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 $\tfrac{1}{2}$, independently of what has happened before. What is the probability that Larry wins the game?
Because Larray throws first, he can only win at an odd turn. The probability for him to knock the bottle off at the $k$'s turn while the bottle was not off in all the previous turns are:
Hence the total probability equals $$\frac{1}{2} + \frac{1}{2^3}+\frac{1}{2^5}+\cdots=\boxed{\frac{2}{3}}$$