In this paper we give a formula for the probability that $n$ random points chosen under the uniform distribution in a disk are in convex position. While close, the formula is recursive and is totally explicit only for the first values of $n$.
Probability that $n$ points are in convex position in a regular $κ$-gon : Asymptotic results
Probability that $n$ points are in convex position in a general convex polygon: Asymptotic results
Random Gaussian Tetrahedra
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