On an attempt to solve this problem I've managed to reduce it to finding the expected number of white balls picked until one black ball is observed (let's call that value $v$). Except that, unlike the geometric distribution, this needs to be done without replacement. Hypergeometric distribution doesn't come to the rescue as the number of black balls picked is immaterial (and of course the white balls must be picked consecutively). So I am looking for a way to compute $v$, or at least its underlying distribution. I've tried to search online but couldn't find a distribution that readily fits this.
So the problem is:
Consider an urn having total of $n$ balls, $w$ of them are white, and the rest is black. Pick $j$ balls without replacement. What is the expected number of white balls that will be picked before a black ball is observed ? Notice that the total number of balls picked, $j$, is fixed.
My guess is that cumulative hypergeometric distribution could be used, while dividing somewhere by the number of ways the black balls intersperse the white ones (after the first black ball is observed). But I have little intuition how to go through with this. I.e. to divide before taking the cumulative distribution or after?
Based on a question in the comments below, to be clear: the desired value is the expected number of white balls observed until either a black ball is observed, or $j$ picks happened, whichever comes earlier.