All dice are unfair because they cannot be manufactured with absolute precision. However, some dice are more unfair than others. Each year hundreds of millions of dice are sold worldwide. Dice commonly used in role playing games are 4-sided (D4), 6-sided (D6), 8-sided (D8), 10-sided (D10), 12-sided (D12), and 20-sided (D20). Most of these are manufactured using plastic mold injection and rock tumbler methods. This method can result in dimensional inaccuracies in the dice and sometimes density inhomogeneities. In 3000-roll tests of eleven D20 dice only three tested fair. In a running chi square test it was shown that for an unfair die the chi square statistic varies linearly with the number of rolls. The chi square statistic for an unfair die will always trend linearly to infinity while there is zero probability that a fair die will trend to infinity. The fair die chi square statistic will oscillate around the number of sides minus 1 (the degrees of freedom). An expression for the slope of the running chi square statistic was derived for unfair dice for both the situation of dimensional inhomogeneities and density inhomogeneities. Using simulations of billions of rolls of unfair dice, a probability distribution for the number of rolls beyond which the chi square statistic stays above the 95 percent critical value was obtained and is fitted well with a 2-parameter gamma distribution.
C. Warren Campbell
Campbell, C. Warren and Dolan, William P., "Dice Mythbusters" (2019). Student Research Conference Select Presentations. Paper 48.