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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.

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    $\begingroup$ Your first example us the hypergeometric distribution, while the second one is the binomial distribution. This approximation is quite common and works well for large N. $\endgroup$
    – MånsT
    Apr 10, 2012 at 14:01

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Partially answered in comments:

Your first example us the hypergeometric distribution, while the second one is the binomial distribution. This approximation is quite common and works well for large N. – MånsT See also Binomial Approximation to Hypergeometric Probability

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