65.31 Theoretical Probabilities for a Cuboidal Die

Two Definite Integrals Arising in Light Transmission Through a Crystal

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PROBLEMS AND SOLUTIONS

Budden [2] gives his experimental results on tossing a cuboidal die of dimensions a x b b and could not offer a theoretic derivation for the observed frequencies. In [3] Singmaster essentially repeats the above treatment by Kranzer, but his results, based on the probabilities being proportional to the solid angles subtended by a face from the centroid, differ markedly with Budden's experimental results. That the probabilities cannot be proportional to the solid angles for convex polyhedra follows from problems 66-12 and 66-13 [4].

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97.16 Probability analysis for rolls of a square cuboidal die

“…The author of that article cut 15 such dice out of a steel bar and had his students roll each of them N times to experimentally determine p. His results are listed in Table 1. A simple explanation was subsequently hypothesized in which p is proportional to the solid angle subtended by a square face [2], but it does not match the data well. In the present paper, an alternative simple model is proposed that better fits the measurements.…”

“…For example, another way to account for the reduced likelihood of the die ending up in an orientation with high centre of mass is to weight the probability by a Boltzmann exponential exp (-{JE), where E is the gravitational potential energy of the centre of mass of the die and {J is taken to be a fit parameter [5,6,7]. The value of the parameter n is found by fitting (2) to the data. The results are graphed in Figure 3 for two values of n that span the range of variation of both Budden's and Heilbronner's experimental points.…”

“…(Consequently even a slight shaving of one face of a die would significantly alter its fairness.) This same model could be applied to the related problem of the probability 1 -p for a tossed coin of thickness x and diameter y to land on edge [2,4,8,9] if one approximates a coin as behaving like the smallest cuboid that would enclose it. In particular, (2) predicts the probability is between 0.12% and 0.83% for a British one-pound coin (of 3.15 nun thickness and 22.5 nun diameter) if n is between 2.5 and 3.5; this range brackets the experimental observation of 6 edge landings out of 1000 tosses [8].…”

Discussion of Professor Kingman's Paper

“…This problem may be related to the following, seemingly simpler, problem posed by Budden (1980): What is the probability that a uniform x x y x y cuboidal die will land with a square (y x y) face uppermost? Theoretical probabilities, obtained by a method corresponding to the orthogonal projection used for the string, have been obtained by Singmaster (1981). Unfortunately they seem to bear little resemblance to the observed relative frequencies.…”