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 $w_0 << k^{-2}$.