A Simple Multiple Integral Solution to the Broken Stick 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 $n\ge 2,$ 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.

Fixed several typos