I was told a puzzle recently, and I can't figure out how to solve it. It went like this:
You are a prisoner. You play a game with the guard many times a day. This game has a unique probability $p$ for you to win, and it is the same every time you play. Each time you win, you "gain a life." Each time you lose, you "lose a life." You begin with 1 life. What does $p$ need to equal for you to stay alive for a long time (many years with you playing multiple times a day)?
I know the probability will have to be greater than 0.5, but how much greater than that does it need to be for you to be sure you'll live a long life?