Dice: Fair or Not Fair? That Is the Question

The issues that students face when rolling dice today are not new but similar to those faced by others throughout history. The ancient astragalus, which comes from the heel bone of a hoofed animal and was probably the earliest chance device, was definitely not fair in the sense that each of its four sides was not equally likely to occur topmost when tossed. This "fairness" was not important at the time, however, because it was believed that a god or fate determined the outcome, not chance based on the shape of the astragalus. Whatever the belief in the mechanism for determining outcomes, the outcome of winning has been an important concept since the advent of gambling in ancient times. The desire to win led to the early "loading" of dice. By the time of Cardano and Galileo, the idea of fairness was firmly entrenched, with Cardano qualifying his analysis of probabilities with "if the die be honest" (Bennett, 1998, p. 77) and Galileo describing a fair die as one with "six faces and when it is thrown it can equally well fall on any one of these" (p. 47). Using trials to test hypotheses about fairness was an idea known, at least theoretically, to Cicero and Cardano.

The issue of fairness is often dismissed in classrooms with questions such as "What is the chance any side will come up when this die is tossed?" Hearing a chorus of "1/6" from students, teachers are likely to move on to issues they consider to be more sophisticated and more likely to challenge their students, such as what happens when two dice are tossed and the outcomes summed, or the fairness of games whose rules are determined based on the outcomes of dice tosses. Even when trials are performed for a single die, the purpose is usually to verify the fairness of the die rather than put fairness to the test, and any empirical deviations from an even distribution are likely to be dismissed as random variation. Researchers have found, however, that the beliefs about dice that students bring with them to school include beliefs that God, fate, or mental powers determine dice outcomes and that such beliefs also include understandings about dice developed through experiences when playing games, such as the importance of rolling technique, experiences of losing games, and the difficulty of obtaining a 6 to start in a game. For many students, these intuitions about dice and probability are resistant to instruction and do not improve with age.