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Imagine a physical biased six-side die. How much and what kind of bias we could possibly introduce by moving it's center of mass? What would be the exact mechanism describing the relation of center of mass and probabilities of landing on each side? How can this be generalized to other dice, e.g. four-side die?

Tim
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    When I was a graduate student at Stanford one of our professor's was Persi Diaconis (famous mathematician / statistician / magician). He asked me and another graduate (Michael Cohen) to roll several dice a large number of times. These dice were shaved and there was a bias to be detected by estimating the probabilities for each side. To our eyes the dice looked like cubes. Exactly how he introduce the bias I don't know. It may have been a combination of shaving along with an modification to the center of mass. – Michael R. Chernick Dec 15 '16 at 08:40
  • I didn't want to take the discussion of the other question into cha. But I do wnt to thank you for pointing out that a pyramid (tetrahedron) has 4 sides. I hadn't thought of that. – Michael R. Chernick Dec 15 '16 at 08:51
  • @whuber I was wondering if posting it here or there, but maybe it fits there better. –  Dec 15 '16 at 16:46
  • The probability of landing on each side follows a categorical distribution. A Dirichlet distribution might be appropriate for the distribution over the biases that would be generated by machine that creates dice. – Neil G Dec 15 '16 at 18:05
  • @NeilG you're right, I already started thinking of unknown probabilities and prior for it. Edited to make it more clear. – Tim Dec 16 '16 at 07:54
  • Change the weight reparation of the die. if center of gravity is closer to one face than to the center, this face will have higher probability to end up at the bottom.

    After that it is a problem of rigid body mechanics, and it will depend on many parameters.

     – user2346536 Dec 16 '16 at 08:09
  • Once you have a mechanical solution with plenty of parameters , giving bounds and assuming uniform probability within these bounds, you may want to compute the probability of each side. Or maybe study it as a chaos theoretic problem given that the sensitivity to initial (launch) conditions is very high. – user2346536 Dec 16 '16 at 08:17
  • If a dice is biased, I doubt its probability distribution stay independent of the surface and how it is rolled. If it is independent, it would be an interesting results by itself. – stochastic Jan 03 '18 at 22:35

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