TopsOJ Summer 2025 - Team Round - Problem 6

A gambler is playing a game of infinite linear roulette. A token is placed at 1 on the number line and the goal of the game is to move the token to 0.

Each turn, the gambler rolls a fair 48-sided disdyakis dodecahedral die, with 14 faces coloured red and 34 faces coloured black. If the die is red, the token moves 1 unit left. Otherwise, the token moves 1 unit right.

Let \frac{p}{q} denote the probability that the gambler ever wins the game, where p,q are coprime. Compute p+q.