On the probability of forming polygons from a broken stick

Abstract

Break a stick at random at n−1 points to obtain n pieces. We give an explicit formula for the probability that every choice of k segments from this broken stick can form a k-gon, generalizing similar work. The method we use can be applied to other geometric probability problems involving broken sticks, which are part of a long-standing class of recreational probability problems with several applications to real-world models.