Sample minimum plus maximum exceeding one

Draw $n$ independent $\text{Unif}(0,1)$ values $X_1,\dots,X_n$, and let $U$ be the smallest and $V$ the largest of them. Find $\mathbb{P}[U+V>1]$.

$1/2$. Replacing every $X_i$ by $1-X_i$ leaves the joint law unchanged but turns $(U,V)$ into $(1-V,1-U)$, which maps the event $U+V>1$ onto $U+V<1$. The two events are equally likely and $U+V=1$ has probability zero, so each has probability $\tfrac12$.