Probability - Gravner-67.b

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.

(b) Compute Probability you win

Answer 1

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

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

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

$\rightarrow \frac{1}{2} (1-(\frac{1}{6})^{n})$ is probability that A wins or player 1 wins