The probabilities are: 3/20 for the first player, 3/17 for the second, 3/14 for the third, and 3/11 for the last.
Since the chance of anyone winning in a round is 3/5 and they all have even chances of winning, they must each have 3/20 chance of winning in a round. The players who don’t go first, however, have a chance that they may not get to take their turns at all. So, the players who don’t go first actually have a better chance to win during their turns, provided that their turns come.
You divide the chance of winning by the chance of getting a turn, and you get the chance of winning during the turn.
2nd player: (3/20)/(17/20)=3/17
3rd player: (3/20)/(14/20)=3/14
4th player: (3/20)/(11/20)=3/11
