After some heavy reordering and canceling of factorials, I discovered that the following experiment is approximately equivalent for $m \ll n < N$ if conducted with or without replacement:
In $n$ turns, draw marbles (with/without replacement) from an urn containing $m$ white and $N-m$ black marbles. Count the white marbles.
Now, the "with replacement" part is a Bernoulli trial:
Throw $m$ times an unfair coin with success probability $n/N$. Count the number of successes.
Are there any pitfalls in this approximation?
I am pretty sure that this is textbook knowledge. Can you recommend a good book where this approximation is derived?
I did the exercise in order to Derive househould weights from a uniformly distributed person sample.