Suppose I have $3f+1$ balls, $f$ are black and $2f+1$ are white. First I choose a random ball, this is the first trial. I continue to draw $f+1$ balls, meaning I have at least one white ball.
My question is, how to calculate the following distribution $X$, such that $P(X = k)$ means what is the probability that the first $k-1$ balls are black, and the $k$-th ball is white.