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