Academic Corner: Omaha Low Hands - Part 1 (page 2)

The ﬁrst fundamental problem we address is looking at the chances of making a low. Of course, a player cannot make a low unless her hand has at least two distinct ranks chosen from {A,2,3,4,5,6,7,8}. Henceforth, we refer to those ranks as low ranks and the other ranks as big ranks. The following fact may be a bit surprising but it is true.

Fact: The probability of a player making an 8-low or better with an Omaha hand is completely determined by the number of distinct low ranks in the hand and no other factor.

What the preceding fact means is that all that matters is how many diﬀerent low ranks you have in your hand and not what the ranks are or how the cards are distributed. Thus, a person holding 4-6-6-6 has the same chances of making a low as does a person holding A-2-Q-K because both hands have two distinct low ranks in them. I am NOT claiming that 4-6-6-6 and A-2-Q-K have the same quality as potential lows; we shall begin to look at that issue in part 2.

The next table gives information on the chances of making lows with various Omaha starting hands. The table actually contains more information than might seem to be the case at ﬁrst sight. So let’s discuss the table and its implications.

The hand A-2-B-B consists of an ace, a deuce and two cards of big rank; the hand A-2-3-B contains an ace, a deuce, a trey and one card of big rank; and the hand A-2-3-4 contains an ace, a deuce, a trey and a four. The symbols down the ﬁrst column indicate the form of particular boards. Lower case letters indicate low ranks, where diﬀerent letters represent diﬀerent ranks, and B represents a card of big rank. The symbol “dup” in parentheses means a board card of low rank duplicating a small rank in the corresponding hand.

Two examples should clarify any questions about the notation. The row for abcBB means those boards with three distinct low ranks and two big cards such that the low ranks do not duplicate any of the low cards in whatever hand you are considering. Hence, the entry 109,640 indicates that there are that many boards for the hand A-2-3-B giving the player a low.

The row for abcBB(dup) means those boards with three distinct low ranks and two big cards such that one of the low ranks duplicates one of the low cards in the hand. Notice that there is no entry for the hand A-2-B-B in this row because if one of the low cards on board has rank A or 2, then the hand does not have a qualifying low.

Therefore, summing the numbers in the columns for the three hands gives the total number of boards giving that hand a low. But now if we refer to the fact written earlier, we can determine the exact probability for any Omaha hand to make a low. We do so by dividing the number of qualifying boards by the total possible number of boards. The number of possible boards for a given Omaha hand is C(48, 5) = 1, 712, 304. Dividing that number into the numbers from the table yields the following useful information.

An Omaha hand with no low cards or only one rank of low cards has a probability of 0 for making a low.

An Omaha hand with two low ranks has a probability of .3562 for making a low.

An Omaha hand with three low ranks has a probability of .4932 for making a low.

An Omaha hand with four low ranks has a probability of .5267 for making a low.

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Comments

FL and PL Omaha 8ob

Tue, 08/04/2009 - 11:21 — Martin (not verified)

Hi Brian
I come from the other side of the poker spectrum. Instead of looking at the game from an academic point-of-view I have learned it simply by grinding. With that said, it’s really interesting for me to see the game broken down into statistics. I think that I could improve significantly by studying the basic maths surrounding Omaha 8 or better.
When I play Omaha 8ob online, I almost exclusively play in pot-limit format. In that game I see a hand’s low potential as a necessity to enter a pot – at least for a raise – but then it often turns into being just added value.
Since players are allowed bet the size of the pot, drawing to lows, in my opinion, rarely receives the proper pot-odds. You could often play the game just as you play Omaha High and see the possible lows as extra value.
In a limit 8ob game, a classic Omaha monster like A-A-K-K is not as good as it might look – especially at a table with lose/passive players. When playing pot-limit Omaha 8ob, however, the hand improves quite a lot. Although it is far from one of the best holdings (when looking at all possible board cards all the way to the river) you could push opponents away from their low hands on the flop with a pot sized bet giving the insufficient pot odds.
I would like to know what you think of the differences between FL and PL Omaha 8ob. Do you agree with me that the low value decreases when playing the latter version? If so, how much and in what way?
Thanks for great thoughts and insights
Martin