Bill and Ben takes turns flipping a coin. Whoever gets the head first is the winner.

If Bill has the first flip, what is the probability that he will win?

2/3.

The probability that he will get a head on his first toss is 1/2, if not, the probability he wins on his second toss is 1/2 + 1/8, etc. The sum of 1/2 + 1/8 + 1/32 + 1/128 = 2/3. QED