A Simple Multiple Integral Solution to the Broken Stick Problem

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.