Following the notation from WP:hypergeometric distribution.
We have an urn with $K$ red and $N-K$ white balls. When we find a red ball, we keep it, removing it from the population. But when we find a white ball, we throw it back. Given a certain number of trials ($n$), how many red balls / successes ($k$) are we likely to find?
What's this distribution called? We can find at most $\min(K, n)$ red balls but are allowed to continue sampling forever.
I can find the probability mass function (in exponential time) and simulate draws from it quite easily but couldn't come up with a closed form.