: An arbitrary hand can have two aces but a flush hand can't. The average number of aces that appear in flush hands is the same as the average number of aces in arbitrary hands, but the aces are spread out more evenly for the flush hands, so set #3 contains a higher fraction of flushes. Aces of spades, on the other hand, are spread out the same way over possible hands as they are over flush hands, since there is only one of them in the deck. Whether or not a hand is flush is based solely on a comparison between different cards in the hand, so looking at just one card is necessarily uninformative. So the other sets contain the same fraction of flushes as the set of all possible hands.