A Simple Multiple Integral Solution to the Broken Stick Problem

TL;DR

This paper derives a simple multiple integral formula to compute the probability that randomly partitioned segments of a stick can form a polygon, providing a clear solution to the classical Broken Stick problem.

Contribution

It introduces a straightforward multiple integral approach to solve the Broken Stick problem, offering a concise derivation of the probability formula.

Findings

Probability that segments form a polygon is 1 - (n+1)/2^n.

Method uses multiple integration for a closed-form solution.

Provides a new perspective on a classical geometric probability problem.

Abstract

Regard the closed interval $[0,1]$ as a stick. Partition $[0,1]$ into $n+1$ different intervals $I_1, \ \dots \ , I_{n+1},$ where $n \geq 2,$ which represent smaller sticks. The classical Broken Stick problem asks to find the probability that the lengths of these smaller sticks can be the side lengths of a polygon with $n+1$ sides. We will show that this probability is $1-\frac{n+1}{2^{n}}$ by using multiple integration.