In order to have a valid low hand in Omaha, two conditions must be met. First, the final board must have at least three unique low cards. A low card is an Ace, Deuce, Trey, Four, Five, Six, Seven or Eight. By unique, we mean non-duplicated cards of differing rank. For example:
|A99QK||invalid board: only one unique low card|
|A4JJQ||invalid board: only two unique low cards|
|A37JK||valid board: three unique low cards|
|A347Q||valid board: four unique low cards|
|A3568||valid board: five unique low cards|
|A337J||valid board: three unique low cards|
|A449K||invalid board: only two unique low cards|
|AA225||valid board: three unique low cards|
Note that just because the board shows three or more unique low cards, that does not mean that there will be a guaranteed hand that wins low. The second condition takes into consideration a player's hand, and is a bit trickier to describe. First, a player must have at least two unique low cards in his hand. Second, he must be able to select cards (two from his hand and three from the board) in a way that gives him a total of five unique low cards. Here are some hand examples:
|A39J||valid hand: two unique low cards|
|A24K||valid hand: three unique low cards|
|A55Q||valid hand: two unique low cards|
|AA23||valid hand: three unique low cards|
|A457||valid hand: four unique low cards|
|299K||invalid hand: only one unique low card|
|44JQ||invalid hand: only one unique low card|
With these ground rules set, we can now go about determining various probabilities of achieving a low hand. First, we will look at the different board combinations. There are 52 cards in a standard deck: 20 of them are "high" (Nine through King), and 32 are "low". Many of the charts I've seen in books and on the web show probabilities and figures based on all 52 cards being unknown. However, a player has additional information that should be considered in all of his calculations -- namely, the four cards he holds in his own hand. He knows what they are, therefore they should be taken into account when determining actual probabilities.
If you have four cards in your hand, then 48 remain as potential board cards. Let us assume we hold an "LLHH" hand; that is, two unique low cards and two high cards. Example hands might be A2QK or 58JJ. Having a "LLHH" hand means there are now 18 high cards and 30 low cards left unseen.
Next, we consider the C(48,5) possible board combinations. There are actually six different high/low board variations:
|HHHHH||all high cards|
|HHHHL||one low card|
|HHHLL||two low cards, not necessarily unique|
|HHLLL||three low cards, not necessarily unique|
|HLLLL||four low cards, not necessarily unique|
|LLLLL||five low cards, not necessarily unique|
Remember, that order does not matter for board combinations, so "HHLLL" is the same as "LLHLH". We can enumerate the total for each board variation as such:
|HHHHL||C(18,4) x C(30,1)||91,800|
|HHHLL||C(18,3) x C(30,2)||354,960|
|HHLLL||C(18,2) x C(30,3)||621,180|
|HLLLL||C(18,1) x C(30,4)||493,290|
Note that for the moment, we are not restricting ourselves that all the low cards be unique. In other words, in the above combinations, we are counting a board such as K2222 as having four low cards (even though no lows are possible with such a board). The tricky part will be to determine which board combinations have three or more unique low cards that give actual lows when holding a "LLHH" hand.
Three of the six board variations are such that no low is possible at all: namely, boards HHHHH, HHHHL, and HHHLL. For the other three, not all of them make for a possible low either (i.e. boards such as K2222 and 445JQ get thrown out). And finally, you must consider what your hole cards are, so you don't get counterfeited. 2349J is a great board for low, unless you happen to be holding A2QK. With that hand, you have no low, even though the board has three unique low cards showing. The Deuce on the board counterfeited the one in your hand. Our goal will be to remove all boards such as the those as well.
Let's use an example hand of A2QK, with the understanding that we will get exactly the same results using similar "LLHH" hands (such as 58JJ or 349K). First, we'll tackle the HHLLL board. If you hold A2QK, you will have valid low with such a board if and only if the three low board cards are a Trey, Four, Five, Six, Seven or Eight, with no duplication. Counting all unseen Treys through Eights give us 24 low cards that help us.
To enumerate these, select your first low card (24 choices), then your second unique low card (20 choices left), and finally your third unique low card (16 choices left). Multiply these together, but then be sure to divide by 3! since order does not matter.
So out of the 621,180 possible HHLLL boards, only C(18,2) x 1280, or 195,840 of them give a player holding A2QK a valid low. By using similar methods for the remaining HLLLL and LLLLL boards, we can likewise determine which combinations give valid lows for A2QK hands. However, they are a bit more trickier to compute. I used enumeration via a computer program to come with the board totals, since all the combinatorics are a pain to deal with. Let's walk through the LLLLL boards to show why. We count the total number of unique low cards, using both the board and our hole cards. Remember, our hand is A2QK.
|Enumeration of all possible LLLLL boards|
|46,080||6||Yes||A4567 or 33568|
|66,240||5||Yes||33356 or 33448 or A2567|
|22,560||4||No||A2335 or A2257|
|1,476||3||No||AA228 or A2444|
|6||2||No||AA222 and AAA22|
Adding up the above valid low boards, we get 6,144 + 46,080 + 66,240, or 118,464 total LLLLL boards which give the A2QK player a valid low. My using the same methodology, I was able to also determine that out of the 493,290 HLLLL boards, the A2QK player has a valid low 311,040 times. Adding them up, we get
We can now finally conclude that for a player holding a "LLHH" hand, if they stay all the way to the river, they will have an actual valid low hand 625,344 out of 1,712,304 times, or 36.52% of the time.
I ran the same calculations using other different starting hands. Here are my findings:
Remember, though, that obtaining a valid low hand and winning with that low hand are two different things. A234 and 5678 are both "LLLL" hands, and both have a 53% shot of making a low hand. However, A234 is clearly the better hand to hold, since if do indeed make your low, you have the "nuts" and are guaranteed half the pot (assuming you don't tie with any other lows).