Simplex Volume: Probability the Sum of Uniforms is ≤ 1
The Problem
Let \( X_1, \dots, X_n \) be independent and identically distributed random variables from the Uniform(0,1) distribution.
We want to compute:
\[ \Pr\left(X_1 + X_2 + \cdots + X_n \le 1\right) \]
Geometric Interpretation
This is equivalent to asking for the volume of the region:
\[ { (x_1, \dots, x_n) \in [0,1]^n : x_1 + x_2 + \cdots + x_n \le 1 } \]
This region is the standard n-dimensional simplex within the unit cube.
Known Result
The volume of this simplex is:
\[ \boxed{\frac{1}{n!}} \]
This result can be derived via iterated integrals or combinatorics, and is a classical formula in probability and geometry.
Final Answer
\[ \boxed{\frac{1}{n!}} \]
This is both the probability that the sum of \( n \) independent Uniform(0,1) variables is at most 1, and the volume of the corresponding simplex.