Poker

In this paper we are going to try to split up the analysis of the game of poker into a number of different pieces. There are lots of interesting issues involving poker, and many of them can be best discussed be separating the issues so that the various parts of the game can receive the focuessed attention needed to understand and comment on some of the complexities.

It should be emphasised at the outset that the interest here is to contribute to philosophy and computer programming, not to accomplish anything useful with respect to the actual play of poker as practiced by all of those folks who so love the game. If we are very lucky, we may be able to develop some technology which will prove to have some use to practicioners. But the realities of playing poker often involve creating betting situations that depend on the fact that real evaluation of the possible outcomes is essentially an impossible task. With modern computer power we can handle some of those situations but it is always easy to increase complexity to push the bounds of analysis.

We will divide the analysis of poker into a number of separate (we hope) stages of analysis. Our first draft is to parse the game into three stages:

In poker each of these is a (partially, at least) separable stage. There is even an important philosophical issue associated with the order in which we can discuss the issues. As we will later see, there is even an implication that in some situations the answer to one of the questions will change the answers to another. This complicates the analysis and the programming materially.

How are hands ranked?

In the analysis presented here we will begin by assuming a simplified poker game that allows us to answer this question without knowing how the ranking that we determine will ultimately be used. Just how appropriate this assumption is in the real game of poker remains to be seen. We will call this Near Poker, and it will, for the most part, be the game of `High Hand Poker' that is perhaps the most common of all of the poker games.

What is the hand?

Although there may be some interesting and important variants of the game that appear to challenge this rule, we will define A poker hand consists of 5 cards. Although some games are called things like `7 Card Stud', in the final analysis the evaluation of the game always depends on evaluating 5 card hands. We may have many different ways of determing which cards make up a hand, but in all cases the final hand to be evaluated will have exactly five cards.

If there are no wild cards in the game, then the answer to this question is both simple and pretty much unequivocal, although, as we will see, there are potential complexities involved if low hands are valuable. When there are wild cards in the game, then there are a number of issues that are associated with this question, and we will see that it is not a trivial issue.

How are winnings determined?

This is another important variable in poker. There is the simple `normal' poker game where the `High Hand' (out of those remaining at the final showdown) is the one single winner. However, there are variants of the basic game that sometimes decree the `lowest' hand to be the winner, and other times split the pot, in some fashion, between the high and low hands.

Cards

Cards are integers C:[0...51] . Denominations are integers D:[0...12] . Suits are integers S:[0...3] .

Mapping

Relationships

Hands

Poker hands are a 5-tuple of integers drawn (without replacement) from C . H:(C1,C2,C3,C4,C5) Ci from C without replacement.

Observation

Order of cards in poker does not matter. Therefore all 5! permutations of any particular hand are identical hands. Therefore there are Comb(52,5) = 2598960 distinct poker hands.

In addition, the concept of suit has only a very restricted significance in poker. There is no notion of rank associated with suit. All are regarded as equal. In addition the only significance of suit in evaluating poker hands is to distinguish a special category of hand, called the flush, consisting of hands which contain cards from only one suit. Such hands have a greater value than they would if they contained mixed suits. However, no significance other than this is attached the number of suits in a hand, so while hands with only one suit are special, hands with 2, 3 or 4 suits are all treated similarly.

Canonical Representation

Because order doesn't matter, we will canonically represent any poker hand H with CH =: /:~ H , the one monotone increasing permutation (there is only one) of the 5! equivalent hands that happen to contain the same cards. We lose nothing important about the game to restrict ourselves to this particular subset of the 52*51*50*49*48 hands that might be obtained if order were to matter.

Ness Enumeration

A mapping exists from the set of Comb(52,5) hands H 1:1 into the integers NE:[0...2598959] . This mapping is (a) known; and (b) programmable. As a result we can unequivocally speak of any hand by giving either its CH 5-tuple or its NE integer. We can always convert back and forth with ease. This enumeration is not particularly useful unless a computer is being used to `count out' some hands, but it is occasionally nice to be able to `define' a hand with a single integer.

Constructing the NE scheme is not a trivial exercise. The process chosen here is reasonably effective. The hand H is ordered into monotone increasing order with /:~H . This 5-tuple is then reduced by subtracting each successive element from one more than the element on its left. This produces a reduced form for H . Let's look at some of the properties of this form.

Each successive number in the 5-tuple can only be drawn from the reduced set of cards that are `larger' (have a higher C value) than the cards to its left. So, there is only one 5-tuple of the form (47 a b c d) Given that the smallest card in the hand is a 47, the rest of the cards must be 48 49 50 and 51 . This particular hand is given the largest NE , namely 2598959.

So, there are Comb(4,51 - 47) = 1 hands that start with 47 . There will be Comb(4,51 - 46) = 5 hands that start with a 46 . They are (46 0 0 0 0) (46 1 0 0 0) (46 0 1 0 0) (46 0 0 1 0) (46 0 0 0 1) . Similarly there will be Comb(4,51 - 45) = 10 hands that begin with 45 . And so forth.

Now this leads to a specific enumeration scheme. The first Comb(4,51 - 0) hands, all begin with a 0 . Then there are Comb(4,51 - 1) hands that begin with a 1 . Then Comb(4,51 - 2) hands that begin with a 2 . It continues this way until we get to the Comb(4,51 - 47) hands that begin with a 47 that we mentioned above.

This description can be applied recursively to enumerate the other cards in the hand. The code necessary to accomplish this has been written, but it turns out to be a little deep to explain in a paper at the level of this exposition.

Denom and Suit Functions

It is easy to define Denom(H) and Suit(H) . They simply apply the denomination and suit calculations to a whole hand, so Denom produces a 5-tuple of [0...12] of 5 denominations and Suit produces a 5-tuple of [0...3] of 5 suits. For all computational purposes the lists produced by Denom and Suit are not ordered, other than they typically retain whatever order is implicit in the H that they are calculated from.

Type

all by the `usual' poker definitions. At this stage all of the definitions are quite unequivocal, although no hand can be categorized as a type `9' unless we allow wild cards, and at this stage of our analysis we do not. We do, however, provide a `category' for such hands, even though we can not yet generate them.

Characteristic Vector

The Characteristic Vector, CV:(T,I1,I2,I3,I4,I5) of a hand CH (or NE ) is a 6-tuple described in two parts.

T:[0...9] is the `type' of a hand. We assume here that we are able to algorithmically determine the type of the hand. We will later show that this is a (fairly trivial) algorithm.

I1,I2,I3,I4,I5 are the Denominations (with ties broken in descending order) of the most frequent card in CH , then the second most frequently occuring card, ..., filling with `unused' denominations (in descending order).

A comment on the CV may help explain a bit better. Depending on the kind of hand, only the first n elements of a CV actually matter. For example, for a full house (or four of a kind) only the first three elements matter. The first element is the type. The second and third elements represent the denominations of the two different values that make up a full house (or, indeed, a four of a kind). For a full house (four of a kind) the fourth thru sixth elements are actually just filler. In our implementation we could bother to build a table to tell us how many members of CV are `relevant'. In our case, though, we have chosen instead just to regularize how we generate the `filler', and in this way we guarantee the proper uniqueness of our representation.

Example

Perhaps an example would help. Consider a hand with [4C, 9S, 4D, 4H, 9C] . This hand is: H = (41,7,28,15,46) This makes CH = (7,15,28,41,46) and NE = 1452225 (Take my word for it). The characteristic vector for this hand is CV = (6,2,7,12,11,10) . This indicates a Full House (6) of 4s (D(2) ) on 9s (D(7) ). The 12, 11, 10 are filler.

Near Poker

Near Poker is a restricted form of poker. It is introduced as an intermediate because:

Near poker does not have wild cards and considers Aces to be `High' in all circumstances, with the exception of that Aces complete 5-4-3-2 straights (and straight flushes). These hands are regarded as `5-high' however.

Ranking

The central issue of Near Poker is ranking the poker hands. In this section we will propose a mathematical form which will rank the hands. A `ranking' is a Many:1 mapping of CH (or, equivalently, NE ) into CV .

Property #1

Hands with equal CV are, in fact, equal poker hands.

Property #2

Equal poker hands will have the same CV .

Property #3

A hand with a higher CV is a better hand.

Property #4

A better hand will have a higher CV .

Adjusting the CV

In order to make this work, we must slightly adjust the definition of the CV given above. This can be accomplished quite easily: CV(4,12,3,2,1,0) is remapped to CV(4,3,2,1,0,12) and CV(8,12,3,2,1,0) is remapped to CV(8,3,2,1,0,12) . This represents ranking low straights containing aces as `5-high' rather than `A-high'.

Five-of-a-Kind

Handling the programming of five-of-a-kind also requires some special consideration. The hands which evaluate to five-of-a-kind would normally be illegal, but are allowed by the specific clause in the rules which allows for a fifth card of the particular denomination of unspecified suit in order to create five-of-a-kind hands.

We have decided, somewhat arbitrarily, to represent all five-of-a-kind hands with a peculiar suit structure, (0 1 2 3 0) , which represents two spades and one each of the rest of the suits. While this hand is technically illegal, there is no situation where the programming involved has any trouble with the concept, and this seems much easier than creating a fifth suit to cover this peculiar (and very rare) situation.

Generating Type

Perhaps it is timely to take a detour to show how we determine type. It is perhaps somewhat easier than it might appear at first. Define a 13-tuple called a TypeCount, as follows: (i0,...,i12) = TC(x) where i0 = # of 0s in x, i1 = # of 1s, ... i12 = # of 12s. For a given hand, H , we can compute DR = TC(Denom(H)) and SR = TC(Suit(H)) >. SR has two important cases. Either it contains a 5, or it does not. If it contains a 5, then there is some suit that has 5 cards. In this case the hand is a flush (or a straight flush). Notice that it cannot have more than one of any particular denomination. If SR does not contain a 5, then there are at least two suits, and so the hand is not a flush. We can calculate 5 = >./SR (i.e.true if the biggest element in SR is 5) to determine a flush.

DR is a little more complex. Look at _5{./:~DR . This is the 5 largest values in DR Theoretically possible cases are:

(0 0 0 0 5)

which would occur if there was 5 of some denomination, but this couldn't happen so far

(0 0 0 1 4)

which would occur if there was 4 of some denomination, and 1 of some other

(0 0 0 2 3)

which would occur if there was 3 of some denomination, and 2 of some other;

(0 0 1 1 3)

which would occur if there was 3 of some denomination, and two other unpaired cards;

(0 0 1 2 2)

which would occur if there were two pairs and an extra card;

(0 1 1 1 2)

which would occur if there was a pair of some denomination, and three different cards;

(1 1 1 1 1)

which would occur if there were no pairs, triples, fours or fives;

So this allows us to determine type, except for the possibility of straights. All straights will clearly have (1 1 1 1 1) , but not all (1 1 1 1 1) will be straights. Straights are detected by looking at TC . Hands that have TC s that are some permutation of ( 1 1 1 1 1 0 0 0 0 0 0 0 0) including the one unusual special case of (1 1 1 1 0 0 0 0 0 0 0 0 1) (this represents A2345) are straights. With this information, and remembering that something can be both a straight and a flush, we can finally determine the T of a hand.

Determining the CV

Once we have this, determing CV isn't very hard. By definition, T of any hand is the first element of its CV . Look at the 5-tuple |:_5{./:DR This differs slightly from the value we used above. There we sorted DR and took the 5 largest values. This quantity is the indexes of the 5 largest values. Therefore it is the index of the largest value in DR , then the index of the second largest value in DR etc. It therefore is the rest of the values in the CV vector, as the indexes in a TC are perfect surrogates for denomination.

This little artifice accomplishes something rather important from the standpoint of ranking hands. The rules of poker are quite regular, from a philosophical viewpoint, in how hands are ranked. Specifying this simply you can summarize the rules in two statements.

First, Ranking by type is absolute. All hands in a particular category beat all hands in any lower category, and lose to all hands in any higher category.

To understand the second rule, which governs the ranking of hands that are in the same category, a observation is necessary first. Hands in the same category will have the same shape. Hands are ranked by comparing the highest value of the most frequently occuring denomination. If they tie then the second, third, ... of the most frequently occuring denomination are used. If still not resolved, then the next most frequently occuring denomination. If hands match, by this rule until all cards are exhaused, then the hands tie.

It is worth noting that this is a complex, but algorithmically oriented, way of saying something that could be described in a sequence of cases. Four-of-a-kinds are ranked by the four, only looking at the fifth card if they tie. Full Houses are ranked by the triple, only looking at the pair if the triples tie. Notice that the rule even effectively covers the No Pair case. No pair hands are ranked by the highest card. If they tie, then the second highest card decides the issue, etc.

The NP Mapping

So now we can at least talk about a game: (i1,i2,i3,i4,i5,i6) = NP(H) . Values of NP can be characterized a bit further in order to understand the mapping.

For example, There are 40 values of H representing straight flushes which produce 10 values of NP . These are (8 12 11 10 9 8) (8 11 10 9 8 7) ... (8 4 3 2 1 0) Notice that there are no other legal NP values that begin with 8, as 8 implies a straight, straight implies that all the rest of the cards differ and are in order. The factor of 4, of course, represents which suit is involved. In poker, all suits are equivalent.

There are 624 values of H that produce 156 values of NP that are four-of-a-kinds and begin with a 7: (7 12 11 10 9 8) (7 12 10 11 9 8) (7 12 9 11 10 8) ... (7 0 2 12 11 10) (7 0 1 12 11 10) The 156 different values are 13 different denominations of the 4 cards with each of the 12 denominations of the single card.

There are 3744 values of H that produce 156 values of NP that represent full houses and that begin with a 6: (6 12 11 10 9 8) (6 12 10 11 9 8) (6 12 9 11 10 8) ... (6 0 2 12 11 10) (6 0 1 12 11 10) The 156 different values are, just as above, 13 different triples with each of the 12 denominations of pairs.

There are 5108 values of H that produce 1277 values of NP that are flushes and therefore begin with a 5: (5 12 11 10 9 7) (5 12 11 10 9 6) (5 12 11 10 9 5) ... (5 5 4 3 2 0) (5 5 3 2 1 0) Each of these is a combination of 5 taken from [0...12], with the 10 particular permutations that happen to also represent straights removed.

There are 10200 values of H that produce 10 values of NP that are straights and begin with a 4: (4 12 11 10 9 8) (4 11 10 9 8 7) (4 10 9 8 7 6) ... (4 5 4 3 2 1) (4 4 3 2 1 0) . Each of these are the usual 10 straights, but they occur in 4*4*4*4*4 combinations of suits with the ones that are also flushes removed.

There are 54912 values of H that produce 858 values of NP that begin with a 3 representing three-of-a-kinds: (3 12 11 10 9 8) (3 12 11 9 10 8) (3 12 11 8 10 9) ... (3 0 3 1 12 11) (3 0 2 1 12 11) The 858 different hands are 13 denominations of triples multiplied by 2!12 combinations of single cards.

There are 123552 values of H that produce 858 values of NP that have two pairs and therefore a CV that begins with a 2: (2 12 11 10 9 8) (2 12 11 9 10 8) (2 12 11 8 10 9) ... (2 1 0 3 12 11) (2 1 0 2 12 11) The 858 different hands, in this case, are 2!13 combinations of pairs multiplied by 11 possible single cards.

There are 1098240 values of H that produce 2860 values of NP that begin with a 1: (1 12 11 10 9 8) (1 12 11 10 8 9) (1 12 11 10 7 9) ... (4 0 4 2 1 12) (4 0 3 2 1 12) The 2860 possibilites come from 13 possible pairs times 3!12 single cards.

There are 1302540 values of H that produce 1277 values of NP that begin with a 0: (0 12 11 10 9 7) (0 12 11 10 9 6) (0 12 11 10 9 5) ... (0 5 4 3 2 0) (0 5 3 2 1 0) These values are very difficult to characterize,

Counting out the hands

Here is a summary of the basic hands in a poker game. In this table, we calculate the number of hands in each category as well as the number of different values within category.

One CV Inconvenience

Notice that there is no particularly convenient simple integer mapping of the values of CV . The 6-tuple is by no means `dense'. For some computational purposes it is convenient to have an integer represent the `value' of a hand, and until we find better, we will simply use the value obtained by interpreting the CV 6-tuple as though they the were `digits' in a base-13 number. Of course, given the way that we have generated CV , there are some numbers in the [0...(13^6)-1] = [0...4826808] which do not reprsent the legal CV of any real NP hand. It remains to be seen if we will discover a more useful mapping of CV into a dense integer space. If we can, quite obviously that space can be quite small as V:[0...7461] will do the job.

Adding Wild Cards to the Mix

The `rules' of poker get much more complicated when wild cards are added to the mix. There are several levels of possible `rules' that govern wild-card play. Let's start with the simple `rules' first.

What can be `Wild'?

Any collection of cards C can be decreed to be `wild'. Probably the most common collection is [2C,2D,2H,2S] [the so-called `Deuces Wild']. Another common collection is [JH,JS,KD] [the so-called `One-Eyed Jacks and Kings with the Axe'] Games with `Baseball' in the title often involve wild collections of 3s and 9s (three strikes, nine innings in baseball) with other special rules governing 4s (base on balls).

What can `Wild Cards' be?

The basic rule of `wild cards' is: A Wild Card can be designated to be any card not already in the hand. It can also be designated to be a card of an unspecified suit in a particular denomination, but only for the purpose of making a hand that consists of five cards of that denomination. Notice that, in this (normal) formulation, the only limitation on what values a wild card can assume are imposed by what is already in the hand. This does not involve other cards that happen to be in other hands.

No `option' on call in NP

In NP there is no `option' about the call of any wild card. Mathematically this is fairly easy to state if we allow ourselves to develop some representations for the concepts involved: Assume a collection (list) of wild cards (W1,...,Wn) . Define NW(CH):[i-tuple of C] which is an i-tuple containg the`non-wild' cards in CH , and W(CH):[5-i tuple of C] which is a 5-i tuple containg the wild cards. Create Gen(NW):[List of 5-i tuples of possible values of the wild cards] . Notice several things about Gen . First, it is only a function of the non-wild cards. Given that NP is a 5 card game, knowing the non-wild cards tells us the number of wild cards. Second, wild cards cannot represent cards already in the hand, so we need to know what they are, but wild cards can, and sometimes do, stand for themselves, so we don't need to know what they are. They will, properly, be in the lists of combinations created by Gen . Finally, it is easy to calculate the size of the list produced by Gen . It will be (5 - i)!(52 - i) elements.

We can now create a function Compose(NW) which makes a list of all possible hands, L , which can be generated from a hand that contains non-wild cards NW . Call this list L . This can be ordered into a monotone non-decreasing list by S =: L/:CV each L . We must choose to make our hand (any one of) the best hands so _1{S will do.

Impact on CV

Adding wild cards clearly implies an important change to CV . Certain hands get promoted to a new category `five of a kind'.This creates 12 or 13 new values in CV and V. It is interesting to note, and we will comment on it further later, that the fact we cannot say whether it creates 12 or 13 new values is a harbinger of the complexity that is introduced by the simple act of creating wild cards.

Near Poker is High-Hand Poker

Up to this point what have called `Near Poker' is actually the real game of poker, if one plays `high hand only'. There is no equivocation in the evaluation or ranking of any hand. We do not need to know anything about what kind of game is being played. We never need consider an `alternative' value for any card, with the sole exception that we choose to admit of the sequence 5-4-3-2-A as a `special case' of straight/straight-flush. We can count every hand, and we have a universal mapping from H (or NE ) into CV (or, by table, V ).

Complications of Low-Hand Poker

Low hand poker is almost the `reverse' of High hand poker. If this were the true reverse, then we would already know enough to handle Low-Hand Poker. We could say: If you'd win at high-hand poker, you'd lose at low hand poker. Notice that this still leaves `a game' if there are wild cards involved. A wild card, or group of wild cards, can be construed `most favorably' for high hand treatment, and `least favorably' for low hand treatment. But, apparently, Low-Hand poker is more complicated than this and requires its own set of ranking computations and rules. And it is in the treatment of the `Ace' that all of the complications ensue.

How can Aces be characterized

It is at this point that I have to pause in ignorance. The issue has to do with how to characterize, in some general way, the `role' of Aces in a `Low Hand'. I await any intelligent exposition. It seems to me, naively, that there is a `High Ace/Low Ace' characterization (depending on what?) or perhaps some other characterization.

Low-Hand `philosophical' questions

Do the `Wild Card' rules need to be adjusted (other than the substitution of `low' for `high').

The argument for the `simple' Low Ace

There is an argument for a very simple rule involving low aces. This rule specifies that a hand that contains 5-4-3-2-A is treated as a straight, no matter what the larger world considerations are. With this rule it is very easy to implement the necessary changes to compute CV , and no complications in determing the ranking of any hands ensue.

Any more complex treatment of the Ace substantially complicates the issues about ranking. The nature of these complications differ depending on the actual rules about the treatment of the Ace. Therefore they have to be discussed in detail when those rules are established.

Some `philosophical' concerns

I think that is it is an unfortunate fact that the rules of conventional poker allow the objective of the game (i.e. High/Low) to enter into the issue of how the hands are mapped into V . This is particularly true if the mapping of H:V in the `high game' is almost identical to the mapping of H:VLOW in the `low game'. This materially complicates the issue how the various parts of the game fit together.

Now notice here that there is actually a quantification of the concept of `almost identical', particularly if V=VLOW . We surely know than many of the mappings in V and VLOW are the same. For example, all hands which do not contain Aces (and perhaps wild-cards)

Computing V

So far we have not found a straightforward way to construct V . However, there is much that we can say about it even failing that. Indeed, we can make some case arguments that will let us determing a large number of the values, and move towards a function that will determine the value of V(H) .

To develop the evaluation rules, we will base the computations on the D of the cards in each hand. We also need some functions to help evaluate the combinations of 2, 3, 4 and 5 cards that are the hands.

We approach this problem by first defining a set of reduction functions that perform a nattering, but necessary, function in our enumerations. We'll start with Rdc(D1,D2) read as the reduction of D2 with respect to D1 . It is defined quite simply with Rdc(D1,D2) = D2 - (D2 > D1) . This is useful because we are very commonly in the situation where D1 can be any denomination, and then D2 can be any denomination other than D1 . But we want to number it on the [0...11] space rather than the [0...12] space, but with D1 left out.

There are Rdc functions of more arguments. Rdc(D1,D2,D3) = D3 - (D3 > D2) - (D3 > D1) , and again this give us values for D3 that are based on the [0...10] space, leaving out values for D1 and D2 . As an example if D1 = 5 and D2 = 8 , then values for Rdc{D1,D2,D3) would map [0 1 2 3 4 6 7 9 10 11 12] into [0,1,2,3,4,5,6,7,8,9,10] .

We can define Two(D1,D2):[0...155] , a function of the denominations of two cards. This function maps the 13 * 12 combinations into the integers in proper order. Two(D1,D2) =: Rdc(D1,D2) + 13 * D1

We will also need Three(D1,D2,D3):[0...857] . This function is a little more complicated than Two ,

For four of a kind we need N4 , the denomination of the four-of-a-kind, and N1 the denomination of the single card. For these hands V = 7296 + Two(N4,N1) .

For full houses we need N3 , the denomination of the triplet, and N2 the denomination of the pair. For these hands V = 7140 + Two(N3,N2) .

For three of a kind V = 4995 + Triple(D(three),D(higher single),D(lower single) .

Recounting the hands in `Deuces Wild'

Here is a summary of the hands in a poker game that are obtained when we allow all four of the deuces to be wild. In this table, we calculate the number of hands in each category as well as the number of different values within category. This particular table was generated by using enumeration, and running a computer for several hours (10h31m on a Pentium 200mhz) to generate and tabulate each and every possible hand.

Observations

Perhaps the most interesting immediate observation is not the expected upward drift in the values of the hands, but the rather substantial decline in the number of different rankings which can be generated. 1300 different ranks of hands have simply disappeared. I suppose the simplest example is all of the hands in regular poker which have solitary deuces in them have been promoted in this accounting into something at least in the one pair category, and perhaps higher.

It will be an interesting exercise to see if there is any broad characterization that we can make of the hands that disappeared.

We should probably also discuss whether the V' that describes the ranking space of `deuces wild' hands should be reorganized into a mapping of the V':[0...5161] variety, or if it should be left to cover the same integer space as V , but now with some known holes.

Type	Number	From	To
Five of a Kind	No Values	No Values	No Values
Straight Flush	10	7452	7461
Four of a Kind	156	7296	7451
Full House	156	7140	7295
Flush	1277	5863	7139
Straight	10	5853	5862
Three of a Kind	858	4995	5852
Two Pair	858	4137	4994
One Pair	2860	1277	4136
No Pair	1277	0	1276

Hand	Number	Number of Distinct
Five of a Kind	672	12
Straight Flush	2552	9
Four of a Kind	31552	132
Full House	12672	132
Flush	14472	784
Straight	62232	9
Three of a Kind	355080	660
Two Pair	95040	660
One Pair	1225008	1980
No Pair	799680	784
Total	2598960	5162

From Cards to Near Poker to Poker David Ness

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Parsing the world of Poker