Poker Card Probabilities

One more piece of the larger puzzle remains to be put in place: One must be sure that all of the cards in the deal or draw have been accounted for in the numerator. If not, the probabilities could be understated. Common points of confusion involve forgetting to describe all of the "other cards" if more than one is present, or to put together a formula for a structure, such as two pair, and then forget to account fully for all four cards involved in that structure.

Two pair is a good illustration. The pairs can be in any two of 13 ranks, and in any 2 of 4 suits. This implies "combin(13;2)*combin(4;2)." Look at this formulation. In the first term the cards are constrained to the same two ranks. This is OK. In fact, that is the idea. But look at the second term. The cards are also constrained to the same two suits! This is not OK, and it is not part of the idea. It may be helpful to remember what "combin" really means: For any given pair, there are combin(4;2) different ways that four suits can used in combinations of two. But there is no requirement at all that whatever combination of suits is used for pair number 1 must also be used for pair number 2. Pair number 2 should be unconstrained to find its own combination of suits from the combin(4;2) of possibilities.

By how much is the more severe constraint limiting the "freedom" of the second pair? The answer is that the number of successful combinations are being shorted by a factor equal to combin(4;2). To resolve this, the second element, the "suit" term, should be repeated for each time suits are to be distributed independently across the hand.

In this case, with two pair, each independent of the other in terms of suits, the "suit" term needs to be repeated. In notational terms, it means raising the term to a power equal to the number of distributions across the hand, but it seems intuitively more appealing to consider it as being repeated a certain number of times.

The full and correct specification for the numerator of the two-pair initial deal (the number of possible hands that contain two pair) is thus: combin(13,2)*combin(4,2)*combin(4,2)*combin(44;1). This equals 123,552. This is the number of possible two-pair combinations in a five-card deal. The chances of that happening are 4.5%.

Note that in this example, the constraint was defined to require that the pairs be of different rank, (this is the "combin(13;2)" term). For this reason, four-of-a-kind is not included in this calculation, even though technically it could be considered a special case of two pairs.

A final example relates to the straight, perhaps the diciest problem, but also the most illustrative, and it helps explain why sometimes one of the constraint terms needs to be repeated. The number of straights is considered by looking first at the rank for a starting point of the first card: combin(10;1). Any one of ten possible ranks is an acceptable starting point. For each rank, there are four available suits, as straights impose no constraints on suits. So the number of starting points for straights is combin(10,1)*combin(4;1). This number is 40. It makes intuitive sense that 40 out of a total of 52 cards can be candidates to start a straight.

Now, the next card has to be a certain rank, so as to be in sequence, but it need not be of any particular suit. This card would be described as combin(1;1) * combin(4;1). The same is also true for cards 3 through 5. As a result, we have a set of five cards described together as combin(10,1)*combin(4;1), with the combin(4;1) term appearing five times in the calculation. If the suit constraint were thus not lifted for cards 2-5, it would require all cards to be of the same suit. In fact, that is the way to describe a straight flush:

combin(10;1)*combin(4;1) = 40

There are exactly 40 instances of straight flushes in the universe of five-card deals. However, for any "normal" straight, any card in the sequence can appear in any one of four suits, so the calculation is:

combin(10;1)*[combin(4;1)]**5 = 984