Abstract
The manufacture of dice for tabletop games is a billion-dollar industry. In gaming circles and online forums, the concept of “cursed dice” is a popular topic. Dice are inherently unfair due to the difficulty of manufacturing them with perfect geometric tolerances and uniform material densities. A common method for testing dice fairness is the chi-square statistic, which is typically assumed to follow the chi-square distribution. However, this assumption is only asymptotically valid.
Exact distributions of the statistic can be computed for dice with few sides and a limited number of rolls—such as 2-sided (D2) and 4-sided (D4) dice—but the computational cost increases rapidly with the number of sides. For example, calculating the exact distribution for a D20 rolled 3000 times is infeasible with current technology. Nevertheless, a lower bound on the deviation between the exact 95th percentile of the statistic and the chi-square approximation can be easily computed for any number of rolls. This bound depends on the number of rolls and only slowly converges to the chi-square distribution as the number of rolls increases.
Disciplines
Civil and Environmental Engineering | Engineering | Mathematics | Physical Sciences and Mathematics | Probability | Statistics and Probability
Recommended Citation
Campbell, Warren, "Dice Math and Probability" (2025). SEAS Faculty Publications. Paper 23.
https://digitalcommons.wku.edu/seas_faculty_pubs/23