Poker Digest Vol. 4, No. 14, June 29 - July 12, 2001
Sometimes I see or hear the question, ``How often should we expect to see a player make a straight flush?'' I'll try to answer this question as best I can. Let's first determine the probability of the occurrence of a semi-deal with at least one player making a straight flush in either hold'em or Omaha.
In order to have a straight flush, there must be at least three suited cards on board. One possibility is for the board itself to be a straight flush. The probability of this happening is 1/64,974, so it is a rare event. A straight flush on board also provides the greatest contrast between hold'em and Omaha. In hold'em, every player remaining in the hand has a straight flush. In Omaha, it is conceivable that no one has a straight flush because players must use two cards from their respective hands.
Let me outline the procedure used to determine the probabilities below. There are three basic cases: The board has three ranks, the board has four ranks or the board has five ranks. Each of the cases is treated separately. When the board has three ranks, it is either two pair or three-of-a-kind. These boards were considered in detail in part III of this series on boards.
When the board has four ranks, there is a single pair on board. These boards were considered in great detail in part IV.
Boards with five ranks are the most common and come in four varieties:
Straight flushes, flushes, straights and high card. The boards with five ranks are the most complex with regard to allowing straight flushes.
There are three forms of completion to a straight flush in hold'em.
One form is a straight flush on board that does not require an additional card to complete it. As we noted above, this rarely happens. Another form is a board that requires only one card to make a straight flush. However, it has two subforms. For example, a board with 10-J-Q-A of spades completes to a straight flush if some player has the king of spades (or the 8-9 of spades). A board with 10-J-Q-K of spades completes to a straight flush with either the nine or ace of spades.
It is clear the probability of someone having a straight flush is bigger for a board with either of two single cards producing a straight flush compared with a board having only one single card producing a straight flush.
The last form requires two cards to produce a straight flush. In this case, there are several different types of boards that can make it happen. Some require a distinct set of two cards, while others provide two choices of two cards. Some boards even offer a total of three choices of two cards to complete the straight flush.
Omaha, of course, requires a player to use two cards from her hand to form a straight flush. The corresponding boards allow one to three choices of two cards to complete it.
The calculation of the probabilities is now straightforward in theory. Count the number of boards of different types allowing completion to a straight flush. For each type determine the probability of some player holding the appropriate cards. Multiply the various probabilities by the numbers of the types of boards and add.
Here are some boards and their probabilities of being completed to a straight flush. Consider a board that allows a straight flush with either of two single cards such as suited 4-5-6-7. The probability of at least one player having a straight flush in hold'em is 730/1081, making it fairly likely. A board such as suited 4-5-7-8 allows a straight flush in hold'em only if a player has the appropriate six. So the probability of some player having this straight flush in hold'em is 20/47.
A board such as suited 4-5-8 allows a straight flush only if some player holds the appropriate 6-7. The probability of this happening in hold'em is 10/1081, and in Omaha is 60/1081. Similarly, a board that allows a straight flush completion with two sets of two cards, has a probability of 20/1081 in hold'em, and a probability of 352/3243 in Omaha that someone has a straight flush. A board that allows a straight flush completion with any of three sets of two cards, has a probability of 329/11891 in hold'em, and a probability of 16986/107019 in Omaha that someone has a straight flush.
Now a few words about counting the number of boards that allow straight flushes. I have a definitive profile of the boards with three ranks and four ranks. Since I do not have a definitive profile for all the boards with five ranks, I have used an approximation, which means there is a potential for error. However, I suspect the error, if any, is small.
After performing the computations, the bottom line is as follows:
The probability of a hold'em semi-deal with at least one straight flush is .0025, or about 1-in-400. The probability of an Omaha semi-deal with at least one straight flush is .0098, or about 1-in-102.
Earlier we questioned the likelihood of players making straight flushes. The numbers we have just reviewed are for semi-deals and not real games. Most of us have heard a player groan and say, ``I would have made a straight flush!'' That is an example of a truncated semi-deal. Can we say anything sensible about the effect of real games on semi-deal probabilities? In other words, what proportion of straight flush semi-deals will actually lead to the player making her straight flush?
One important factor is the way the board develops. A given board with five cards is dealt in 20 ways. To see this, note there are C(5,3) = 10choices for the flop and two different orders for the remaining two cards to be dealt. Thus, if the board has three suited cards, which together with a hold'em or Omaha player's hand make a straight flush, there is a probability of .1 that the player will flop the straight flush. There is a probability of .6 that the player will flop two of the cards she needs, and there is a probability of .3 that she will flop just one. So there is a probability of .7 that the player will flop at least two of the three cards required for the straight flush.
So in a loose game where many players play any two suited cards, might .7 be a reasonable factor to determine the likelihood of a straight flush?
It is important to consider the looseness of the game. Many players routinely throw away hands with two suited cards before the flop unless the hand has other redeeming traits. There are C(13,2) = 78 combinations of two suited cards. Ten of them involve two big cards. There are eight suited aces, eight suited connectors and eight suited one-gappers that have not been counted. And that is still less than half of the possible combinations of two suited cards in hold'em.
Depending on the looseness of the game, I would be inclined to use a factor of .2 to about .7 to give an estimate of how often straight flushes should occur in hold'em. I find there are even fewer hands worth playing in Omaha. However, all of this is speculative and it depends on how you play.