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Index > Poker games and rules > Odds and probabilities

Why are ace-hi flushes ranked highest, when it's much harder to get a seven-hi flush? And similarly for two pairs?

Author: Michael Maurer, Giancarlo DiPierro
Last updated: 1998
Copyright © 2004 Michael Maurer, Giancarlo DiPierro
The official and up-to-date version of this answer is here.

[Michael Maurer's original answer:] Only the classes themselves (flush, straight, etc) are ranked by the probability of getting them in five cards. Within each class we use an arbitrary system to rank hands of the same type. For example, our arbitrary system ranks four aces higher than four deuces, even though the hands occur with the same frequency. Similarly, flushes are ranked by the highest card, with the next highest card breaking ties, and so on down to the fifth card. This has the curious effect of creating many more ace-hi flushes than any other kind, because any flush that contains an ace is "ace-hi", regardless of the other cards. Thus, although 490 of the 1277 flushes in each suit contain a seven, only four of them are seven-hi flushes: 76542, 76532, 76432, and 75432. The median flush turns out to be KJT42.

A similar situation occurs for two pair hands. There are twelve times as many ways to make two pair with aces being the high pair ("aces up") as there are to do it with threes as the high pair ("threes up"). While the aces can go with another other rank of pair, the threes must go with twos, or we would reverse the order and call them, for instance, "eights up". Note that it is fruitless to alter the relative rankings to try to account for this imbalance, since as soon as we do the cards will be reinterpreted to make the best hand under the new system. For example, if we decide to make "threes up" the best possible two pair hand, now all the hands like "eights and threes" will be interpreted as "threes and eights", and the population of "threes up" hands will soar twelve-fold. The median two pair hand turns out to be a tie between JJ552 and JJ44A.

[Giancarlo DiPierro suggests a fresh interpretation:] You've figured it out. Flushes are not correctly ranked according to their mathematical probability. The ranking of flushes and no-pair hands by the highest card (hence the term "high-card" for no-pair hands) that is commonly used around the world today is an arbitrary system that likely dates back to when someone first started betting on poker hands.

The correct way to rank these hands according to how hard they are be dealt becomes clear if you have ever played lowball or any high-low split game. In those games, low hands are ranked by the worst card, not the best card. Any 6-high low hand is ranked higher than any 7-high low hand because a 6-high is dealt three times less frequently than a 7-high. It doesn't matter if the lowest card in the 7-high hand is an ace and the lowest card in the 6-high hand is only a deuce, the 6-high wins.

Applying that principle to flushes and no-pair hands in high poker, a 9-low hand is dealt about three times less frequently than an 8-low and about seven times less frequently than a 7-low. So the 9-low should ranked higher, even if the 7-low contains an ace and the 9-low does not. In any situation where unpaired cards are determining the ranking of a hand, whether it is a flush, no-pair, or the side cards in hands with trips of equal rank, the worst card -- the lowest one -- should be used for the ranking.

This concept also applies to two pair hands -- the mathematically correct way of ranking them would be to use the value of the lower pair. Kings-under-aces is twice as rare as any queens-under hand, three times are rare as jacks-under, four times as rare as tens-under, and twelve times as rare as dueces-under -- the easiest two pair to make. The next time your queens-under-kings loses to a pair of aces that turns into aces-and-dueces on the river, you can feel justified that mathematically, at least, you had the better hand!