TxO Math Bowl 2024 - Individuals A - Problem 8

Alice and Bob play a game with a fair die numbered $1$ to $6$. In this game, someone possesses the advantage, and this alternates between players each round. The only way for the game to end is if the player with the advantage wins the round. To win the round, the player must roll a number strictly higher than their opponent’s.

For example, if Alice has the advantage and she and Bob both roll a $5$, Bob now has the advantage in the next turn and Alice does not win. If, in the turn immediately after this, Bob rolls a $6$ and Alice rolls a $4$, Bob wins the game. If Alice starts with the advantage, let the probability that she wins the game be $\frac{m}{n}$, where $m$ and $n$ are relatively prime, positive integers. Compute $m + n$.