Poker stats.

Hola members.

In traditional poker, the various hands and their order is as follows (as is probably well known):
* High Card
* One Pair
* Two Pairs
* Three of a Kind
* Straight
* Flush
* Full House
* Four of a Kind
* Straight Flush

Supposing one were to play with say X decks of cards shuffled together.
I would intuitively assume, that the order of the hands is the same as above except that Five of a Kind will be added as the highest hand.
(For instance a hand like 5c 5c 5c 6c 6c will be interpreted as a Full House, 5c 5c 6c 7c 7c a Flush, 5c 5c 6s 7c 7c as Two Pairs etc.)
Is this assumption correct?

Thank you in advance for any clarification.
Tom

I would say that Five of a Kind would be lower than a Straight Flush, as that is probably harder to achieve.

It is questionable whether multiple of the 'same' card (such as 5c 5c 5c) should rank more highly than an equivalent (5c 5d 5s).

I could compute/tell you how to compute the probability that a randomly selected 5-card hand using X standard decks is a five of a kind/full house/etc. I don't know how relevant those computations would be to real-world poker (there are a lot of variants, too). Figuring in details like how people bet and strategize based on the cards they can see would be extremely difficult, so a quick-and-dirty computation is probably as good as you're going to get.

Quote:

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Originally Posted by si_the_geek

I would say that Five of a Kind would be lower than a Straight Flush, as that is probably harder to achieve.

It is questionable whether multiple of the 'same' card (such as 5c 5c 5c) should rank more highly than an equivalent (5c 5d 5s).

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If you're sure that 5 of a Kind is below a Straight Flush, I'll add it there instead.
I doubt, I'll implement multiple versions of pairs, tripples, houses, quads or quintets based on suit though. I need the game to be as similar to Texas Hold'em (or other poker variations) as possible.

Quote:

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Originally Posted by jemidiah

I could compute/tell you how to compute the probability that a randomly selected 5-card hand using X standard decks is a five of a kind/full house/etc. I don't know how relevant those computations would be to real-world poker (there are a lot of variants, too). Figuring in details like how people bet and strategize based on the cards they can see would be extremely difficult, so a quick-and-dirty computation is probably as good as you're going to get.

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I am not going to show probabilities while flops, turns and rivers are displayed (as is traditional in Texas Hold'em in an allin), since the game I am designing has some complexities connected to that approach (similar but a bit more difficult than the problem of predicting odds for Omaha hands on the fly). I have thought about a couple of ways to decrease the complexity, by distributing decks and cards in specific ways but I'm a bit concerned that randomness might be affected. I'm going to create a post here about just that in the near future, if I decide to go down that road.
If you have some links or material on how to implement an AI for a Texas Hold'em computer-player (preferably C#, java or VB), that would be highly appreciated; though I'm nowhere near implementing that just yet. The same for other poker variants, especially Omaha.

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Thank you both for your help
Will keep this open for a little while longer, though I doubt anyone can elaborate further without knowing my design in depth (which I'm not going to share at this time).

Regards Tom

Just to comment on the five of a kind vs. straight flush question, given X decks, the number of five-card hands which are five-of-a-kind is just (4X choose 5), since there are 4X fives to choose from. The number of straight flushes is 40*X^5---choose a suit (4 ways); choose a low card from ace to 10 (10 ways); for each of the five cards, pick one of its versions from one of the decks (X ways 5 times). The binomial coefficient expands to (128 X^5)/15 - (64 X^4)/3 + (56 X^3)/3 - (20 X^2)/3 + (4 X)/5. The leading coefficient is 8.533..., so for large X there are a factor of 40/8.533... = 4.6875 more straight flushes than five-of-a-kind's. For small X, this ratio is even larger.

So, by this measure, si was correct.

You forgot to choose rank (deuce to ace) on five-of-a-kind. There are 13 * (4X choose 5) five-of-a-kind in X decks. This is fewer than the number of straight flushes when playing with two decks, but greater with three or more. That is to say, five-of-a-kind outranks straight flush with two decks, but comes in second otherwise.

Another thing to consider is that a straight outranks a flush when playing with more than four decks. (Assuming I counted correctly.)

There may be other, less intuitive changes in the ranking.

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Originally Posted by Logophobic

You forgot to choose rank

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How silly, thanks!

It seems I have been moving on a slippery slope. Those very informative posts have convinced me, that no poker-player would ever play a game with more decks than one.

Two points especially:
1) The fact that hands aren't ordered the same way for any number of decks will cause confusion.
2) The fact that 5a5a intuitively will be considered higher than 5a5b will cause requests for a multitude of different hands (multiple pairs, trips, houses etc.).

But thanks for all your help anyways; at least you spared me the time programming it all only to discard it later.

Regards
Tom

To Wikipedia!!!

And yes, poker players often do play variants, though not often at actual casinos (casinos prefer simple standardized games that require very little rules interpretations, in order to avoid confusion and arguments, which slow the game down and thus reduce the house's take). At house games, however, folks often play a variety of games, chosen by the rotating dealer. These often include wild cards, which can lead to five-of-a-kind hands. The exact structure of a particular game can flip the probabilities of one particular hand... in some games straights are harder to get than flushes, for example. Nevertheless, the ranking of hands almost never changes, as it is easier to remember that a flush outranks a straight than it is to program a simulator every time the next dealer invents a variant. And so it goes that five-of-a-kind is the highest valued hand unless specifically stated otherwise.

Interestingly, the biggest source of confusion is if one is going for the lowest hand (common in high/low games). In such a case, there are two common sources of conflict that need to be specified before the deal:
1. Does an ace count as a one (lowest card) or a thirteen (highest card)
2. Is the lowest hand A-2-3-4-5 (some say no, because that would be a straight) or A-2-3-4-6 (substitute 2-3-4-5-6 and 2-3-4-5-7 if the ace is always high)?

Also, one should specify whether a flush counts as a high hand when playing low.