Range beats the midpoint

HardProbability Random Variables~8m

Let $X_1, X_2 \sim \text{Exponential}(1)$ be independent. The range is $|X_1 - X_2|$ and the midpoint is $\tfrac{X_1 + X_2}{2}$ . What is the probability that the range exceeds the midpoint?

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Assume $X_1>X_2$ : the condition becomes $X_1>3X_2$ . Add the mirror case.
$\mathbb{P}(X_1>3X_2)=\int_0^\infty e^{-y}e^{-3y}\,dy=\tfrac14$ ; double it.

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Range beats the midpoint

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