I want to find an average rate for a changing probability: the probability for A is 1/420 but the probability doubled after you reach 420 tries without getting A yet, if you get A the counter will reset and the chance will back to 1/420 and it will be doubled if you didn't get A in the next 420 tries (if you got A in try number 100 for example the counter will back to 0), first I tried to get how many tries I need to do for a 99% chance to get A, I tried several equation and I came up with this (419/420)^420 * (418/420)^(x-420) = 0.01
while 0.01 is the chance of not getting it in x tries, now I want calculate the average of how many times I get A per tries, the answer is indeed more than 1 time per 420 tries, but I have no idea how to calculate it, or if there is a law for questions like this.