1. Until someone wins, isn't the probability on any turn 4/12 that that person will win (because the black balls are replaced)? If so, then the probability anyone will win is the probability that he'll win on that turn and that everyone has lost before.
2. A brute force way to figure out that probability is to consider the probability that you'll have drawn 2 cards from the same suit, or 3 cards, or 4 cards and subtract from the # ways of drawing 4 cards in general.