Brian Alspach

Poker Digest Vol. 4, No. 10, May 4 - 17, 2001

Let's now continue to partition all possible hold'em and Omaha boards in order to answer a variety of questions about boards. Below is a table describing all 2,598,960 boards according to several properties.

 Type Number Allows Low Clumping Hands Sequential Hands SF1 4 no board plays board plays SF2 12 no board plays straight flush SF3 24 yes board plays straight flush Q1 52 no board plays board plays Q2 572 no bigger quads board plays FH 3,744 no quads, full house board plays F1 3,540 yes board plays straight flush F2 300 yes board plays bigger flush F3 1,268 no board plays straight flush S1 600 no board plays board plays S2 1,800 no board plays bigger straight S3 1,680 no board plays straight flush S4 2,520 yes board plays straight flush S5 3,600 yes board plays bigger straight T1 2,016 yes quads flush T2 8,736 yes quads T3 8,280 no quads flush T4 35,880 no quads D1 6,048 yes quads flush D2 18,144 yes quads D3 24,840 no quads flush D4 74,520 no quads P1 63,888 yes quads flush P2 121,968 yes quads none P3 93,984 no quads flush P4 179,424 no quads none P5 230,832 yes quads straight P6 188,496 no quads straight P7 50,712 yes quads straight flush P8 70,200 yes quads flush, straight P9 43,096 no quads straight flush P10 55,640 no quads flush, straight H1 31,500 yes 3-of-a-kind flush H2 45,000 yes 3-of-a-kind none H3 531,000 yes 3-of-a-kind straight H4 190,200 no 3-of-a-kind straight H5 100,552 yes 3-of-a-kind straight flush H6 271,148 yes 3-of-a-kind flush H7 41,328 no 3-of-a-kind straight flush H8 91,812 no 3-of-a-kind flush

This table and those that evolve from it are going to provide the basis for subsequent articles. Please keep this table as a reference. Let's first discuss how to read the table starting with the column headings. The type of a part (frequently I shall use the word part to mean a collection of boards which are being treated as equivalent when drawing the table) is just a designation we are giving in order to keep track of what is under discussion. Parts starting with SF contain straight flushes, those starting with Q contain four-of-a-kinds, those starting with FH contain full houses, those starting with F contain flushes, those starting with S contain straights, those starting with Tcontain three-of-a-kinds, those starting with D contain two pairs, those starting with P contain a single pair, and those starting with H contain high card hands.

The next column simply tells us the number of boards of that type.

The third column is an indication of whether or not boards of the particular type allow the possibility of some player making a low, given that the low requirement is an eight-or-better.

The fourth column refers to clumping hands -- poker hands whose values are determined from clumping of cards by ranks, that is, high card, one pair, two pair, three-of-a-kind, a full house and four-of-a-kind. Sequential hands are straights, flushes and straight flushes. The entry in each column is the best hand of the particular kind which can be formed using the board and beating the board.

The latter point is important. For example, if there is a straight on board, then no player can beat the board using a clumping hand since three-of-a-kind is the best clumping hand which can be formed with a straight on board. Consequently, in the clumping column, we write ``board plays'' because the board is better than any clumping hand. Some entries are left blank since further refinement is required. This is because some boards allow better hands than others.

In subsequent articles we shall go into details about the parts as we consider various questions about boards. This will clear up what each part really is, but it is possible to do some detective work and figure out what many of them are. For example, let's look at the five parts S1, S2, S3, S4 and S5 containing all 10,200 straights. Note that S1 contains 600 straights and under the sequential hand column we see that the board plays. This means the boards in S1 do not allow flushes or bigger straights. That tells us the ranks of the straight must be 10-J-Q-K-A, and there cannot be three or four suited cards. Hence, the suit distribution is either 2-1-1-1 or 2-2-1. In the case of a 2-1-1-1 suit distribution, there are 10 choices for the ranks with two suited cards, four choices for the suit and six choices for assigning the suits to the remaining ranks. This gives 240 straights. For a suit distribution of 2-2-1, there are five choices for the rank with one suit, three ways of chopping the remaining four ranks into two sets of two each, four choices for the suit appearing only once, and six ways to assign suits to the other four ranks. Multiplying here gives 360 straights with suit distribution 2-2-1. Adding produces the 600 straights for which the board plays.

Note that S2 contains the straights not allowing a low and not allowing a flush. Of course, they do allow a bigger straight. S3 then contains the straights not allowing a low but allowing a straight flush. This simply means the straight has either three or four suited cards, because as soon as we have a straight on board with three or four suited cards, it allows a player to have a straight flush. Similarly, we see that S4 and S5 are the straights which allow a low and have been split into two parts depending on whether or not they have three or more suited cards.

Let's conclude this article with examples of two easy questions answered by using the table. Recently on RGP, someone asked for the probability of obtaining a board in hold'em such that the board plays -- that is, it is impossible for a player to have a better hand than the board itself. Looking at the table, we see this happens for parts SF1, Q1 and S1. The total number of boards in these three parts is 656. Dividing by the total number of boards gives us a probability of .000252 which is about 1 in 3,968.

The second question is to determine the probability of a low board occurring. Adding the numbers of boards in the parts which have a ``Yes'' in the column headed ``Allows Low'' yields 1,561,728. This gives a probability of .601 that a low board occurs.

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