BIAS Fair (i.e. symmetric) nsided 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 noncubic shaped dice have nonzero bias. 
FLATNESS 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 "16 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 16 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 
LOADING 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.
The statistical variations in the bottom row totals are not statistically significant, (15% likelihood of random occurence) so there is no evidence of bias. With this many rolls, a bias of as little as 0.006 of one face occuring for all the dice would have been statistically resolvable (less than 2% of random occurence). Conclusion: All six faces have a probability of one in six, p = 0.1667 to within plus or minus 0.0010. 
Low Quality Dice A variety of dice were purchased from toy stores and variety stores in Kelowna, Canada. One group of dice came in a package priced at US $0.10 each. One was selected at random and rolled 21543 times. The table was covered with 21 ounce mali cloth (wool felt) as on a casino craps table. The results were as follows: 

Figure 1 shows a plot of bias versus flatness.
Zero flatness corresponds to cubic dice, and zero
bias is the result. The slope of the linear fit
is 1.91. Error bars show one standard deviation. Casino dice are high quality plastic cubes made to high precision. When measured with a micrometer, all three axes always had dimensions that agreed to within 0.01 mm. The dice used to make this graph were produced by using a milling machine to mill off part of the "1" face of casino dice. Lower quality dice are not perfect cubes. This graph can be used to determine what tolerances are acceptable in the manufacture of dice, according to the amount of bias which is tolerable. 
Figure 2 shows bias versus loading, L. Cubic
dice are used. L=0 means the center of mass is
at the center. L=1 means the center of mass
touches the "5" face. Loading is accomplished
by drilling a 9.5 mm diameter hole in the center
of the "2" face. Undrilled casino dice had zero
bias. The slope of this graph is 3.37. Error
bars show one standard deviation. Lower quality dice could have nonzero loading due to poor design, such as hollow pips. This graph can be used to determine what tolerances in the location of the center of mass are acceptable. For this graph, the dice were dropped onto a 10 cm by 10 cm table made of 12 mm thick acrylic plastic. 

