Generalized Approach to Poker Probabilites

Even though generalization of the approach makes everything somewhat more abstract at first, it really allows calculations to be made with much more confidence in all situations. It will not be necessary, then, to memorize peculiarities of specific combinations. This generalized approach applies equally to an initial five-card deal in five-card draw, as well as to partial draws, and to deals, flops and board cards in stud and Hold'em. For starters, the denominator in the generalized approach is the same as ever. It is the "combin" function of the number of unseen cards in the deck ("n") and the number of cards to be dealt or drawn ("k"). In probability theory, this is the "sample space" or the universe, in so far as the calculations are concerned.

The numerator can have one or more different scenarios, as follows:

Simple Element

Often, only one term is required to specify correctly the number of cases of "success" in a given draw or deal. An example would be holding three aces and drawing two cards in hopes of a fourth. The simple element is combin(1;1)*combin(1;1)*combin(46;1). (In a 47 card deck, there is only one ace, and the other card can be any one of 46 others, so the total is 46 "winning" combinations out of combin(47;2) or 1081 possibilities, or 4.26%). The combin(1;1) term is repeated because it reminds us there is only one card of interest in terms of rank out of a universe of 1 ranks, and there is only one card of interest in terms of suit, out of a universe of 1 suit. As combin(1;1) is 1, and 1 does not change the other calculations, it is included for the sake of formality, to indicate that in identifying the one combination that will work (the one ace), we have specified it as the only available card in both its rank and its suit.

Additive Elements

Look back to the example of calculating the second pair on a three card draw. There was the scenario with the three ranks that had only three mates in the deck, and the scenario with the nine ranks that had all four mates in the deck. There the calculation needed to be divided into different types of cases, and the results added together. Another example might be to calculate the chance of drawing a third king to a pair on a three card draw, but ruling out a full house. The full house possibilities would need to be calculated and then subtracted from the total of the three-of-a-kind combinations. To calculate the "net" probability of a hand, sometimes it is required to calculate more than one scenario and then subtract one from the other.

In the seven-card stud example about improving the hand by getting a match on the next card to one of three candidates, each candidate was really a separate calculation, and the results added. They can be added after the probabilities are calculated, as percentages, or before they are divided by the common divisor.

Multiplicative or Compound Elements

Look at the example of the odds of being dealt a full house. The calculation is [combin(13,1)*combin(4,3)*combin(12,1)* combin(4,2)]. Here the first two terms describe the three of a kind (all of the same rank, and in three out of the four suits), while the second two terms describe the pair (all of the same rank, but a different rank from the first one, and of two of the four suits).

Correct calculation requires thinking through the proposition to be sure that all additive (and subtractive) elements have been considered, and whether any compound elements are involved.