Probability - Gravner-67.c

Question

You and your opponent both roll a fair die. If you both roll the same number, the game is repeated, otherwise whoever rolls the larger number wins. Let $N$ be the number of times the two dice have to be rolled before the game is decided.

(c) Assume that you get paid 10 dollar for winning in the first round, 1 dollar for winning in any other round, and nothing otherwise.Compute your expected winnings .

Answer 1

E(A wins)=$\frac{5}{12}*10+(\frac{1}{6}*\frac{5}{12}+\frac{1}{6^{2}}*\frac{5}{12}....$ N-1 terms )*1

$\rightarrow \frac{5}{12}*10 +\frac{5}{12} (\frac{1}{6}+\frac{1}{6^{2}}...)$*1

$\rightarrow \frac{5}{12}*10 + \frac{5}{12}(\frac{1- (\frac{1}{6})^{n-1} }{1 - \frac{1}{6}})$*1

$\rightarrow \frac{25}{6}+\frac{1}{2} (1-(\frac{1}{6})^{n-1})$*1 is Expected  money A wins or player 1 wins