Let F be a face of a bounded convex polyhedron K in R^3. Assume K is solid with constant density. Then: ----- What is the probability p = p(F; K) that if K is randomly thrown onto a horizontal surface, it will land on the face F ? ----- My guess back in high school was that p = (1/4π) * (solid angle(F), subtended from the center of gravity of K). I recently saw for sale some novelty dice that are irregular hexahedra with planar faces, advertised as having equal probabilities for all faces. So I'm wondering if this is just an issue of that solid angle fraction, or whether there is more to it. Maybe this depends on how "randomly thrown" is defined? A priori it seems that this might be possible, say, if some faces would be unlikely to be landed on if our polyhedral die K were rolling on the horizontal surfaces with enough velocity (or end-over-end angular momentum). So the random throws might have different statistical outcomes if there are many with high energy. Or not? —Dan