Expected number of selected subset for a weighted random sampling

Question

We have a set of $\sum n_k$ items. The set includes $n_i$ items of $w_i$. And we will randomly choose $m$ items in the set according to the weights. Is there a formula for the expectation of number of chosen item of kind $i$?

For example, we have a set $\{A,A,B,B,B,B\}$, with each $A$ having weight 10 and $B$ having weight 1. If we pick 3 items among them considering their weights, what is the expected number of $A$ or $B$ picked?

Answer 1

Assuming

1. you sample without replacement, and
2. by "according to the weights" you mean that at the $j$-th draw, an item of weight $w$ is sampled with probability $\frac{w}{\sum_k n_k^{(j)} w_k}$, where $n_k^{(j)}$ is the number of items of weight $w_k$ at the moment the draw is made,

then your variable follows Wallenius' noncentral hypergeometric distribution, which is sadly a bit hard to analyze. As the Wikipedia page indicates, in the general case the mean can only be approximated.