Academic Corner: Omaha Low Hands - Part 2

The essence of Part 1 is captured by paraphrasing George Orwell: All lows are created equal but some lows are more equal than others. That is, we saw that the chances of making a low depend only on the distinct number of low ranks in the player's hand. On the other hand, it takes little common sense to know that there is something better about the hand A-2-4-K as compared to 2-5-7-K with regards to potential low hands. One huge difference is captured by the notion of a nut low, that is, a low hand that cannot be beaten. That is what we now explore.

Players must be able to recognize quickly what the nut low is that the board affords. This makes sense because a player does not want to be raising and/or calling under the belief she or he has a nut low when in fact they do not. A lot of chips may disappear in such
a case.

If a board contains three or more ranks chosen from {A,2,3,4,5}, then the nut low is a wheel, that is, A-2-3-4-5. Otherwise, take thethree smallest ranks appearing on board, say x,y,z, and let a and b be the two smallest ranks not appearing on board. The nut low then is a-b-x-y-z.

Let's take a quick look at two examples. If the board is 2-3-4-5-Q,then the nut low is a wheel. If the board is 2-2-4-7-J, then the nut low is A-2-3-4-7 because we add A-3 to 2-4-7.

A nice way to show that two sets have the same number of elements, without actually having to count the number of elements in each set, is to establish a one-to-one correspondence between them. We do this as far as nut lows are concerned for A-3-B-B and 2-3-B-B. Here is a one-to-one correspondence between their boards that makes it explicit. We also use the fact that changing suits does not affect the numbers of boards giving lows and nut lows. Assume the low cards are clubs and the two big cards are identical in the two
hands.

Given a board for A-3-B-B, replace any cards of rank 2 with an ace in the same suit and replace any ace with a deuce in the same suit to get the corresponding board for 2-3-B-B. It is easy to
see that the boards are in one-to-one correspondence and that boards giving nut lows also are in one-to-one correspondence. What is happening is that each hand has a single rank that must appear on board in order for the player to have a nut low. Similarly, A-4-B-B and 2-4-B-B are identical for nut lows because they have two
forced ranks.

Continue to page 2

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