The probability that n random points in a disk are in convex position
Résumé
Pick n random points x(1), ... , x(n) uniformly and independently in a disk and consider their convex hull C. Let P-D(n,m) be the probability that exactly m points among the x(i)'s are on the boundary of the convex hull of {x(1), ... , x(n)} (so that P-D(n,n) is the probability that the x(i)'s are in a convex position).
In the paper, we provide a formula for P-D(n,m).