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



Sequence of p draws without replacement with biased probabilities
This probability is always bounded from below by the probability with replacement, which is
$1-(1-w)^k$ where $w$ is the probability to pick the white marble in a single draw, and $k$ is the number of draws (changed from $P$ in your question which is rather unorthodox choice).


The probability of drawing the white marble at the $i$-th stage is bounded from above by $w/(1-\sum\_{j=0}^i w\_j)$, where $w\_0,\ldots$ are ordered by descending weight, and sum to 1. So the probability of drawing the white marble can be bounded from above by
$$1-\prod\_{i=0}^k(1-\frac{w}{1-\sum\_{j=0}^i w\_j})$$


In particular, we get the same asymptotic bound when $w\_0 << k^{-2}$.


