First marked item in a random permutation ($\mathrm{Pr}(K=k)=\left(\genfrac{}{}{0ex}{}{N-k}{m-1}\right)/\left(\genfrac{}{}{0ex}{}{N}{m}\right)$)

When $m\ge 1$ items among $N$ are marked, inspect a uniformly random permutation until the first marked item. For its first position to be $k$, the other $m-1$ marked positions must lie among the last $N-k$ slots. This yields the displayed law with support $1\le k\le N-m+1$. It is the first-success form of sampling without replacement. For one marked item the position is uniform, while its conditional success chance after $r$ failures increases to $1/(N-r)$.