Hi

I have a problem which i find hard to modelize.

Suppose i have an urn with $N$ marbles. Among these marbles, one is white and all the other ones are black. I draw $P$ marbles without replacement. If the probability of drawing one marble is uniform, then the hypergeometric distribution tells me that the probability $P_W$ of having the white marble among the $P$ marbles is:

$P_W=\frac{\binom{1}{1} \binom{N-1}{P-1}}{\binom{N}{P}}=\frac{P}{N}$

That was the easy case.

Now suppose we have different weights for each marble. One marble $i$ is assigned a weight $w_i$, and the probability of drawing the marble $i$ with one and only one draw is $p_i=\frac{w_i}{\sum_i w_i}$

Now let's go back to ou $P$ draws. What is the probability $P_W$ according to $P$,$N$,$w_i$.

All the ideas are welcome even with limit cases such as $P << N$

Thanks