On average, how many trials does it take to get X successes given that successes increase the success rate? Specific Scenario:
What is the average number of trials it takes to reach 4 success in a scenario with a 9/20 success rate, where each success increases the success rate by 1/20?
 A: It should be intuitively clear that the time from the first success to the second success is independent of the time from the start to the first success, and similarly for all non-overlapping combinations of success start and ends.
This enables us to break the problem up into four independent sub-problems as follows:
$$\mathbb{E}[\text{time to 4th success}] = \mathbb{E}[\text{time to 1st success}] + \\ \mathbb{E}[\text{time from 1st to 2nd success}] + \\ \mathbb{E}[\text{time from 2nd to 3rd success}] + \\ \mathbb{E}[\text{time from 3rd to 4th success}]$$
Since each trial within the four sub-problems is independent of the other trials and has constant probability (within the sub-problem,) the distribution of the number of trials to reach the next success is Geometric. 
 Using basic properties of the Geometric distribution, the time to the first success given a probability $p$ of success is equal to $1/p$.
So we have:
$$\mathbb{E}[\text{time to 4th success}] = 20/9 + 20/10 + 20/11 + 20/12 = 7.70707...$$
