On the probability of forming polygons from a broken stick

TL;DR

This paper derives an explicit formula for the probability that randomly broken stick segments can form a polygon, generalizing previous work and applicable to various geometric probability problems.

Contribution

It provides a new explicit formula for polygon formation probability from broken sticks, extending prior results and demonstrating broader applicability.

Findings

Derived an explicit formula for polygon formation probability

Generalized previous specific cases to arbitrary segment counts

Applicable to broader geometric probability problems

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.