What is the algorithm called for the increase in how many times you will roll successfully as you decrease the chances for failure

Question

Let's say you have a 20 sided dice and every side is considered bad, but each time you roll a bad side it will no longer be bad for future rolls. I am looking for the mathmatical term for the scale of increase on the expected amount of times you will be able to roll success as bad sides are removed

Answer 1

You are looking at the distribution of the count for rolls of a twenty sided dice until some number appears a second time.   Who knows what this is called, but anyway:

Let $X$ denote this random variable.   The event $\{X=k\}$ is that of showing some permutation of $(k-1)$ distinct numbers, followed by a repetition of one of these.   There can be between at least $2$ and at most $21$ rolls before some number is encountered a second time.   Thus:

$$\begin{align}\mathsf P(X=k) ~=~& \left(\frac {{^{20}{\rm P}_{k-1}}}{20^{k-1}}\right)\dfrac{(k-1)}{20}~\mathbf 1_{k\in\{2,..,21\}} \\[1ex] =~& \dfrac{20!~(k-1)}{(21-k)!~20^k}~\mathbf 1_{k\in\{2,..,21\}} \\[3ex] \mathsf E(X)~=~&\dfrac{4027894135040576041}{640000000000000000} ~~\approx~~6.{\small 2\tiny 9}\end{align}$$