What is the probability that $HH$ occurs before $TH$ in an infinte sequence of coin flips? This is one of the questions of a set of exam review questions that don't have solutions to them. I can't get my head around this but it seems so simple.

By flipping a fair coin repeatedly and independently, we obtain a sequence of
  H's and T's. We stop flipping the coin as soon as the sequence contains either HH or TH.
  Two players play a game, in which Player 1 wins if the last two symbols in the sequence
  are HH. Otherwise, the last two symbols in the sequence are TH, in which case Player 2
  wins.
A = "Player 1 wins"
  and
  B = "Player 2 wins."
Determine Pr(A) and Pr(B)


First flip is either heads or tails. If the second flip is heads we have a winner no matter what. If the second flip is tails we have no winner, but it follows that Player 2 must win. Why?
Flip three is either heads or tails. If it is heads, player 2 wins. Tails, no one wins. Flip four and each afterward either results in a heads and player 2 wins or a tails and no one wins. Given the last flip was tails, HH will never occur before TH.
Using this information, we have a $\frac{1}{2}$ chance of the game ending on the second flip. Assuming it ends on the second flip, each player wins $\frac{1}{2}$ of the time (HH or TH). If the game does not end on the second flip, Player 2 wins.
This must mean $\operatorname{Pr}(A) = \frac{1}{4}$ and $\operatorname{Pr}(B) = \frac{3}{4}$.
