suppose we have $k$ points placed uniformly at random in the unit cube in $\mathbb{R}^n$. what is the probability that their convex hull has all of the $k$ points as extreme points?

[if it would be easier, "unit cube'' can be replaced by "unit ball".]

*this question was posed on yahoo.answers [of all places] a while ago. it seemed a bit out of place there. hopefully this site is a better match.*

(the problem i quoted is what was originally posted on yahoo.)– ronaf Sep 3 '10 at 16:57