A Simpler Expression for $K$th Nearest Neighbor Coincidence Probabilities

Abstract

Given $N$ points distributed at random on $\lbrack 0, 1)$, let $p_n$ be the size of the smallest interval that contains $n$ points. Previous work finds $\operatorname{Pr}(p_n \leqq p)$ for all $n, N$, and rational $p$. The present note derives a new and considerably simplified formula for $\operatorname{Pr}(p_n \leqq p)$ for all $n, N$, and $p$.