I've seen the Coupon collector's problem and I believe this is a variant of it but I can't quite wrap my head around it. This is not a homework assignment.
I have a set of k elements. I randomly sample s elements from the set and then replace them. If I do this n times, what is the probability that I will have sampled every single element at least once?
I don't yet have the solution in closed form, but I'll leave this here as it gives a computable expression for this probability, and perhaps can be used to find a more ideal solution.
Let $A_i$ be the event that the $i$th element of the $k$ is never sampled in the course of this process. We wish to compute $P((\bigcup A_i)^c) = 1 - P(\bigcup A_i)$. This quantity can be written, by the principle of inclusion exclusion, as $$ \sum_{i=0}^k (-1)^{i} \sum_{J\subset [1,\ldots,k], \lvert J\rvert = i} P(A_J), $$ where $A_J = \bigcap_{j\in J} A_j$. First we need to compute the $P(A_J)$. This is the probability that a fixed set $J$ of the elements is never sampled throughout the process. This happens in one trial with probability $\frac{\binom{ k-\lvert J\rvert}{s}}{\binom{k}{s} }$. The trials are independent, so the probability that it happens in all trials is the $n$th power of this. Substituting into the above expression, the result is given by $$ \frac{1}{\binom{k}{s}^n } \sum_{i=0}^k (-1)^{i} \sum_{J\subset [1,\ldots,k], \lvert J\rvert = i} \binom{ k-\lvert J\rvert}{s}^n $$ $$ = \frac{1}{\binom{k}{s}^n } \sum_{i=0}^k (-1)^{i} \binom{k}{i} \binom{ k-i}{s}^n $$
