The problem: In this game, if two dices give: \begin{cases} 7 \text{ or } 3, \text{ then the player wins} \\ 2,\space 11 \text{ or } 12, \text{ then the player loses} \\ \text{else, then the game continues with the following rule} \end{cases} New rule for tie, continue throwing the dices until the one of the following condition is met. \begin{cases} \text{the number from first round}, \text{ then the player wins} \\ 7, \text{ then the player loses} \\ \end{cases} Find the probability of winning.
Attempt: I thought the most straight forward way of doing this problem is find the conditional probability of each case after the tie. This gives $\frac {244}{495}$ as the probability of winning.
Question: The problem is that I know my professor loves using trick to solve problems. Thus there must be a faster way to do this. It actually took about a page to work this out.