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Thinking about dice: for all the Platonic solids, it's very easy to figure out the odds of a particular face landing face-up in a roll of the die.

If I have an arbitrary 6-sided solid, how do you determine the probability of a specific face landing face-up?

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I suppose the question is not too well defined (how exactly do you roll it? On a flat surface? Friction? Gravity? Air? etc etc) and will probably be more suited to physics.se after you do define it better. –  Aryabhata Jun 29 '11 at 22:49
    
I think it can be interpreted fairly reasonably as follows: choose a random uniform vector on the sphere defining the orientation of the solid, and assume it's moving downwards towards a flat surface. Which face would hit the surface? With that interpretation, I guess it's solvable for any given solid by projecting the faces outwards towards a sphere centered at the solid's center of mass, and measuring the areas of the resulting regions. Need to think about it more to be sure though... –  Alon Amit Jun 29 '11 at 22:58
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This question was discussed on MathOverflow. See http://mathoverflow.net/questions/46684/fair-but-irregular-polyhedral-dice

The question on MathOverflow mentions a paper by Persi Diaconis and Joseph Keller, titled "Fair Dice", in which Diaconis and Keller argue that, by continuity, there are fair dice that are not simply fair by symmetry. Knowing a bit about Diaconis's philosophical stance on the interpretation of probability (I believe he is very strongly Bayesian), I was surprised by this.

But if you look at the paper, it states in the first paragraph of Section 3 that the shape of the die which would make it fair will depend upon the mechanical properties of the die and the table. This seems to be the essence of Matt Fayers's answer in the MathOverflow page, which states in part that "the only possible notion of a fair die is an isohedral one, because for any other die, it depends how you throw it."

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