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# Poker scoring system

Here is a puzzle I've been thinking of for a while. Assume that there are five people playing "five-card draw" poker ( http://en.wikipedia.org/wiki/5_card_draw ) using all of the 52 cards in the set. Each is given 5 cards, then they are allowed to either bet or fold. After they all agree on an amount of bet, they change from 0 to 5 cards of their hands with those coming from the remaining cards in the stack. In the end, the player with the best combination wins. ("Best", here, means a hand with lowest probability.)

If there was no "draw" phase, then the probability of each combination would be as described here: http://en.wikipedia.org/wiki/List_of_poker_hands

But because there is a draw phase where people can change upto 5 cards of their hands, (here is the question) would the probabilties change? For example, (and as some poker players say) would this draw phase cause a "flush" to happen less often than a "full house", and so make the flush a "better" hand? ("Better" is defined as less probable, again)

I really am stock with the calculations. As an alternative, I thought of writing a program which would use Monte Carlo simulation to assess this hypothesis, but I got stock again!

Please advise
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huji
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4 Solutions

Commented:
This is very difficult (if not impossible) to put into numbers.

What you can be sure about (assuming the players have a certain level of experience with the game), is that due to the draw phase, the probabilities of the higher hands will be raised at the cost of the probabilities of the lower hands. The reason is that a player will be more inclined to change cards when he has a bad hand, giving him the chance to get a better hand. However, as the chance of getting a better hand than the one the player currently has is getting lower, he will be less inclined to change cards. So, on average (again, assuming a certain experience of the players), the hands will get better due to the draw phase. How much better is difficult to say.
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Commented:
Not to mention that it will also depend on the players (whether they like to take chances or prefer to play safe, whether they like bluffing or not).
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Author Commented:
The question is if the increased/dicrease in the probablities will cause a "different" rank for hands (like Flush going above Full House, or Three going above Straight, etc)
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Author Commented:
Oh, and disregard the players. Suppose that they'll change the cards only randomly.
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Commented:
What you could calculate for example is the odds of getting a better hand, given your current hand, and the number of cards you change.

Theoretically, you could perform that calculation for every possible hand, and every possible amount of cards switched. This would give you probabilities for the final hands. However, they will be inaccurate, because there's one piece of information that's missing : the probability that a player will decide to change cards and how many. And that probability is not bound by rules, so very difficult to put into numbers (as I said ;) )
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Author Commented:
Yeah, that's exactly when I really don't know where to go. All I know is, some poker players "do" change the order of the hands when playing five-card draw in small groups or with a proportion of the cards (like from 6 to Ace)
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Commented:
If they'll change the cards only randomly, then the odds do not change,
If they play with a particular strategy, then the odds can change depending on the strategy,
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Commented:
Yeah,
Treat odds the same, or it is all poppyconck.
> Oh, and disregard the players. Suppose that they'll change the cards only randomly.
> After they all agree on an amount of bet, they change from 0 to 5 cards of their hands with those coming from the remaining cards in the stack. In the end, the player with the best combination wins.
> they are allowed to either bet or fold

This does not compute.
Where people are involved, it is not random, some may bluff and some with a good hand will drop out while others will insist on drawing to an inside straight.

A person with four kings can fold when another in one game has four queens and in another game another person has four Aces.

In addition to strategy, random does not account for inside knowledge either, such as where a person had three queens, then tossed a king and an Ace and picked up another Queen and Ace.

When dealing with 'random', one can exchange four Aces for four Deuces

I agree with the other expert comments above

> What you could calculate for example is the odds of getting a better hand, given your current hand, and the number of cards you change.

Another point missed there is that there are other 'givens' that the human has, that 'random' does not. Suppose you have a pair of tens, or jacks, (or whatever) and someone else raises heavily before you drop out. You are thinking of drawing three cards. The big raiser goes first. Can you decide what cards to dump differently based upon your observation of how many cards (0-5 is six possibilities) just a one of your opponents discard for swapping? Similar example can be for first bet after they decide to go with their initial five (no new cards).

I have tendency to use inside knowledge when coding programs, or when playing cards. Little help from playing random.
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Author Commented:
>> Where people are involved, it is not random, some may bluff and some with a good hand will drop out while others will insist on drawing to an inside straight.

Yes. But I want to do the calculations based on the assumption that they drop and take the cards "randomly".

Let me put it this way: In some of the most commonly played forms of poker, like 7-card Stud, the order of hands used is the exact order I linked above. They do so, regardless of the fact that during several betting rounds, human players may act non-randomly and cause the odds of each hand to change. They use the probabilities of each hand happening randomly, nothing more!

Now, I want to calculate the probability of each hand coming randomly, in a "draw" game. That, is my question.
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Commented:
>> Yes. But I want to do the calculations based on the assumption that they drop and take the cards "randomly".

I think ozo answered that in http:#21824104, no ?

In any case, it's a very poor strategy to assume that players will change cards randomly. You might as well not have a strategy at all ;)
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Author Commented:
>> If they'll change the cards only randomly, then the odds do not change,

Well, this is exactly what I think. But I want to prove it in a mathematical way.
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Commented:
You start by getting 5 random cards. This first hand is thus randomly chosen, and the probability of having a certain hand can be found in the list you posted earlier.

If you randomly change cards, then the randomness of the resulting hand is not impacted. And the probability of having a certain hand is therefore unchanged (and can be found in the list you posted earlier).

That's the thing with "random" ;)
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Commented:
Ignoring the personal choices a person makes there is still probability in making a certain hand. For example if you were dealt a royal flush and opt to get rid of one card then there is a probability of drawing to high card, one pair, straight and flush. This has nothing to do with personal choice other than it eliminates the probability for certain outcomes.

The overall math doesn't quite work out though. We'll take less complex problem. The probability of getting heads twice in a row. This math would be expressed by multiplying the probability of each seperate instance. In this case 50% X 50%. So the overall probability is 25%. The most important part to keep in mind though is that after you have flipped heads once, the probability of flipping heads again is 50%. So it can be shown with multiple attempts the probability goes down.

The problem with figuring these probabilities is the result can occur without drawing to the hand. We'll take for example a four of a kind. According to your link there is about a .024% chance of getting a 4 of a kind. The theoretical numbers if you had to draw to it when having three of a kind to start would be 2.1% (The number shown in the link of getting a three of a kind) then 2/47 (The chances of getting the last card. 2 since you discard 2 cards and 47 the amount of potential cards remaining. (Naturally there aren't that many cards in the deck left but you can't rule them out) multiplying these together you would get an overall probability of .089%. Now going back a paragraph, once you have been dealt three to start with the probability isn't that bad. You then have 2/47 chance of getting the last card you need to make 4 of a kind.

So like they show odds when you watch poker on tv, before looking at your hand you have a .024% chance of getting four of a kind. After you look at your hand and see 3 of a kind you then have 4.3% chance of getting it. Naturally these numbers change if you didn't get 3 of a kind to start with. Anything less and your chances get reduced. Normally one would think that if you discard all 5 your chances would go up but in that case you would have 5 less for of a kind possibilities so out of 13 total possibilities you almost reduced your chances by half.
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Commented:
Closed? I got no Notify. Obvious discrimination, since the odds change, you did not get them, and now that I reviewed Wiki it seems they were well ignored as well as others here as well. Too bad, come to my table, I beat cheaters too.
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Commented:
>> Closed?

It hasn't been closed yet ;) Maybe you got confused by the new color scheme ? It gave me a scare too initially lol.
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