PREMISE
Suppose we have an urn with $N$ balls, with $m$ white balls and $N-m$ black balls. At every iteration, a ball is chosen at random. If a white ball is chosen, we add an extra white ball and remove a black ball, so that there are always $N$ balls. If a black ball is chosen, then we simply put it back in the urn. The process terminates when $m=N$, since it can no longer continue with the same rules.
QUESTIONS
- What is the expected value of the number of white balls after $k$ iterations? Alternatively, how many iterations does it take on average to reach $m=N$?
- Is this a well known probability distribution? Maybe a variant of the hypergeometric distribution?
WHAT I KNOW
In a particular simulation, if we have $s$ successes out of $k$ iterations, then the probability of this event is $$Pr(S=s) = \frac{(m+s)!/(m-1)!}{N^k} \times Pr(k-s \ \text{failures})$$
always. The term in the expression representing the probability of successes is always the same/independent of the order of successes and failures.
The probability of having a failure at any iteration is only dependent on the number of previous successes; order matters here.