A bag contains four red balls, three blue balls, and three green balls. Jim plays a game in which he bets $1 to draw a ball from the bag. If he draws a red ball, he wins $1; otherwise he loses $1....

A bag contains four red balls, three blue balls, and three green balls. Jim plays a game in which he bets $1 to draw a ball from the bag. If he draws a red ball, he wins $1; otherwise he loses $1. Assume that the balls are drawn with replacement and that Jim starts with $50 with the hope of reaching $100 at which point he stops playing. However, if he loses all his money before this, the game also ends and he becomes bankrupt. What is the probability that the game ends with Jim being $50 richer than he was at the beginning?