Last of sixteen uniforms smallest, first largest

Let $X_1,\dots,X_{20}$ be independent $\text{Unif}(0,1)$. Given that $X_1$ is the largest of $X_1,\dots,X_{16}$ and that $X_{17}<X_{18}$, what is the probability that $X_{16}$ is the smallest of $X_1,\dots,X_{16}$?

$\tfrac1{15}$. The event $X_{17}<X_{18}$ involves different variables and is independent of how the first 16 are ordered, so drop it. Given $X_1$ is the maximum, the remaining 15 variables are in uniformly random relative order, so $X_{16}$ is the minimum with probability $\tfrac1{15}$.