This is a problem from Barry Leung’s Math Games.

“Larry and Julius are playing a game, taking turns throwing a ball at a bottle sitting on a ledge.  Larry throws first. The winner is the first person to knock the bottle off the ledge.  At each turn the probability that a player knocks the bottle off the ledge is ½, independently of what has happened before.  What is the probability that Larry wins the game?”

Answer.

See Knock the Bottle Problem for a solution.