I do this experiment: I flip fair coin, if it comes heads on first toss I win. If it comes tails, I flip it two times more and if both heads I win. Else, I flip it 3 more times, if it comes heads all three I win. On the n'th turn I flip it n times, and if I get n heads I win and quit the game.
Before I start, I calculate probability I win at some point: 1/2+1/2^2+1/2^3... = 1. So I am guaranteed to win this game always.
However I started to play, and it came tails first toss, now I calculate probability I win 1/2^2+1/2^3+1/2^4...=1/2.
Before I start I was guaranteed to win, but now only 50% chance I will win? What went wrong with the mathematical reasoning?