Fair (i.e. symmetric) n-sided dice have equal probabilities of landing on all n sides. Label the sides j = 1, 2, ... n. Call p(j) the probability of landing on side j. For convenience, we defined the "bias" of side j to be b(j) = n p(j) - 1. Fair dice have zero bias on all sides.
Example: If a six sided die was so heavily loaded (asymmetrically weighted) that the six came up half the time (instead of 1/6 of the time) we would say that b(6) = 2.
Findings: In this study, we found that high quality casino dice of the kind used in gambling casinos have no detectable bias. However, low quality dice, loaded dice and non-cubic shaped dice have non-zero bias.
A common method for dice cheating us to use dice that are not perfect cubes. Consider dice which have six rectangular faces. Let d1:6, d2:5 and d3:4 be the lengths of the sides of the dice.
Dice for which d1:6 < d2:5 = d3:4 are called "1-6 flats" by gamblers. On such dice the "1" and "6" come up more often.
Definition: It is convenient to define the "flatness", f1:6 of 1-6 flat dice as
f1:6 = (d2:5 + d3:4 - 2 d1:6)/(d2:5 + d3:4)
Example: If a gambling cheat takes a 20.00 mm cubic die and saws 2 mm off the "1" face, then d1:6 is now 18 mm, so that the resulting die has a flatness of f1:6 = 0.1
Loaded dice are made by adding mass so as to move the center of mass away from the geometric center. Dice can be face loaded, edge loaded or corner loaded. We looked only at face loading.
Drilling a hole in the "2" face brings the center of mass closer to the "5" face. We define loading, L(5) so that L(5) = 0 for fair dice and L(5) = 1 for the limiting case where the center of mass is shifted all the way to the "5" face.
Let CM(5) denote the location of the center of mass, using the geometric center of the die as the origin. Let d denote the length of the three identical sides. Then the loading is defined as:
L(5) = 2 CM(5) / d
(...after 640934 rolls)|
High Quality Dice Are Fair
The high quality dice used by gambling casinos are typically only used for 8 hours and then taken out of play. Those discarded dice are sold in game supply stores. Those dice were always found to have the same dimensions on all three axes to the limit of a micrometer caliper (0.01 mm). These dice look like new, even on close inspection.
A number of these dice were randomly selected and rolled many times to check for fairness. All rolls were made on acrylic sheet (plexiglass) tables of size approximately 30 cm by 30 cm, held up at the corners. The surface on which the dice landed was 10 cm by 10 cm.
For information contact:
Dan Murray, Associate Professor, Mathematics, Statistics and Physics Unit, University of British Columbia Okanagan
Email: daniel "dot" murray "at" ubc "dot" ca