Omaha Probability Calculations

Hello Everyone,

I am currently trying to figure out the probability of hitting each hand in standard Omaha. I was able to find the probability of WizardofOdds, but after much searching, I cannot find the calculations anywhere, only the final results. I am planning on deriving the results on my own in the near future, but if someone is aware of the derivations, can you share where they were originally derived? This will save me a lot of time and effort if so.

Quote: pgambill

Hello Everyone,

I am currently trying to figure out the probability of hitting each hand in standard Omaha. I was able to find the probability of WizardofOdds, but after much searching, I cannot find the calculations anywhere, only the final results. I am planning on deriving the results on my own in the near future, but if someone is aware of the derivations, can you share where they were originally derived? This will save me a lot of time and effort if so.
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Are you trying to go from any starting hand to every possible final hand, or are you just trying to find the probability of each final hand from a random player hand and random board? The derivation is jus running out the cards and using a fast hand evaluator.

Googling 'Omaha Odds Calculator' will find you tools.

Well, the way I would do it is "brute force."
There are 2,598,960 sets of 5-card hands for the community cards, although if you're not interested in specific suits (e.g. there's no bonus for a Royal Flush in spades), there are only really 134,459 different hands. (For example, all four Royal Flushes are really the same hand.) For each hand, there are 178,365 sets of 4 cards for your hole cards. Now, for each of the 134,459 x 178,365 combinations, determine what the best hand is (you need to check each of the 10 sets of 3 community cards and 6 sets of 2 hole cards).

I mean of the set of all (52 choose 4) * (48 choose 5) possible Omaha hands, how many combinations of cards yield a royal flush, how many yield a straight flush, ect. Once the number of combinations are known, I can compute the probability of a hand chosen uniformly at random having a particular value.

My only worry with using this method is the different hands for the 5 cards on the board are not all equally likely. For example, 23456 all of spades is one of the four possible board with these exact values all of the same suit. For 2d3d4s5c6h, we can permute the suits 4! different ways so we would get 24 possible ways (up to suits) of this occurring. Wouldn't we need to also account for the number of permutations when calculating using a brute force method like the one you described?

Quote: pgambill

My only worry with using this method is the different hands for the 5 cards on the board are not all equally likely. For example, 23456 all of spades is one of the four possible board with these exact values all of the same suit. For 2d3d4s5c6h, we can permute the suits 4! different ways so we would get 24 possible ways (up to suits) of this occurring. Wouldn't we need to also account for the number of permutations when calculating using a brute force method like the one you described?
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If you use 2,598,960 as the number of community sets, then you are already counting all 24 hands of 2a3a4b5c6d. If you are using 134,459, then the results will be the same for all 24 of the permutations; just multiply it by 24.

("But...but...if the 5 cards are 2s 3s 4s 5h 6d, then having 5s 6s in your hole cards makes a straight flush, but if the 5 cards are 2h 3h 4h 5d 6c, then 5s 6s doesn't!" True, but then again, 5h 6h makes the straight flush with the second hand but not the first. The number of each hand you will get over all 178,365 sets of hole cards will be the same for both.)

Quote: pgambill

I mean of the set of all (52 choose 4) * (48 choose 5) possible Omaha hands, how many combinations of cards yield a royal flush, how many yield a straight flush, ect. Once the number of combinations are known, I can compute the probability of a hand chosen uniformly at random having a particular value.
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Okay, since this is poker not video poker, I would have thought you would want to know separately the number of 5-hi straights, the number of 6-high straights, etc. Your classification question is much simpler.

The brute force method ThatDonGuy mentions would take a fraction of a second. But, the OP mentions 'derivations'. I am sure the Wiz used a computer to get the counts. That is not a derivation. It is difficult to provide combinatorial equations for Omaha because you only use three cards from the board. I could provide tips if you want to code the brute force algorithm yourself. I could also calculate the answer quickly, but you say you already have the answer.