Pardon my statistical illiteracy and feel free to correct any silly/wrong terminology I use in this post.
There is an RPG system called Storytelling where the vast majority of dice rolls are
Roll x 10 sided dice Re-roll any dice with a value z or higher A success is a die with a value of t or higher You need y successes
So one might call this equation P(x,z,y,t), where typically it would be P(x,y,10,8);
I believe that the default (P(x,y,10,8)) is exactly 1/3rd chance of success per die, but I'm not really sure where to go from there, and more importantly, it would be better to be able to express this accurately with numbers than having to resort to Monte-Carlo.
Oh, and for what it's worth (surely a good statistician will know this already) my guess is that to accurately handle the recursive re-rolling one must use a limit as re-roll approaches infinity.