Alice, Bob, and Carol play a game in which each of them starts with three coins. The game consists of three rounds. In each round, three balls are placed in an urn---one green, one red, and one white. The players each draw a ball at random without replacement. Whoever gets the green ball gives one coin to whoever gets the red ball. What is the probability that, at the end of the third round, each of the players has three coins?
