After 1 success of a series of trials, the probability of success increases from 1/2 to 21/36. How do I calculate the expected number of trials until I reach 5 successes?
Naively, it seems to be a geometric distribution and a negative binomial distribution: I expect the first 2 trials to give 1 success, and I expect 7 trials to give just over 4 successes afterwards, so 9 trials leads to 5 successes.
To get a better answer, it seems I should calculate the probability of getting the first success on the n-th trial, then multiply by a binomial distribution of some sort. However, there is a (very small) chance of still not getting a single success in n+1 trials, and 4 more successes in arbitrarily many trials, and I don't see how to integrate across the two infinities.