Range versus midpoint of two exponentials

Draw two independent $\text{Exp}(1)$ random variables $X_1,X_2$. What is the probability that their range, $|X_1-X_2|$, exceeds their average $\tfrac{X_1+X_2}{2}$?

$\tfrac12$. Writing $m$ and $M$ for the smaller and larger value, the condition $M-m>\tfrac{m+M}{2}$ simplifies to $M>3m$, i.e. $M-m>2m$. By memorylessness $m\sim\text{Exp}(2)$ and $M-m\sim\text{Exp}(1)$ independently, so the probability is $\mathbb{E}[e^{-2m}]=\tfrac{2}{2+2}=\tfrac12$.