Four friends, Emma, Finn, Grace, and Harry, play a game where each starts with four coins. The game consists of three rounds. In each round, four balls are placed in an urn---one green, one red, and two white. The friends 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 friends has four coins?