die lands on 1, 2, 3, or 4, Player A gets one point (and Player B gets 0). If the die lands on 5 or 6, Player B gets one points (and Player A gets 0). Continue rolling the die. The first player to get ten points is the winner.
(1) Is this a fair game? Why or why not?
(2) Play the game with a partner. Do the results of playing the game support your answer? Explain.
(3) If you think the game is unfair, how could you change it so that it would be fair?
Chris’ Game:
Roll two dice. If their sum is 2, 4, 6, 8, 10 or 12, Player A gets one point (and Player B gets 0). If the sum of the dice is 1, 3, 5, 7, 9, or 11, Player B gets on point (and Player A gets 0).

Video References:
B82, 42a, Probability problems: Dice games for two players part 1 of 2 (Student view), Grade 6, April 29, 2004, raw footage