Consider $n$ IID random variables $X_1, \ldots, X_n \sim U(0,1)$. What is the probability that $\max(X_1, \ldots, X_n) - \min(X_1, \ldots, X_n) \leq 0.5$.
Denote $Z_1, Z_n$ as the min and max respectively. Then by symmetry, I believe $E[Z_1] = 1 - E[Z_n]$. I am unsure how to find $P(Z_n - Z_1 \leq 0.5)$. I think I can find the distribution for $P(Z_n), P(Z_1)$ individually, how how do I go about finding the distribution of the difference between the 2?