# Probabilities of Non-Regular Dice

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?

-
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