How Many Rounds Will I Last?
In my favorite game, I start with 3 lives. In each round, I have a 10% chance of dying, losing a life. If I don't die, I am given an extra life, but I cannot ever have more than 3 lives.
What is the expected value for the number of rounds I will play before losing the game (when I lose my final life)?
Note: Each round is counted, even if you lose a life in the round.