### Chris and Jerel’s Initial Idea of Fairness in Ordinary Dice Games (Part 2 of 3)

PurposesLesson activity; Student collaboration; Reasoning
DescriptionThis analytic is the second 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, the second of the three analytics, sixth-grade students Chris and Jerel are solving a problem about fairness with a game using two fair dice. Gameplay involves rolling two dice and assigning points to either Player A or Player B based on the sum of the rolled dice. Player A gets one point (and Player B gets 0) for sums of 2, 3, 4, 10, 11 and 12. Player B gets one point (and Player A gets 0) for sums of 5, 6, 7, 8 and 9. 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’s initial response to this game is to highlight that Player A has three “small” numbers (2, 3, and 4) and three “large” numbers (10, 11, and 12) while Player B has all “large” numbers (5, 6, 7, 8, and 9). This suggests that they are not relying on the number of outcomes assigned to each player (Player A having 6 assigned outcomes to Player B’s 5 assigned outcomes) as the measure of fairness for the game. They are considering the numbers assigned to each player (Player A’s 2, 3, 4, 10, 11, and 12 compared to Player B’s 5, 6, 7, 8, and 9) as important to the fairness of the game, demonstrating they are no longer exhibiting an equiprobability bias in the outcome of rolling two dice (as seen in Part 1). Instead, the pair begins to see that each outcome does not have the same probability of occurring.
During gameplay, Shay (2008) explains that researchers had encouraged students to record the outcome of each roll of the dice. This naturally led to students beginning to list out the sample space for the game. We see in the second event, Chris and Jerel begin writing a list of sums for each outcome (i.e. 5=4+1, 5=2+3). During the interview, Chris and Jerel state they started writing out their sample space to figure out why Player B was winning. It is important to note that Chris and Jerel have a sample space of 21 outcomes rather than 36. Shay (2008) attributes this to students not considering symmetric pairs as separate events (for example, rolling a 3 and 4 versus rolling a 4 and 3). In the complete sample space of 36 outcomes, Player A has a 12/36 (approximately 0.333) chance of winning a single event, while Player B has a 24/36 (approximately 0.667) chance of winning a single event. With their sample space of 21 outcomes, Chris and Jerel conclude that Player B has 13 possible outcomes while Player A has 8 possible outcomes. This would mean Player B has a 13/21 (approximately 0.619) chance of winning a single event while Player A has an 8/21 (approximately 0.381) chance of winning a single event. This explains why the game is unfair in favor of Player B.
Throughout the discussion of the sample space for the game, Chris and Jerel mention that 7 came up the most when they were playing. Researcher Powell asks the pair why 7 comes up more than 6 when Chris and Jerel’s sample space shows an equal number of possibilities for rolling a sum of 6, 7, or 8. Chris suggests that a sum of 7 is rolled more frequently than 6 because 7 is comprised of sums of what he calls “big numbers.” These “big numbers” refers to rolling a 4, 5, or 6 on an individual die, while “small numbers” refers to rolling a 1, 2, or 3 on an individual die. This big and small number theory is explored in Part 3.

Dice game 2:
Roll two dice. If their sum is 2, 3, 4, 10, 11, or 12, player A gets one point (and player B gets 0). If their sum is 5, 6, 7, 8, or 9, player B gets one point (and player A gets 0). Continue rolling the dice. The first person 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 could be fair?
[Note: The game favors Player B with a ⅔ probability of winning a point and a probability of approximately .935 of winning a game.]

Videos Referenced:
Title: B89, Probability strand: Dice games with two players (Student View), grade 6, May 05, 2004, raw footage.