Gambler's Ruin and First Passage Time

Our first objective is to find the probability that our player wins the game, ending up with a+ b dollars. To do this, we have to imagine that the game has been going on for some time, and the player has reached the point of having exactly i dollars in his pocket, so that his opponent has a+ bi. Given that, we denote our player’s probability of winning the game by wi. If one more round is played, one can see that