Academic Corner: Omaha Low Hands - Part 1
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Omaha does not enjoy the same level of popularity or devotion as hold’em but I must confess I enjoy Omaha more. Omaha is not spread in many of the card rooms I have visited but my experience has been restricted to card rooms in Australia, Canada and the USA; perhaps the situation is different in Europe. However, the most popular game in the home game I played in Vancouver was Omaha high-low with the standard 8 or better qualifier for low. I believe we played that game about forty per cent of the time.
I played in that home game from 1975 until 1998 so that I have been playing Omaha high-low for more than thirty years. It is my game of choice online as well. The main difference is that I play pot limit online, whereas, our old home game was strictly limit.
I mention my long history with Omaha in order to indicate that I have developed a feeling about the values of various starting hands. In a home game, where you play with a small number of people many times over a long period, one develops a good understanding of how everyone plays. A natural outcome of this is that play becomes reasonably predictable and surprises are rare. Needless to say, I have been surprised on many occasions by some online play I have observed. In particular, I have been surprised by some of the hands with which players have thrown in pre-flop raises. Of course, there is the possibility that some of the notions I developed from my old home game are incorrect. Consequently, I have decided to take a fresh look at hands for the game. This article is the first in a series devoted to trying to analyze the strength of Omaha hands. I start by looking at potential low hands.
Just in case some readers are a little fuzzy about terminology or low rules, here is a quick primer on a few basics. Frequently there is a qualifying rule for poker games in which a pot is split between the best high hand and the best low hand. A standard rule for both Omaha and seven-card stud is that a player must have an 8-low or better in order to qualify for the low half of the pot. The way one determines a low hand is by taking the five cards with smallest distinct ranks, where an ace is considered a “one” for purposes of low hands, and placing them in increasing order. For example, if a seven-card stud player has a hand consisting of A-A-Q-5-9-J-7, then her five smallest distinct ranks written in increasing order are: A-5-7-9-J. We would then say she has a “jack low” for her best low hand. In other words, a player’s best low hand is the biggest rank appearing in the five cards with smallest ranks.
As another example, consider a player with a seven-card stud hand K-K-Q-Q-10-8-8. This player does not even have five distinct ranks in his hand. The best low hand he has is 8-10-Q-K-8. It, and the preceding example, do not qualify for an 8-low or better.
Deciding which of two low hands is better is another feature of low hands that sometimes causes confusion. Write both hands in order of increasing rank as described earlier. Suppose the hands are a-b-c-d-e and A-B-C-D-E. Everyone is familiar with dictionary order, that is, one compares words from left to right. We do just the opposite when comparing poker hands for low, that is, we compare the ranks from right to left. The first time we encounter one hand with a lower rank than the other, the hand with the lower rank is considered the better of the two. An example should clarify the issue.
Let hand A be A-2-5-6-8 and hand B be 2-3-4-6-8. As we scan the two hands from right to left, we find that both start with an 8 followed by a 6. In the next position hand A has a 5 while hand B has a 4. Thus, hand B is a better low than hand A.
The last pertinent rule to consider for Omaha is the rule that a player must use precisely two cards from her hand to make a hand, with the caveat that she may use different cards for high and low. This implies that a board without three distinct ranks from {A,2,3,4,5,6,7,8} does not allow a low. So even though a player may hold a hand like A-2-3-4, if the board is 5-6-10-10-Q, then the player does not have a qualifying low because she may use only two of the cards in her hand towards a low.
A basic question then becomes: How often does an Omaha board allow a low? It turns out the answer is 60%: that is, the probability an Omaha board allows a low is .60. We shall not derive that number here, but the details of the derivation are available in the Poker Computations directory at http://math.sfu.ca/~alspach/.