I thought of an interesting probability problem but I'm not sure how to approach it. My knowledge of statistics is not terrible, but this is complex enough that I'm not even sure how to start.
Any insight would be appreciated (such as if this is similar to a well-known problem) or how to start on solving this.
Say Bob really wants to prove to someone that he is a top player in Video Game. In order to do so, his gamer profile needs to have a win rate greater than or equal to X% with at least Y games played. That is to say, if W is the number of wins on Bob's profile and L is the number of losses, Bob wants W/(W+L) ≥ X% and W+L ≥ Y.
For example, assume X% = 80% and Y=50.
At any point between games, Bob can choose to reset his profile, setting both W and L to 0 (Bob has no knowledge of the outcome of future games when choosing to do this). Say Bob's true chance of winning any given game is P%, and he knows this number. Bob wants to strategically reset his profile such that he minimizes the expected number of games he needs to play to attain his goal.
What is Bob's optimal strategy for when to reset his profile? How many games will he be expected to play before achieving his goal, and how many times will he be expected to reset his profile?
This problem is a bit more complicated than just finding the first occurrence of a sequence of 50 games with a win percentage higher than 80% because that sequence might have started out Win-Loss-Loss, for example, and Bob might have chosen to reset his profile after those two consecutive losses without knowing that he would achieve many wins in the next 47 games.
I'm not even sure what the optimal resetting strategy is for Bob. Clearly you're supposed to reset if your profile is 0-1, since that's strictly worse than being 0-0. But if Bob's record is currently at 39-11, for example, is it better to continue onwards in hopes that his next five games will be wins (thereby pushing his record up to 80% win rate with only five additional games played), or reset and be guaranteed to play at least 50 more games?
Does this problem even have a nice solution?