The probability that $n$ random points in a disk are in convex position

Abstract

Pick $n$ random points $x_{1},\dots,x_{n}$ 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 $\{x_{1},\ldots,x_{n}\}$ (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 $P_{D}^{n,m}$.