Abstract
Dice are cursed or blessed; that is, they roll low or high, but they are never fair. They cannot be manufactured with uniform density and geometric precision. This is particularly true of 20-sided dice or D20s. Faces are smaller than 6-sided dice, and manufacturing tolerances are similar. However, some dice are fairer than others. In our studies of plastic-mold dice about 1 in 4 test fair in 3000 rolls. We have used different statistical tests, including chi-square, modified Kolmogorov Smirnov, and double binomial tests. Of these, the method that consistently performed better is the chi-square goodness of fit test. The probability distribution of the χ2 statistic for a fair die only asymptotically approaches the chi-square distribution. The exact distribution is the multinomial distribution that is computationally intensive for many rolls of dice, especially those with high numbers of faces. The χ2 statistic for unfair dice asymptotically approaches the noncentral chi-square distribution. Both of these distributions are continuous, but the multinomial distribution is a discrete distribution. Furthermore, χ2 for dice is a rational number. Ninety-five percent values for the chi-square distribution do not match those from the exact distribution. For up to 100 rolls, they are significantly different. For χ2, the minimum interval between possible values is 2 divided by the expected value for a fair die. Some exact distributions of fair and unfair dice are presented with exact values of the power of the test.
Disciplines
Civil and Environmental Engineering | Manufacturing | Physical Sciences and Mathematics | Statistical Models | Statistics and Probability
Recommended Citation
Campbell, Warren and Miller, Cameron, "Dice Are Blessed or Cursed" (2024). SEAS Faculty Publications. Paper 21.
https://digitalcommons.wku.edu/seas_faculty_pubs/21
Included in
Civil and Environmental Engineering Commons, Manufacturing Commons, Statistical Models Commons