The question:

Tom and Bob play a game by each tossing a fair coin. The game consists of tossing the two coins together, until for the first time either two heads appear when Tom wins, or two tails appear when Bob wins.

a) Show that the probability that Tom wins at or before the $\displaystyle n^{th}$ toss is $\displaystyle \frac{1}{2} - \frac{1}{2^{n + 1}}$

b) Show that the probability that the game is decided at or before the $\displaystyle n^{th}$ toss is $\displaystyle 1 - \frac{1}{2^{n}}$

My attempt:

I'm quite confused as to how to attempt this. I tried drawing a tree to get my head around it, but I'm sure I'm missing something obvious here. Any assistance would be most appreciated.