I find those two puzzles to be rife with ambiguities. First, a “force” is not going affect the attitude; only a torque will do that, and as another poster has noted, there’s a bit of a difference between “random force” and “random attitude”. There’s also the question of just what “random” means. The word “random” just means that there is some probability distribution; it does nothing to tell us what it is. Finally, there’s the issue of starting orientation. If the needles start out vertical, then the most likely result is that they will end up vertical.
I know—all of us over-precise folk had the same frustrations. But if you allow the problem to be “the things are in a (uniformly) random configuration (‘attitude’ - nice word choice!) and haven’t bounced or even started in some known orientation”, it’s a fun problem to think about. A uniform orientation seems appropriate since it’s maximum entropy given the word-problem constraints.
I find those two puzzles to be rife with ambiguities. First, a “force” is not going affect the attitude; only a torque will do that, and as another poster has noted, there’s a bit of a difference between “random force” and “random attitude”. There’s also the question of just what “random” means. The word “random” just means that there is some probability distribution; it does nothing to tell us what it is. Finally, there’s the issue of starting orientation. If the needles start out vertical, then the most likely result is that they will end up vertical.
I know—all of us over-precise folk had the same frustrations. But if you allow the problem to be “the things are in a (uniformly) random configuration (‘attitude’ - nice word choice!) and haven’t bounced or even started in some known orientation”, it’s a fun problem to think about. A uniform orientation seems appropriate since it’s maximum entropy given the word-problem constraints.
http://math.stackexchange.com/questions/87230/picking-random-points-in-the-volume-of-sphere-with-uniform-probability