# Where am I going wrong in interpreting this problem as a gambler's ruin problem?

I was trying to solve this problem (Strategic Practice Week 3, Homework problem 4 in Harvard's Stat 110 class), by framing it as a gambler's ruin problem:

Calvin and Hobbes play a match consisting of a series of games, where Calvin has probability $p$ of winning each game (independently) and $q = 1-p$. They play with a “win by two” rule: the first player to win two games more than his opponent wins the match. Find the probability that Calvin wins the match (in terms of $p$), by interpreting the problem as a gambler's ruin problem.

Here's how I approached the problem:

Let,

$W:$ Calvin wins the match;

$D_i:$ (Wins by Calvin) $-$ (Wins by Hobbes) $= i$