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

PurposesLesson activity; Student collaboration; Reasoning
DescriptionThis analytic is the third 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 third of three analytics), Chris and Jerel are sixth-grade students discussing a problem about fairness with a game using two fair dice during the interview with Researcher Powell. 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?
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. During the interview, Chris and Jerel show the sample space they developed. 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. This is significant because in the sample space Chris and Jerel have created 6, 7, and 8 all have a 3/21 (approximately 0.143) chance of occurring, while in the sample space of 36 outcomes 6 and 8 have a 5/36 (approximately 0.129) chance of occurring and 7 has a 6/36 (approximately 0.167) chance of occurring. Due to the incomplete sample space, Chris and Jerel are investigating why the sum of 7 occurs more frequently, leading to a question in the fairness of rolling a single die.
At the end of the second analytic. Chris and Jerel have expressed the idea that rolling a 7 occurs more frequently than rolling a 6 even though, according to their notes, each number can be the result of exactly three outcomes. The boys claim that this is because the sums of 7 have more “large numbers” referring to rolling a 4, 5, or 6 on one of the two dice. During the third analytic Chris and Jerel carry out two trials to investigate the claim that when rolling a single die the “large numbers” (4, 5, and 6) occur more frequently than the “small numbers” (1, 2, and 3). The first trial results in more small numbers being rolled than large numbers, while the second trial results in more large numbers being rolled than small numbers. Combining the results of both trials reveals that 12 small numbers were rolled while 10 large numbers were rolled. This evidence causes Chris and Jerel to tentatively reject their claim that large numbers are more likely to occur than small numbers.
After viewing the analytic, it is worth reflecting on what might be a follow up task to challenge Chris and Jerel to find ALL possible outcomes.