  For a probability distribution $P$ on an at most countable alphabet $\mathcal
A$, this article gives finite sample bounds for the expected occupancy counts
$\mathbb E K_{n,r}$ and probabilities $\mathbb E M_{n,r}$. Both upper and lower
bounds are given in terms of the counting function $\nu$ of $P$. Special
attention is given to the case where $\nu$ is bounded by a regularly varying
function. In this case, it is shown that our general results lead to an
optimal-rate control of the expected occupancy counts and probabilities with
explicit constants. Our results are also put in perspective with Turing's
formula and recent concentration bounds to deduce bounds in probability. At the
end of the paper, we discuss an extension of the occupancy problem to arbitrary
distributions in a metric space.
