tossing the two coins together, until for the first time either two heads appear ....

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.

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

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