This was presented as a poster at the 6th Experimental Chaos Conference, July 22-26, 2001, Potsdam, Germany.

Experimentally obtained statistics of dice rolls

Dean Christie, Ryan Glasheen, Chris Hamilton, Masahiro Imoto, Philip Matthews, James Moffat, Thalat Monajemi, Daniel B. Murray, John Nelson and Arne Sturm
Department of Physics and Astronomy, Okanagan University College
Kelowna, British Columbia, Canada

ABSTRACT: The first ever extensive experimental investigations of dice rolls have been carried out and are reported here for the first time. An automatic mechanism repeatedly rolls a die, photoraphs it, counts the number of dots and records the outcome. In this way, one dice roll can be made every 2.4 seconds. It is then easy to study the fairness or degree of bias of dice by rolling them many times. The dice freely fall at least 10 cm and bounce on a level plexiglas surface. We found that high quality dice of the kind used in Nevada gambling casinos showed no evidence of bias in a study involving 640934 rolls. Non-cubic dice were also studied. It was found that the non-cubic dice were biased in proportion to the amount by which they deviated from perfect cubicity. Dice that have the distance between the 1 and 6 face shortened by 3% have 1's and 6's come up 6% more often than one-sixth of the time. Dice shortened by 1.5% to 13% were studied with up to 36,000 rolls each.

BIAS

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.

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 "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

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.

Table 1: Rolls of used casino dice
Dice
ID code
Table
thickness
1 2 3 4 5 6 Total
             
aag 6 mm 7896 8065 8108 8137 7951 7922 48079
abd 6 mm 7084 7022 7112 7071 7097 7166 42552
abu 6 mm 13559 13714 13577 13386 13521 13536 81293
aem 12 mm 21642 21393 21500 21860 21603 21336 129334
aem 3 mm 25013 26007 25490 25702 25591 25267 153070
aes 3 mm 20324 19861 19996 19905 20432 20010 120528
aes 12 mm 10858 10999 11067 11139 10973 11042 66078
Total  106376 107061 106850 107200 107168 106279 640934

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:

1 2895
2 4214
3 3389
4 3347
5 4383
6 3315

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 non-zero 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.

Application: Loading Dice

Problem: A dice cheater wants to make dice that land with the 1 up 5% less often than normally. The usual probability is 1/6 = 0.1667, and 5% less is 0.1583. Assume normal 19.1 mm cubic dice with density 1.3 g/cm3 are used.

Solution: The desired bias is b = 0.05. Referring to table 2, bias / loading = 3.37. So the loading required is L = 0.0148. The mass of the unloaded die is 9.0 grams. So a loading mass of 0.13 grams must be added to the "6" face.

Normal casino dice dots are 5.5 mm in diameter. If a small round platinum (21.4 g/cm3) disc is hidden behind each dot, then each disc must be 0.04 mm thick.
Application: Dice Manufacturing

Suppose a company is ordering dice to be made. Cubic dice of 19.1 mm × 19.1 mm × 19.1 mm are ordered (traditional 3/4 inch cubes). It is desired that all six faces have the same probability of 1/6 to within 0.1%, that is p = 0.16667 ± 0.00017. What tolerance in the dimensions is allowable?

For simplicity, assume that two of the three dimensions are exactly 19.10 mm. The bias allowed is b = 0.001. Referring to figure 1, bias/flatness = 1.91. Therefore, the allowable flatness is f = 0.00053. So the dimensions must be accurate within 0.010 mm. (1/2500 of an inch; 0.4 mils)


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