Musings on Hand Rankings

Brian Alspach

Poker Digest, Vol. 5, No. 3, January 27 - February 7, 2002

I had some recent correspondence with a player about hand rankings for seven-card stud. As a departure point for this article, let me give you a table with the numbers of different kinds of hands.

Hand Number
Straight Flush 41,584
Four-of-a-Kind 224,848
Full House 3,473,184
Flush 4,047,644
Straight 5,180,020
Three-of-a-Kind 6,461,620
Two Pair 31,433,400
One Pair 58,627,800
High Card 23,294,460

The person with whom I was corresponding noticed there are fewer high-card hands than either one pair hands or two pair hands. He then asked why a high card hand is ranked lower than either of the two categories listed above them in the table.

Let's take a look at the hand rankings and we'll see that they do make sense as commonly given and represented by the table above.

The three most commonly played games in cardrooms -- seven-card stud, hold'em and Omaha -- all involve ranking hands with many cards. Seven-card stud and hold'em are played with seven cards, while Omaha is played with nine cards. A player forms the best five-card poker hand he can, and then uses that hand for determining the winner of the pot. In Omaha, the player has the additional restriction that he must use precisely two cards from his four hole cards and three cards from the five board cards in forming his five-card hand

The common thread woven through all of the best hand selection methods in the three games is the following: The player looks at the cards he has available -- seven in seven-card stud and hold'em, and nine in Omaha -- and chooses the best five card poker hand he can (subject to the additional restriction in Omaha) to declare as his hand. Everything is then decided on the basis of this five-card hand.

The preceding method is precisely what is used to determine the table given above. There are $C(52,7) = 133,784,560$ possible seven-card combinations chosen from 52 cards. Each combination is examined for the best possible five card poker hand it contains. The latter is then used to determine the value of the hand. We even use the names of five-card hands for the seven card-combinations.

There are good reasons for doing it the way we do. Everyone is familiar with five-card hands, so people become adept at finding the best five-card hand among the cards being examined. Everyone is familiar with ranking five-card hands, so there are no disagreements as to which hand is best. By agreeing to use the best five-card hand among the combination, it means the five cards we use as a representative are essentially unique. Most importantly, weird inconsistencies are avoided by using the standard method under discussion.

What do I mean by weird inconsistencies?

Let's consider two alternatives. One is to rank seven-card combinations as seven-card hands. If we chose to do this, we would have to invent new categories and discard others. For example, a hand with four queens and three 7s would be in a new category. A hand with three distinct pairs would be a new category. On the other hand, a hand with five hearts, two spades and no pairs would no longer be anything special.

Ranking seven-card combinations as seven-card hands requires a complete reshuffling of classifications. We could all learn the new rankings, but they would be quite different than what we are accustomed to.

Another alternative is to maintain the practice of choosing a subset of five cards to represent a seven-card combination, but try to allow some five-card subset other than the best five-card hand. This produces a nightmare. For example, you look at the table above and notice there are fewer high-card seven-card combinations than one-pair combinations. If you attempt to let five high cards beat five cards with a pair, players who have been dealt seven cards with a pair will simply ignore the pair and choose five cards without a pair as the hand representing their seven cards. Thus, the category of high card hands becomes considerably enlarged and an inconsistency has been created.

Let me conclude this article with an amusing look at five-card hands. The table above for seven-card hands is derived by choosing the best five-card hand among seven cards and using that as the representative for the seven cards.

So, instead of ranking five card combinations as five-card hands, let's use three-card poker hands to rank five-card combinations. This is an exact analogy of what we do for seven-card hands. Thus, for each five-card combination, let's use the best three-card poker hand contained in the five cards to label and rank it.

For those of you who have forgotten the rank order of three-card poker hands, in descending order they are: straight flush, three-of-a-kind, straight, flush, one pair, and high card. Using these categories and the ranking of three card poker hands, we get the following table for the 2,598,960 five-card combinations.

Hand Number
Straight Flush 52,492
Three-of-a-Kind 58,548
Straight 542,736
Flush 720,560
One Pair 699,624
High Card 525,000

Judging five-card combinations based on the best three card poker hands they contain leads to some bizarre differences compared with how they stack up as five-card poker hands. For example, the five-card combination 2-3-4-7-J, where the first three cards are in the same suit, is considered a straight flush when judged on the basis of the best three-card poker hand it contains. Meanwhile, the hand K-K-K-K-A is only a three-of-a-kind when judged on the same basis. Thus, the latter hand is ranked lower than the former.

Upon seeing an example like that, is it any surprise seven-card combinations sometimes suffer a similar fate when judged on the basis of the best five-card hand they contain?

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