Problems

Calamity Jane sits down to play the game of craps with Buffalo Bill. In this game she rolls two fair dice. If, on the first throw, the sum of the dice is \(2,3\) or \(12\) she loses, while if it is \(7\) or \(11\) she wins. Otherwise Calamity continues to roll the dice until either the first sum is repeated, in which case she wins, or the sum is \(7\), in which case she loses. Find the probability that she wins on the first throw. Given that she throws more than once, show that the probability that she wins on the \(n\)th throw is \[ \frac{1}{48}\left(\frac{3}{4}\right)^{n-2}+\frac{1}{27}\left(\frac{13}{18}\right)^{n-2}+\frac{25}{432}\left(\frac{25}{36}\right)^{n-2}. \] Given that she throws more than \(m\) times, where \(m>1,\) what is the probability that she wins on the \(n\)th throw?