Poker Digest Vol. 4, No. 13, June 15 - 28, 2001
As a result of working through many details about boards, I have formed the opinion that the most interesting boards are those with a single pair.
One reason is that such boards allow all of the clumping hands: A pair, two pair, three-of-a-kind, full houses and four-of-a-kind. A second reason is that there are four ranks on board as well. This gives enough flexibility for rank and suit choices, so that many of these boards allow all the sequential hands as well: straights, flushes and straight flushes.
A board with a single pair provides many challenges to players trying to put other players on a hand. A final and compelling reason is that boards with a single pair are common. They occur more than 42 percent of the time and cannot be ignored.
Boards with three-of-a-kind or two pair certainly allow all of the big clumping hands, but they lose interest because the chances of them allowing sequential hands are greatly diminished. Also, they are much rarer in that combined, they occur with a frequency only one-sixth of that of a single pair. A board with five distinct ranks does not allow a clumping hand bigger than three-of-a-kind, although there is a greater chance of sequential hands being allowed. Straights and flushes are rare, though high card boards do occur about one-half of the time. They are interesting but, in my opinion, not as interesting as boards with a single pair.
Now let's take a close look at boards with a single pair. There are four ranks present. Given a rank set with four elements, there are four choices for the rank of the pair, there are six choices for the pair, and there are four choices for each of the other three ranks. Multiplying all these numbers gives us 1,536 boards with one pair for each given rank set of four elements. The number of rank sets with four elements is C(13,4) = 715. Multiplying 715 and 1,536 yields 1,098,240 boards with a single pair. Since there are 2,598,960 boards altogether, dividing yields a probability of .4226 for a board with a single pair.
Of the 715 rank sets with four elements, 350 allow a low (we are assuming a player needs an eight or better for low) and 365 do not allow a low. Thus, there are 537,600 boards with a single pair which also allow a low and there are 560,640 boards with a single pair not allowing a low.
The preceding paragraph provides a partition of boards with a single pair based on whether they allow a low. Let's refine this partition according to whether a flush is allowed. A flush is possible whenever there are three or four suited cards on board.
For a given rank set, there are four choices for the rank of the pair and six choices for the pair of the chosen rank. Once they have been chosen, if there are four suited cards on board, then there are two choices for the suit of the three remaining ranks. If there are three suited cards, there are two choices for the suit of the remaining ranks when the three suited cards do not include one of the pair cards. If the three suited cards include one of the paired cards, there are two choices for the suit, three choices for the other two cards in that suit, and three choices for the suit of the other card. This yields 24(2 + 2 + 18) = 528boards allowing a flush for the given rank set. This allows us a further breakdown of the previous boards. There are 184,800 boards with a single pair allowing both a low and a flush, there are 352,800 boards with a single pair allowing low but not allowing a flush, there are 192,720 boards with a single pair not allowing a low but allowing a flush, and there are 367,920 boards with a single pair allowing neither a low nor a flush.
We can refine the partition further by considering straight possibilities. A rank set set such as allows a player to make a straight using only one card. However, a player holding 6-8 makes a bigger straight than a player holding, say, A-6, so the rank set also allows straights using two cards. A rank set such as allows a straight but two cards are needed, namely, 4-6. The next table summarizes this information. The number refers to the number of boards with a pair having the properties indicated. A yes in the Low column means the boards allow a low, and a yes in the Flush column means the boards allow flushes. A yes in the One Card for Straight column means a player can make a straight using only one card, and similarly for a yes in the last column.
|Number||Low||Flush||One Card for Straight||Two Cards for Straight|
Of interest is the probability that some player holds one or two cards necessary for the existence of a certain hand. For example, if the rank set is , then anyone holding a five in hold'em has a straight.
For Omaha, any player holding a five plus any one of ace, deuce, three, four, six or seven has a straight. If, in addition, there are four hearts on board, then anyone holding 5-7 of hearts has a straight flush for either hold'em or Omaha.
These probabilities and the occurrences of certain hands will be explored in future articles.