Coins and Colored Balls

Four players each start with $4$ coins. A bag contains one green ball, one red ball, and two white balls. In each round, the four players draw balls from the bag one at a time without replacement (all four balls are drawn), and the player who draws the green ball gives one coin to the player who draws the red ball; the balls are then returned for the next round. After four rounds, what is the probability that every player still has exactly $4$ coins?