Suppose we have ordered numbers $a_1 < a_2 < \dots < a_n$. Now we sample $k$ of them with equal probability and with replacement and compute the median of these and call it $m$. (Let us assume that $k$ is odd for the sake of simplicity.) What is the probability the $m = a_i$ for each $i$?

I want to know this to construct the probability mass function for the median of $k$ random draws from a sample of $n$ numbers.

(Already asked here but does not seem to be simple.)