When the extremes sum above one

Let $X_1, \dots, X_n \sim \text{Unif}(0,1)$ be IID, with $U = \min_i X_i$ and $V = \max_i X_i$ . Compute $\mathbb{P}[U + V > 1]$ .

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Replace every $X_i$ by $1-X_i$ : same distribution, but min and max swap.
That shows $U+V$ is symmetric about $1$ , so $\mathbb{P}[U+V>1]=\tfrac12$ , for all $n$ .

Answer

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0.5 (± 0.001)

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