On Sep 21, 2012, at 12:32 AM, meekerdb <meekerdb@verizon.net> wrote:
On 9/20/2012 9:02 PM, Dan Asimov wrote:
Back in H.S. I wondered about this for any polyhedron, and guessed a face's probability of landing down is proportional to the solid angle it subtends from the center of gravity.
And whether the projection of the CG onto the plane of the face falls within the face. :-)
Brent That's close to my physical model. It came up in connection with estimating the probability of a coin landing on its side. I had my students design a cylindrical dice that has equal odds for heads, tails and "side". But because this was a physics course we did not make the ad hoc assumption that the rotation group is sampled uniformly in a toss. Instead we used the coin tossing model where the cylinder axis rotates rapidly about a horizontal axis. Non-uniform sampling is especially valid for symmetric objects, where the three moments of inertia are equal and there is no free precession. So I don't find Michael Kleber's "fairness by symmetry" very compelling. I bet if one "rolled" a cubic dice more like the standard coin toss one could strongly bias the outcome (against the faces intersected by the rotation axis).
-Veit