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I'm trying to figure out how I can work out the number of possible valid hand combinations of a poker game that a player (opponent) could possibly have when the flop has been dealt on the table.

So I'm guessing that I have to take the probability that each hand can occur and then see which one is the highest one that could possibly be?

Let's say that the opponent has bet $10 and now I have to make my next move. Is there some math way, (with use of computer) that I can work out the number of valid combinations of hands that this opponent could still have and which he'll most likely pick?

If anyone could give some pointers as to what I should look into?

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Yes, there is, but it takes some work; you would take into account the cards you can see (your hand, the cards on the table, etc), work out how many combinations are still "out there", and then figure out which one shows up the most. Though I'm not clear what "he'll most likely pick" means. Also: there are multiple variants of poker, you need to specify which variant you are looking at. –  Arturo Magidin Feb 23 '11 at 19:44
    
@Arturo: Texas Hold'em –  Tony The Lion Feb 23 '11 at 20:23
    
As in, "in your question, to make it both clear and self-contained"... –  Arturo Magidin Feb 23 '11 at 20:27
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2 Answers

up vote 2 down vote accepted

The number of hands is $\frac{47\cdot 46}2=1081$ as you see five cards (your two plus the flop) so there are $47$ left to choose from. If you are looking for the highest, there is no probability involved, just look at all of them and find the highest. Ignoring any information from the betting, the chance that an opponent has three of a kind is just (number of holdings that make three of a kind)/1081. For this example, if none of the five cards you see match in value, there are nine hands that make your opponent three of a kind (three pairs of each value that matches one of the flop cards).

None of this speaks to which of the hands are reasonable for your opponent to behave as you have seen so far. There are many books on the subject (most of which give similar advice) that your opponent may well have read, so you can take some guidance there.

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I'm not sure what you mean by this: "None of this speaks to which of the hands are reasonable for your opponent to behave as you have seen so far", could you explain? –  Tony The Lion Feb 23 '11 at 20:24
    
Just that the fact the opponent is betting makes it less likely that he has some garbage hand (tho he could be bluffing). –  Ross Millikan Feb 23 '11 at 20:41
    
The analysis is much, much deaper than this depending on the player's style, if he is competent, what the limits are, if he is on tilt, etc. I don't think a simple statement such as that one is accurate with all do respect –  fdart17 Feb 23 '11 at 20:47
    
@Fdart: I agree. This was a first example –  Ross Millikan Feb 23 '11 at 21:42
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There is no software that can accurately take into consideration what move your opponent has done (i.e., assuming you were in position and he bet before you, take into account his bet or check) and tell you what likely has he has.

There are way too many variables to account for here, including but not limited to, how aggressive, how tight he is playing pre-flop, how likely he is bluffing given the cards if they look scary or not, etc.

That said, the best one can do after the flop is rely on previous experience and their own judgement calls.

There is software that takes into account number of players at the table, chip counts, etc, that will give you the mathematically correct play late in tournaments. This is usually when you are in push or fold mode. If you wish to see how this works, please see some software such as sit and go wizard (sngwiz.com)

As to answering your original question, I don't think the hands a player could possibly has would be useful to you at all, but of course, it is not hard to see what these are based on the cards you see.

EDIT:
In answer to your question about where you should look into, if you are looking to improve your game, which seems to be the case, the best thing you can do in my opinion is sign up at a forum such as 2 plus 2 or any really, that has some good players willing to help you out. Post hand histories and see the players will be more than happy to comment on your play.

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