Probability that \(n\) random points are in convex position
Suppose that \(n\) points are chosen independently and uniformly from a parallelogram. The paper shows that the probability that they are the vertices of a convex polygon is \(\left( \left( \begin{smallmatrix} 2n - 2\\ n - 2\end{smallmatrix} \right)/n!\right)^ 2\). The proof proceeds by reduction to the unit square and a combinatorial treatment of approximating grid versions. A note added in proof states the following probability for the analogous problem in a triangle: \(2^ n(3n - 3)!/((n - 1)!)^ 3(2n)!\) (proof to be given elsewhere).
combinatorial treatment of approximating grid versions
scientific article; zbMATH DE number 764252