Number of points on the interior of the convex hull of a random subset
Computing the probability that a point is inside a random convex-hull is quite interesting, and got some attention recently (under a slightly different sampling model). The exact probability can be computed (but not too quickly). Good estimates follow by looking on the Tukey depth of the points of interest. See the following recent paper, and references therein.

http://sarielhp.org/p/14/prob_ch/