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

Here is a sample of what needs to be done. We examine the hand A-2-B-B and refer back to the table in Part 1. All of the boards corresponding to the first three entries for A-2-B-B give the
player a nut low. For the fourth entry of the table, the board has the form abcdB(dup) and now the player must have a wheel in order to have a nut low. This yields 6,912 boards instead of
138,240.

The next three entries in the table correspond to nut lows. The remaining entries correspond to boards having cards on board of rank A or 2, thereby, forcing the player to have a wheel in order
to have a nut low. This leads to 424,512 boards that give A-2-B-B a nut low.

We perform the same calculations for the other hands shown in Table 2. Here are a few comments about the table. The first column shows the hand, where B denotes an actual card whose rank
is chosen from {9,10,J,Q,K}. The column headed ``low'' gives the probability of making a low, the column headed ``nut low boards'' gives the number of boards producing a nut low, and the column headed ``nut low'' is the probability of making a nut low.

The reason we have to be fussy about the B designation is that the probability of making a low with A-2-10-J and A-2-2-J is the same, but the probability of making a nut low for those two
hands is not the same. This may sound counter-intuitive so let's establish this fact. We again use one-to-one correspondences.

Suppose both hands have the ace, deuce and J of clubs, one also has the deuce of hearts and the other has the 10 of hearts. Given a board for A-2-2-J, to get the corresponding board for A-2-10-J leave the board the same if it does not contain the 10 of hearts, whereas, if it contains the 10 of hearts replace that card by the deuce of hearts. It is easy to see that this establishes a one-to-one correspondence between boards for the two hands. It also is easy to see it establishes a one-to-one correspondence between low boards for the two hands. In addition, a board for A-2-10-J that gives it a nut low, also gives the corresponding board for A-2-2-J a nut low. Now look at the board 4-7-8-Q with the 10 of hearts which gives A-2-2-J a nut low. The corresponding board for A-2-10-J is 4-7-8-Q with the deuce of hearts. It does not give A-2-10-J a nut low. Thus, A-2-10-J has fewer nut lows than does A-2-2-J.

The hands have been placed in Table 2 in decreasing probability of making a nut low, but this is not the end of the story. We complete our look at Omaha low hands in Part 3.

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