Poker probability (Omaha)

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In poker, the probability of many events can be determined by direct calculation. This article discusses how to compute the probabilities for many commonly occurring events in the game of Omaha hold 'em and provides some probabilities and odds^{[note 1]} for specific situations. In most cases, the probabilities and odds are approximations due to rounding.

When calculating probabilities for a card game such as Omaha, there are two basic approaches.

Determine the number of outcomes that satisfy the condition being evaluated and divide this by the total number of possible outcomes.
Use conditional probabilities, or in more complex situations, a decision graph.

Often, the key to determining probability is selecting the best approach for a given problem. This article uses both of these approaches, but relies primarily on enumeration.

[edit] Starting hands

The probability of being dealt various starting hands can be explicitly calculated. In Omaha, a player is dealt four down (or hole) cards. The first card can be any one of 52 playing cards in the deck; the second card can be any one of the 51 remaining cards; the third and fourth any of the 50 and 49 remaining cards, respectively. There are 4! = 24 ways (4! is read "four factorial") to order the four cards (ABCD, ABDC, ACBD, ACDB, ...) which gives 52 × 51 × 50 × 49 ÷ 24 = 270,725 possible starting hand combinations. Alternatively, the number of possible starting hands is represented as the binomial coefficient

${52 \choose 4} = 270,725$

which is the number of possible combinations of choosing 4 cards from a deck of 52 playing cards.

The 270,725 starting hands can be reduced for purposes of determining the probability of starting hands for Omaha—since suits have no relative value in poker, many of these hands are identical in value before the flop. The only factors determining the strength of a starting hand are the ranks of the cards and whether cards in the hand share the same suit. Of the 270,725 combinations, there are 16,432 distinct starting hands grouped into 16 shapes. Throughout this article, hand shape is indicated with the ranks denoted using uppercase letters and suits denoted using lower case letters. For example, the hand shape XaXbYaYc is any hand containing two pair (XX and YY) that share one suit (a), but not the other suits (b and c). The 16 hand shapes can be organized into the following five hand types based on the ranks of the cards.

Rank type	Shapes	Distinct hands	Combos	Probability	Odds
XXXX: Four of a kind	1	13	13	0.0000480	20,824 : 1
XXXY: Three of a kind	2	312	2,496	0.00922	107 : 1
XXYY: Two pair	3	234	2,808	0.0104	95.4 : 1
XXYZ: One pair	5	5,148	82,368	0.304	2.29 : 1
XYZR: No pair	5	10,725	183,040	0.676	0.479 : 1
TOTAL	16	16,432	270,725	1.0	0 : 1

There are also five types of hands based on the suits of the cards that mirror the five rank types: aaaa, aaab, aabb, aabc, and abcd. The following are the probabilities and odds of being dealt each suit type.

Suit type	Shapes	Distinct hands	Combos	Probability	Odds
aaaa	1	715	2,860	0.0106	93.7 : 1
aaab	2	3,718	44,616	0.165	5.07 : 1
aabb	3	3,081	36,504	0.135	6.42 : 1
aabc	5	7,098	158,184	0.584	0.711 : 1
abcd	5	1,820	28,561	0.105	8.48 : 1
TOTAL	16	16,432	270,725	1.0	0 : 1

Unlike the rank types, the suit types can be absolutely ranked in terms of starting hand value because suits in poker hands only factor in flushes and straight flushes. From best to worst starting hand value the suit types are: aabb, aabc, aaab, aaaa, and abcd.

The relative probability of being dealt a hand of each given shape is different. The following shows the probabilities and odds of being dealt each shape of starting hand.

Rank type	Hand shape	Distinct hands		Suits for each hand		Hand combos	Dealt specific hand		Dealt any hand
Rank type	Hand shape	Derivation	Number	Derivation	Combos	Hand combos	Probability	Odds	Probability	Odds
Four of a kind	XaXbXcXd	$\begin{matrix} {13 \choose 1} \end{matrix}$	13	$\begin{matrix} {4 \choose 4} \end{matrix}$	1	13	0.00000369	270,724 : 1	0.0000480	20,824 : 1
Three of a kind	XaXbXcYa	$\begin{matrix} {13 \choose 1}{12 \choose 1} \end{matrix}$	156	$\begin{matrix} {4 \choose 3}{3 \choose 1} \end{matrix}$	12	1,872	0.0000443	22,559 : 1	0.00691	144 : 1
Three of a kind	XaXbXcYd	$\begin{matrix} {13 \choose 1}{12 \choose 1} \end{matrix}$	156	$\begin{matrix} {4 \choose 3} \end{matrix}$	4	624	0.0000148	67,680 : 1	0.00230	433 : 1
Two pair	XaXbYaYb	$\begin{matrix} {13 \choose 2} \end{matrix}$	78	$\begin{matrix} {4 \choose 2} \end{matrix}$	6	468	0.0000222	45,120 : 1	0.00173	577 : 1
	XaXbYaYc	$\begin{matrix} {13 \choose 2} \end{matrix}$	78	$\begin{matrix} {4 \choose 2}{2 \choose 1}^2 \end{matrix}$	24	1,872	0.0000887	11,279 : 1	0.00691	144 : 1
	XaXbYcYd	$\begin{matrix} {13 \choose 2} \end{matrix}$	78	$\begin{matrix} {4 \choose 2} \end{matrix}$	6	468	0.0000222	45,120 : 1	0.00173	577 : 1
One pair	XaXbYaZa	$\begin{matrix} {13 \choose 1}{12 \choose 2} \end{matrix}$	858	$\begin{matrix} {4 \choose 2}{2 \choose 1} \end{matrix}$	12	10,296	0.0000443	22,559 : 1	0.0380	25.3 : 1
	XaXbYaZb	$\begin{matrix} {13 \choose 1}{12 \choose 2} \end{matrix}$	858	$\begin{matrix} {4 \choose 2}{2 \choose 1} \end{matrix}$	12	10,296	0.0000443	22,559 : 1	0.0380	25.3 : 1
	XaXbYaZc	$\begin{matrix} {13 \choose 1}{12 \choose 2}{2 \choose 1} \end{matrix}$	1,716	$\begin{matrix} {4 \choose 2}{2 \choose 1}^2 \end{matrix}$	24	41,184	0.0000887	11,279 : 1	0.152	5.57 : 1
	XaXbYcZc	$\begin{matrix} {13 \choose 1}{12 \choose 2} \end{matrix}$	858	$\begin{matrix} {4 \choose 2}{2 \choose 1} \end{matrix}$	12	10,296	0.0000443	22,559 : 1	0.0380	25.3 : 1
	XaXbYcZd	$\begin{matrix} {13 \choose 1}{12 \choose 2} \end{matrix}$	858	$\begin{matrix} {4 \choose 2} \times 2! \end{matrix}$	12	10,296	0.0000443	22,559 : 1	0.0380	25.3 : 1
No pair	XaYaZaRa	$\begin{matrix} {13 \choose 4} \end{matrix}$	715	$\begin{matrix} {4 \choose 1} \end{matrix}$	4	2,860	0.0000148	67,680 : 1	0.0106	93.7 : 1
	XaYaZaRb	$\begin{matrix} {13 \choose 4}{4 \choose 3} \end{matrix}$	2,860	$\begin{matrix} {4 \choose 1}{3 \choose 1} \end{matrix}$	12	34,320	0.0000443	22,559 : 1	0.127	6.89 : 1
	XaYaZbRb	$\begin{matrix} {13 \choose 4}{4 \choose 2} \div 2 \end{matrix}$	2,145	$\begin{matrix} {4 \choose 2}{2 \choose 1} \end{matrix}$	12	25,740	0.0000443	22,559 : 1	0.0951	9.52 : 1
	XaYaZbRc	$\begin{matrix} {13 \choose 4}{4 \choose 2} \end{matrix}$	4,290	$\begin{matrix} {4 \choose 1}{3 \choose 2} \times 2! \end{matrix}$	24	102,960	0.0000887	11,279 : 1	0.380	1.63 : 1
	XaYbZcRd	$\begin{matrix} {13 \choose 4} \end{matrix}$	715	$\begin{matrix} 4! \end{matrix}$	24	17,160	0.0000148	67,680 : 1	0.0634	14.8 : 1

[edit] Starting hands for straights

In addition to the rank type and suit type of a starting hand, each starting hand also has a sequence type that is useful for estimating the possibility of improving to a straight or straight flush. The sequence type is based on the sequential proximity of the ranks in the hand—the number of different ranks in the hand that can be combined to fill a straight on the board. The ace is a special case in the sequence type because it can be either high or low (i.e. can make a straight with A-2-3-4-5 or 10-J-Q-K-A), and is also both the high and low card in the rank sequence from which straights are formed: A-2-3-4-5-6-7-8-9-10-J-Q-K-A.

The sequence type of the hand is only relevant in determining the probability of making a straight or straight flush. In order to make a straight, exactly three community cards must be combined with exactly two cards from the starting hand. Thus the sequence shape of a hand is the number of different combinations of three cards that can make a straight when combined with two cards from the hand. There are 20 different sequence shapes ranging from hands like 2-2-8-K that can't make a straight (0 straight combinations) to hands like 8-9-10-J that can make a straight with 20 different combinations of three ranks (5-6-7, 6-7-8, 6-7-9, 6-7-10, 7-8-9, 7-8-10, 7-8-J, 7-9-10, 7-9-J, 7-10-J, 8-9-Q, 8-10-Q, 8-J-Q, 9-10-Q, 9-J-Q, 9-Q-K, 10-J-Q, 10-Q-K, J-Q-K, Q-K-A). The 20 sequence shapes can be organized by the number of ranks in the starting hand. This is similar to the rank type of the hand, the only difference being that both the rank types two pair (XXYY) and three of a kind (XXXY) have two ranks. The following table shows the four sequence types based on the number of distinct ranks in the starting hand.

Distinct ranks	Shapes	Distinct hands	Combos	Probability	Odds
1	1	13	13	0.0000480	20,824 : 1
2	5	546	5,304	0.0196	50.0 : 1
3	12	5,148	82,368	0.304	2.29 : 1
4	18	10,725	183,040	0.676	0.479 : 1
TOTAL	36	16,432	270,725	1.0	0 : 1

Note that the table above shows 36 sequence shapes because although there are only 20 different sequence shapes, the sequence shapes overlap with the sequence types. For example, the sequence shape where two different combinations of three cards can make a straight occurs for hands with two ranks (e.g. 3-6 makes a straight with 2-4-5 or 4-5-7), three ranks (e.g. 2-J-A makes a straight with 10-Q-K or 3-4-5), and four ranks (e.g. 3-9-K-A makes a straight with 2-4-5 or 10-J-Q).

The relative probability of being dealt a hand of each sequence shape is different. The following shows the probabilities and odds of being dealt starting hands of each sequence shape.

Sequence shape	Distinct hands by ranks in hand				Distinct hands	Combos by ranks in hand				Total combos	Probability	Odds
Sequence shape	1 rank	2 ranks	3 ranks	4 ranks	Distinct hands	1 rank	2 ranks	3 ranks	4 ranks	Total combos	Probability	Odds
0	13	224	72		309	13	2,176	1,152		3,341	0.01234	80.03 : 1
1		112	468		580		1,088	7,488		8,576	0.03168	30.57 : 1
2		91	1,314	240	1,645		884	21,024	4,096	26,004	0.09605	9.41 : 1
3		70	972	480	1,522		680	15,552	8,192	24,424	0.09022	10.08 : 1
4		49	1,278	1,290	2,617		476	20,448	22,016	42,940	0.15861	5.30 : 1
5			108	1,980	2,088			1,728	33,792	35,520	0.13120	6.62 : 1
6			108	1,380	1,488			1,728	23,552	25,280	0.09338	9.71 : 1
7			396	1,320	1,716			6,336	22,528	28,864	0.10662	8.38 : 1
8			72	885	957			1,152	15,104	16,256	0.06005	15.65 : 1
9			216	870	1,086			3,456	14,848	18,304	0.06761	13.79 : 1
10			36	660	696			576	11,264	11,840	0.04373	21.87 : 1
11			108	720	828			1,728	12,288	14,016	0.05177	18.32 : 1
12				375	375				6,400	6,400	0.02364	41.30 : 1
13				60	60				1,024	1,024	0.00378	263.38 : 1
14				30	30				512	512	0.00189	527.76 : 1
15				30	30				512	512	0.00189	527.76 : 1
16				150	150				2,560	2,560	0.00946	104.75 : 1
17				150	150				2,560	2,560	0.00946	104.75 : 1
19				30	30				512	512	0.00189	527.76 : 1
20				75	75				1,280	1,280	0.00473	210.50 : 1
TOTAL	13	546	5,148	10,725	16,432	13	5,304	82,368	183,040	270,725	1.0	0 : 1

As the table indicates, there is a 98.8% chance that a starting hand will have at least one straight draw, but only a 3.3% chance that it will have more than 12 ways to make a straight.

[edit] Starting hands for straight flushes

The set of starting hands that can make a straight flush are a subset of the intersection of the set of hands that can make a straight and the set of hands that can make a flush. The hands that can make a straight flush can be organized similar to the parent superset of hands that can make a straight.

The straight flush sequence shape of a hand is the number of different combinations of three cards that can make a straight flush when combined with two cards from the hand. There are 9 different straight flush sequence shapes ranging from hands that can't make a straight flush to hands like 8♥ 9♥ 8♠ 9♠ that can make a straight flush with 8 different combinations of three cards ({5♥ 6♥ 7♥}, {6♥ 7♥ 10♥}, {7♥ 10♥ J♥}, {10♥ J♥ Q♥}, {5♠ 6♠ 7♠}, {6♠ 7♠ 10♠}, {7♠ 10♠ J♠}, {10♠ J♠ Q♠}).

As with straights, the relative probability of being dealt a hand of each straight flush sequence shape is different. The following shows the probabilities and odds of being dealt starting hands of each straight flush sequence shape.

Straight flush sequence shape	Distinct hands by ranks in hand				Distinct hands	Combos by ranks in hand				Total combos	Probability	Odds
Straight flush sequence shape	1 rank	2 ranks	3 ranks	4 ranks	Distinct hands	1 rank	2 ranks	3 ranks	4 ranks	Total combos	Probability	Odds
0	13	362	2,060	2,877	5,312	13	2,820	33,168	64,224	100,225	0.37021	1.70 : 1
1		48	794	1,686	2,528		768	13,752	30,264	44,784	0.16542	5.05 : 1
2		55	870	2,113	3,038		720	13,872	32,768	47,360	0.17494	4.72 : 1
3		30	690	1,782	2,502		480	10,920	26,960	38,360	0.14169	6.06 : 1
4		34	594	1,730	2,358		414	8,976	24,044	33,434	0.12350	7.10 : 1
5			68	300	368			816	2,592	3,408	0.01259	78.44 : 1
6		10	38	165	213		60	456	1,444	1,960	0.00724	137.13 : 1
7			28	56	84			336	560	896	0.00331	301.15 : 1
8		7	6	16	29		42	72	184	298	0.00110	907.47 : 1
TOTAL	13	546	5,148	10,725	16,432	13	5,304	82,368	183,040	270,725	1.0	0 : 1

[edit] Low starting hands

Omaha Hi-Low is a high-low split variant where the best qualifying low hand, if any, splits the pot with the high hand. Different cards can be used to form the high and low hands, each using two cards from the player's hand and three from the board, and a single player can win both the high and low pots. In Omaha/8, the most common form played in American casinos, a qualifying low hand is 8-high or lower (8-7-6-5-4 or lower). A less common variant of Omaha Hi-Lo uses a qualifying low hand of 9-high or lower (9-8-7-6-5 or lower).

Suits and cards higher than the maximum qualifying low hand do not factor into low hands and neither do straights and flushes. Based on the ranks of cards, low starting hands in Omaha Hi-Lo are grouped into 12 different low-hand shapes, seven of which have the possibility of making a qualifying low hand. The low hand shapes can be organized by the number of distinct low card ranks in the hand: 0 or 1 low ranks (no low hand possible), 2 low ranks, 3 low ranks and 4 low ranks. The number of distinct low hands depends on the low-hand qualifier.

Low ranks	Shapes	8-high qualifier				9-high qualifier
Low ranks	Shapes	Distinct hands	Combos	Probability	Odds	Distinct hands	Combos	Probability	Odds
0–1	5	33	51,093	0.189	4.30 : 1	37	29,045	0.107	8.32 : 1
2	4	168	113,904	0.421	1.38 : 1	216	99,216	0.366	1.73 : 1
3	2	224	87,808	0.324	2.08 : 1	336	110,208	0.407	1.46 : 1
4	1	70	17,920	0.0662	14.1 : 1	126	32,256	0.119	7.39 : 1
TOTAL	12	495	270,725	1.0	0 : 1	715	270,725	1.0	0 : 1

The preceding table shows that with an 8-high low qualifier, a random hand has an 81.1% chance of having at least two low card ranks to make a low hand possible, and that with a 9-high low qualifier the chance increases to 89.3%.

If $r$ represents the low hand qualifier (8 or 9), there are $\begin{matrix} (13 - r) \times 4 = 52 - 4r \end{matrix}$ cards with a rank higher than the low hand qualifier (20 high cards in 8-high, 16 in 9-high). Using * to represent any high card and lower case letters to represent low card ranks, the following gives the probability of being dealt the various low hand shapes.

Low shape	Derivations		8-high qualifier (r = 8)						9-high qualifier (r = 9)
	Distinct hands	High card & low suit combinations	Distinct hands	Hand combos	Dealt specific hand		Dealt any hand		Distinct hands	Hand combos	Dealt specific hand		Dealt any hand
	Distinct hands	High card & low suit combinations	Distinct hands	Hand combos	Probability	Odds	Probability	Odds	Distinct hands	Hand combos	Probability	Odds	Probability	Odds
****	$\begin{matrix}{r \choose 0}\end{matrix}$	$\begin{matrix}{52 - 4r \choose 4}\end{matrix}$	1	4,845	0.0179	54.9 : 1	0.0179	54.9 : 1	1	1,820	0.00672	148 : 1	0.00672	148 : 1
x***	$\begin{matrix}{r \choose 1}\end{matrix}$	$\begin{matrix}{4 \choose 1}{52 - 4r \choose 3}\end{matrix}$	8	36,480	0.0168	58.4 : 1	0.135	6.42 : 1	9	20,160	0.00827	120 : 1	0.0745	12.4 : 1
xx**	$\begin{matrix}{r \choose 1}\end{matrix}$	$\begin{matrix}{4 \choose 2}{52 - 4r \choose 2}\end{matrix}$	8	9,120	0.00421	236 : 1	0.0337	28.7 : 1	9	6,480	0.00266	375 : 1	0.0239	40.8 : 1
xxx*	$\begin{matrix}{r \choose 1}\end{matrix}$	$\begin{matrix}{4 \choose 3}{52 - 4r \choose 1}\end{matrix}$	8	640	0.000296	3,383 : 1	0.00236	422 : 1	9	576	0.000236	4,229 : 1	0.00213	469 : 1
xxxx	$\begin{matrix}{r \choose 1}\end{matrix}$	$\begin{matrix}{4 \choose 4}\end{matrix}$	8	8	0.00000369	270,724 : 1	0.0000296	33,839 : 1	9	9	0.00000369	270,724 : 1	0.0000332	30,080 : 1
xy**	$\begin{matrix}{r \choose 2}\end{matrix}$	$\begin{matrix}{4 \choose 1}^2{52 - 4r \choose 2}\end{matrix}$	28	85,120	0.0112	88.1 : 1	0.314	2.18 : 1	36	69,120	0.00709	140 : 1	0.255	2.92 : 1
xxy*	$\begin{matrix}{r \choose 1}{r - 1 \choose 1}\end{matrix}$	$\begin{matrix}{4 \choose 2}{4 \choose 1}{52 - 4r \choose 1}\end{matrix}$	56	26,880	0.00177	563 : 1	0.0993	9.07 : 1	72	27,648	0.00142	704 : 1	0.102	8.79 : 1
xxxy	$\begin{matrix}{r \choose 1}{r - 1 \choose 1}\end{matrix}$	$\begin{matrix}{4 \choose 3}{4 \choose 1}\end{matrix}$	56	896	0.0000591	16,919 : 1	0.00331	301 : 1	72	1,152	0.0000591	16,919 : 1	0.00426	234 : 1
xxyy	$\begin{matrix}{r \choose 2}\end{matrix}$	$\begin{matrix}{4 \choose 2}^2\end{matrix}$	28	1,008	0.000133	7,519 : 1	0.00372	267 : 1	36	1,296	0.000133	7,519 : 1	0.00479	208 : 1
xyz*	$\begin{matrix}{r \choose 3}\end{matrix}$	$\begin{matrix}{4 \choose 1}^3{52 - 4r \choose 1}\end{matrix}$	56	71,680	0.00473	211 : 1	0.265	2.78 : 1	84	86,016	0.00378	263 : 1	0.318	2.15 : 1
xxyz	$\begin{matrix}{r \choose 1}{r - 1 \choose 2}\end{matrix}$	$\begin{matrix}{4 \choose 2}{4 \choose 1}^2\end{matrix}$	168	16,128	0.000355	2,819 : 1	0.0596	15.8 : 1	252	24,192	0.000355	2,819 : 1	0.0894	10.2 : 1
xyzr	$\begin{matrix}{r \choose 4}\end{matrix}$	$\begin{matrix}{4 \choose 1}^4\end{matrix}$	70	17,920	0.000946	1,057 : 1	0.0662	14.1 : 1	126	32,256	0.000946	1,057 : 1	0.119	7.39 : 1

The probability of making a low hand depends on the number of low card ranks in the hand. However, although both are important, the probability of having the lowest hand depends more on the ranks of the low cards than on the number of low cards.

[edit] Hand selection

Beginning hand selection is critical in Omaha. Exactly two hole cards are combined with three community cards to form a hand in Omaha. The most favorable hand shapes have two suits with two cards in each suit, giving the hand two flush draws; have card ranks that are consecutive, giving the hand straight possibilities; and have one or more pairs, giving the hand a pair, and draws to three of a kind, full house and four of a kind possibilities. This makes the hands with the shape XaXbYaYb, with the ranks of X and Y adjacent, great starting hands in Omaha. Against one opponent, AaAbKaKb is the strongest starting hand in Omaha (against multiple opponents, the strongest starting hand is AaAbJa10b), while in Omaha Hi-Low the best starting hand is AaAb2a3b, which gives A-2-3 for making a low hand and straights, two suited aces for nut flushes, and a pair of aces for high. The best low starting hand is A-2-3-4, which makes the nut low hand more than 92% of the time when a qualifying low hand is possible and has a better than 50% chance to win at least a portion of the pot at showdown with an 8-high low qualifier, and at least ⅔ of the time will win at least a portion of the pot with a 9-high qualifier.

Contrary to most poker variants, more is not necessarily better (or the same) in Omaha, because only two hole cards are used. Because of this limitation, hands with more than two of the same suit or more than two of the same rank are weaker than the hand would be with exactly two of the suit or rank. The extra cards of the same suit remove outs for the flush draw and the extra cards of the same rank remove valuable outs for three of a kind, a full house, and four of a kind. The suit type aaaa is only about half as likely to make a flush as aabc. Paradoxically, the worst hand in Omaha hold 'em is four of a kind deuces (twos), because this hand can only make a pair of deuces plus the community cards. A much more common poor starting hand has the shape XaYbZcRd with the ranks of the cards spaced, such as 2♥ 6♦ 9♣ K♠—this hand has no flush draw, limited straight possibilities, and no pairs, although it has many more possibilities than 2-2-2-2 and considerably more than 2♥ 2♦ 2♣ 9♠.

Some professional poker players have created point systems for evaluating starting hands in Omaha, with the decision to raise, fold or call based on the number of points assigned to the starting hand.^[1]^[2] However, because of the necessary simplifications point systems make, there is disagreement regarding the value of particular point systems and point systems in general.^[3]

[edit] The flop

There are

${52 \choose 3} = 22,100$

possible flops assuming a random starting hand. By the turn the total number of combinations has increased to

${52 \choose 4} = 270,725$

and on the river there are

${52 \choose 5} = 2,598,960$

possible boards. For a given starting hand there are four known cards, which leaves

${48 \choose 3} = 17,296$

possible flops. At the turn the number of combinations is

${48 \choose 4} = 194,580$

and on the river there are

${48 \choose 5} = 1,712,304$

possible boards to go with the hand.

An Omaha poker hand consists of two cards from the player's hand and three cards from the board. Therefore, there are

${3 \choose 3}{4 \choose 2} = 6$

ways to form a poker hand from a starting hand after the flop and

${4 \choose 3}{4 \choose 2} = 24$ and ${5 \choose 3}{4 \choose 2} = 60$

ways at the turn and river, respectively. By contrast, in Texas hold 'em there are only $\begin{matrix}{5 \choose 5} = 1\end{matrix},$ $\begin{matrix}{6 \choose 5} = 6\end{matrix}$ and $\begin{matrix}{7 \choose 5} = 21\end{matrix}$ ways to form a poker hand on the flop, turn and river, respectively. This increase in opportunities to make a hand means that the average strength of the winning hand in Omaha is higher than in Texas hold' em and other 7-card poker variants.

[edit] Making a low hand

See the section "Low starting hands" for a description of low hands in Omaha.

The first question regarding making a low hand in Omaha Hi-Lo is "how often does a qualifying low hand occur?" In order for any hand to qualify for low, the community cards must include at least three cards to a qualifying low hand. If $r$ is the maximum rank (8 or 9) of a qualifying low hand, then assuming random starting hands, the probability $P f$ of the flop containing three cards to a qualifying low hand is

$P_f = \frac{{r \choose 3}{4 \choose 1}^3}{{52 \choose 3}}.$

Three ranks from the available low ranks are chosen and each rank can have one of four suits. There are $\begin{matrix}{8 \choose 3} = 56\end{matrix}$ and $\begin{matrix}{9 \choose 3} = 84\end{matrix}$ ways to choose three low ranks. The number of hands that can make a qualifying low are divided by the $\begin{matrix}{52 \choose 3} = 22,100\end{matrix}$ possible flops. This gives the following low hand combinations and probabilities on the flop.

Making low on flop		for 8-high (r = 8)			for 9-high (r = 9)
Making low on flop		Combos	Probability	Odds	Combos	Probability	Odds
three low ranks	$\begin{matrix} {r \choose 3}{4 \choose 1}^3 \end{matrix}$	3,584	0.1622	5.116 : 1	5,376	0.2433	3.111 : 1

Calculating the probability of three low ranks on the board for the turn and river is slightly more complicated because there are multiple possibilities for the fourth card. By the turn a qualifying low is possible with either four low ranks, three low ranks and a pair, or three low ranks and a high card. The probability $P t$ of making at least three cards to a qualifying low hand by the turn is the sum of the probabilities for each of these configurations. Each probability is calculated by dividing the number of combinations that satisfy the conditions by the $\begin{matrix}{52 \choose 4} = 270,725\end{matrix}$ possible boards on the turn.

Making low by turn		for 8-high (r = 8)			for 9-high (r = 9)
Making low by turn		Combos	Probability	Odds	Combos	Probability	Odds
four low ranks	$\begin{matrix} {r \choose 4}{4 \choose 1}^4 \end{matrix}$	17,920	0.06619	14.11 : 1	32,256	0.1191	7.393 : 1
three low ranks + a pair	$\begin{matrix} {r \choose 1}{4 \choose 2}{r - 1 \choose 2}{4 \choose 1}^2 \end{matrix}$	16,128	0.05957	15.79 : 1	24,192	0.08936	10.19 : 1
three low ranks + high card	$\begin{matrix} {r \choose 3}{4 \choose 1}^3{52 - 4r \choose 1} \end{matrix}$	71,680	0.2648	2.777 : 1	86,016	0.3177	2.147 : 1
$P t$		105,728	0.3905	1.561 : 1	142,464	0.5262	0.9003 : 1

Finally, at the river there are seven ways to make at least three cards to a low hand and $\begin{matrix}{52 \choose 5} = 2,598,960\end{matrix}$ possible boards on the river, giving the following probability for $P r$ .

Making low by river		for 8-high (r = 8)			for 9-high (r = 9)
Making low by river		Combos	Probability	Odds	Combos	Probability	Odds
five low ranks	$\begin{matrix} {r \choose 5}{4 \choose 1}^5 \end{matrix}$	57,344	0.02206	44.32 : 1	129,024	0.04964	19.14 : 1
four low ranks + a pair	$\begin{matrix} {r \choose 1}{4 \choose 2}{r - 1 \choose 3}{4 \choose 1}^3 \end{matrix}$	107,520	0.04137	23.17 : 1	193,536	0.07447	12.43 : 1
four low ranks + high card	$\begin{matrix} {r \choose 4}{4 \choose 1}^4{52 - 4r \choose 1} \end{matrix}$	358,400	0.1379	6.252 : 1	516,096	0.1986	4.036 : 1
three low ranks + three of a kind	$\begin{matrix} {r \choose 1}{4 \choose 3}{r - 1 \choose 2}{4 \choose 1}^2 \end{matrix}$	10,752	0.004137	240.7 : 1	16,128	0.006206	160.1 : 1
three low ranks + two pair	$\begin{matrix} {r \choose 2}{4 \choose 2}^2{r - 2 \choose 1}{4 \choose 1} \end{matrix}$	24,192	0.009308	106.4 : 1	36,288	0.01396	70.62 : 1
three low ranks + a pair + high card	$\begin{matrix} {r \choose 1}{4 \choose 2}{r - 1 \choose 2}{4 \choose 1}^2{52 - 4r \choose 1} \end{matrix}$	322,560	0.1241	7.057 : 1	387,072	0.1489	5.714 : 1
three low ranks + two high cards	$\begin{matrix} {r \choose 3}{4 \choose 1}^3{52 - 4r \choose 2} \end{matrix}$	680,960	0.2620	2.817 : 1	645,120	0.2482	3.029 : 1
$P r$		1,561,728	0.6009	0.6642 : 1	1,923,264	0.7400	0.3513 : 1

As the last table indicates, there is a 60% chance that a low hand is possible at the river with an 8-high qualifying hand, and a 74% chance with a 9-high qualifier. The actually probability that there will be a qualifying low hand is less because there is a very real possibility that even with three cards to a low hand on the board, no player remaining in the hand at showdown can make a qualifying low hand.

[edit] Making a low hand based on low hand shape

Any hand starting with at least two different qualifying low ranks has a chance to make a low hand. The more different low ranks a hand has, the better the chances of making a low hand. Also, the more pairs and trips among the low ranks, the less chance there is of a low hand being possible and not being able to make a low hand. The following tables show the probability and odds of making a low hand, missing a low hand when one is possible (at least three qualifying low ranks on the board), and having no low hand possible for each of the starting low hand shapes on the flop, turn and river.

8-high qualifier:

Making low hand on flop, 8-high qualifier										Make low when low possible
Low shape	Make low			Miss low			No low			Make low when low possible
Low shape	Combos	Probability	Odds	Combos	Probability	Odds	Combos	Probability	Odds	Probability	Odds
****	0	0	∞	3,584	0.2072	3.83 : 1	13,712	0.7928	0.26 : 1	0	∞
x***	0	0	∞	3,248	0.1878	4.33 : 1	14,048	0.8122	0.23 : 1	0	∞
xx**	0	0	∞	2,912	0.1684	4.94 : 1	14,384	0.8316	0.20 : 1	0	∞
xxx*	0	0	∞	2,576	0.1489	5.71 : 1	14,720	0.8511	0.18 : 1	0	∞
xxxx	0	0	∞	2,240	0.1295	6.72 : 1	15,056	0.8705	0.15 : 1	0	∞
xy**	1,280	0.0740	12.51 : 1	1,656	0.0957	9.44 : 1	14,360	0.8302	0.20 : 1	0.4360	1.29 : 1
xxy*	1,280	0.0740	12.51 : 1	1,344	0.0777	11.87 : 1	14,672	0.8483	0.18 : 1	0.4878	1.05 : 1
xxxy	1,280	0.0740	12.51 : 1	1,032	0.0597	15.76 : 1	14,984	0.8663	0.15 : 1	0.5536	0.81 : 1
xxyy	1,280	0.0740	12.51 : 1	1,056	0.0611	15.38 : 1	14,960	0.8649	0.16 : 1	0.5479	0.83 : 1
xyz*	2,080	0.1203	7.32 : 1	567	0.0328	29.50 : 1	14,649	0.8470	0.18 : 1	0.7858	0.27 : 1
xxyz	1,920	0.1110	8.01 : 1	438	0.0253	38.49 : 1	14,938	0.8637	0.16 : 1	0.8142	0.23 : 1
xyzr	2,272	0.1314	6.61 : 1	108	0.0062	159.15 : 1	14,916	0.8624	0.16 : 1	0.9546	0.05 : 1

Making low hand on turn, 8-high qualifier										Make low when low possible
Low shape	Make low			Miss low			No low			Make low when low possible
Low shape	Combos	Probability	Odds	Combos	Probability	Odds	Combos	Probability	Odds	Probability	Odds
****	0	0	∞	91,392	0.4697	1.13 : 1	103,188	0.5303	0.89 : 1	0	∞
x***	0	0	∞	85,008	0.4369	1.29 : 1	109,572	0.5631	0.78 : 1	0	∞
xx**	0	0	∞	78,288	0.4023	1.49 : 1	116,292	0.5977	0.67 : 1	0	∞
xxx*	0	0	∞	71,232	0.3661	1.73 : 1	123,348	0.6339	0.58 : 1	0	∞
xxxx	0	0	∞	63,840	0.3281	2.05 : 1	130,740	0.6719	0.49 : 1	0	∞
xy**	40,320	0.2072	3.83 : 1	38,484	0.1978	4.06 : 1	115,776	0.5950	0.68 : 1	0.5116	0.95 : 1
xxy*	40,320	0.2072	3.83 : 1	31,968	0.1643	5.09 : 1	122,292	0.6285	0.59 : 1	0.5578	0.79 : 1
xxxy	40,320	0.2072	3.83 : 1	25,140	0.1292	6.74 : 1	129,120	0.6636	0.51 : 1	0.6159	0.62 : 1
xxyy	40,320	0.2072	3.83 : 1	25,680	0.1320	6.58 : 1	128,580	0.6608	0.51 : 1	0.6109	0.64 : 1
xyz*	59,520	0.3059	2.27 : 1	13,284	0.0683	13.65 : 1	121,776	0.6258	0.60 : 1	0.8175	0.22 : 1
xxyz	56,000	0.2878	2.47 : 1	10,515	0.0540	17.50 : 1	128,065	0.6582	0.52 : 1	0.8419	0.19 : 1
xyzr	64,464	0.3313	2.02 : 1	2,565	0.0132	74.86 : 1	127,551	0.6555	0.53 : 1	0.9617	0.04 : 1

Making low hand on river, 8-high qualifier										Make low when low possible
Low shape	Make low			Miss low			No low			Make low when low possible
Low shape	Combos	Probability	Odds	Combos	Probability	Odds	Combos	Probability	Odds	Probability	Odds
****	0	0	∞	1,174,656	0.6860	0.46 : 1	537,648	0.3140	2.18 : 1	0	∞
x***	0	0	∞	1,116,780	0.6522	0.53 : 1	595,524	0.3478	1.88 : 1	0	∞
xx**	0	0	∞	1,052,520	0.6149	1.60 : 1	659,784	0.3853	0.67 : 1	0	∞
xxx*	0	0	∞	981,540	0.5732	0.74 : 1	730,764	0.4268	1.34 : 1	0	∞
xxxx	0	0	∞	903,504	0.5277	0.90 : 1	808,800	0.4723	1.12 : 1	0	∞
xy**	625,344	0.3652	1.74 : 1	432,480	0.2526	2.96 : 1	654,480	0.3822	1.62 : 1	0.5912	0.69 : 1
xxy*	625,344	0.3652	1.74 : 1	367,320	0.2145	3.66 : 1	719,640	0.4203	1.38 : 1	0.6300	0.59 : 1
xxxy	625,344	0.3652	1.74 : 1	295,644	0.1727	4.79 : 1	791,316	0.4621	1.16 : 1	0.6790	0.47 : 1
xxyy	625,344	0.3652	1.74 : 1	301,464	0.1761	4.68 : 1	785,496	0.4587	1.18 : 1	0.6747	0.48 : 1
xyz*	847,944	0.4952	1.02 : 1	150,264	0.0878	10.40 : 1	714,096	0.4170	1.40 : 1	0.8495	0.18 : 1
xxyz	810,784	0.4735	1.11 : 1	121,808	0.0711	13.06 : 1	779,712	0.4554	1.20 : 1	0.8694	0.15 : 1
xyzr	908,976	0.5308	0.88 : 1	29,376	0.0172	57.29 : 1	773,952	0.4520	1.21 : 1	0.9687	0.03 : 1

9-high qualifier:

Making low hand on flop, 9-high qualifier										Make low when low possible
Low shape	Make low			Miss low			No low			Make low when low possible
Low shape	Combos	Probability	Odds	Combos	Probability	Odds	Combos	Probability	Odds	Probability	Odds
****	0	0	∞	5,376	0.3108	2.22 : 1	11,920	0.6892	0.45 : 1	0	∞
x***	0	0	∞	4,928	0.2849	2.51 : 1	12,368	0.7151	0.40 : 1	0	∞
xx**	0	0	∞	4,480	0.2590	2.86 : 1	12,816	0.7410	0.35 : 1	0	∞
xxx*	0	0	∞	4,032	0.2331	3.29 : 1	13,264	0.7669	0.30 : 1	0	∞
xxxx	0	0	∞	3,584	0.2072	3.83 : 1	13,712	0.7928	0.26 : 1	0	∞
xy**	2,240	0.1295	6.72 : 1	2,268	0.1311	6.63 : 1	12,788	0.7394	0.35 : 1	0.4969	1.01 : 1
xxy*	2,240	0.1295	6.72 : 1	1,848	0.1068	8.36 : 1	13,208	0.7636	0.31 : 1	0.5479	0.83 : 1
xxxy	2,240	0.1295	6.72 : 1	1,428	0.0826	11.11 : 1	13,628	0.7879	0.27 : 1	0.6107	0.64 : 1
xxyy	2,240	0.1295	6.72 : 1	1,456	0.0842	10.88 : 1	13,600	0.7863	0.27 : 1	0.6061	0.65 : 1
xyz*	3,440	0.1989	4.03 : 1	675	0.0390	24.62 : 1	13,181	0.7621	0.31 : 1	0.8360	0.20 : 1
xxyz	3,200	0.1850	4.41 : 1	522	0.0302	32.13 : 1	13,574	0.7848	0.27 : 1	0.8598	0.16 : 1
xyzr	3,640	0.2105	3.75 : 1	108	0.0062	159.15 : 1	13,548	0.7833	0.28 : 1	0.9712	0.03 : 1

Making low hand on turn, 9-high qualifier										Make low when low possible
Low shape	Make low			Miss low			No low			Make low when low possible
Low shape	Combos	Probability	Odds	Combos	Probability	Odds	Combos	Probability	Odds	Probability	Odds
****	0	0	∞	120,960	0.6216	0.61 : 1	73,620	0.3784	1.64 : 1	0	∞
x***	0	0	∞	114,240	0.5871	0.70 : 1	80,340	0.4129	1.42 : 1	0	∞
xx**	0	0	∞	107,072	0.5503	0.82 : 1	87,508	0.4497	1.22 : 1	0	∞
xxx*	0	0	∞	99,456	0.5111	0.96 : 1	95,124	0.4889	1.05 : 1	0	∞
xxxx	0	0	∞	91,392	0.4697	1.13 : 1	103,188	0.5303	0.89 : 1	0	∞
xy**	63,840	0.3281	2.05 : 1	43,722	0.2247	3.45 : 1	87,018	0.4472	1.24 : 1	0.5935	0.68 : 1
xxy*	63,840	0.3281	2.05 : 1	36,624	0.1882	4.31 : 1	94,116	0.4837	1.07 : 1	0.6355	0.57 : 1
xxxy	63,840	0.3281	2.05 : 1	29,106	0.1496	5.69 : 1	101,634	0.5223	0.91 : 1	0.6869	0.46 : 1
xxyy	63,840	0.3281	2.05 : 1	29,624	0.1522	5.57 : 1	101,116	0.5197	0.92 : 1	0.6830	0.46 : 1
xyz*	87,840	0.4514	1.22 : 1	13,122	0.0674	13.83 : 1	93,618	0.4811	1.08 : 1	0.8700	0.15 : 1
xxyz	83,520	0.4292	1.33 : 1	10,449	0.0537	17.62 : 1	100,611	0.5171	0.93 : 1	0.8888	0.13 : 1
xyzr	92,340	0.4746	1.11 : 1	2,133	0.0110	90.22 : 1	100,107	0.5145	0.94 : 1	0.9774	0.02 : 1

Making low hand on river, 9-high qualifier										Make low when low possible
Low shape	Make low			Miss low			No low			Make low when low possible
Low shape	Combos	Probability	Odds	Combos	Probability	Odds	Combos	Probability	Odds	Probability	Odds
****	0	0	∞	1,407,168	0.8218	0.22 : 1	305,136	0.1782	4.61 : 1	0	∞
x***	0	0	∞	1,359,568	0.7940	0.26 : 1	352,736	0.2060	3.85 : 1	0	∞
xx**	0	0	∞	1,305,248	0.7623	0.31 : 1	407,056	0.2377	3.21 : 1	0	∞
xxx*	0	0	∞	1,243,760	0.7264	0.38 : 1	468,544	0.2736	2.65 : 1	0	∞
xxxx	0	0	∞	1,174,656	0.6860	0.46 : 1	537,648	0.3140	2.18 : 1	0	∞
xy**	903,504	0.5277	0.90 : 1	405,804	0.2370	3.22 : 1	402,996	0.2354	3.25 : 1	0.6901	0.45 : 1
xxy*	903,504	0.5277	0.90 : 1	348,866	0.2037	3.91 : 1	459,934	0.2686	2.72 : 1	0.7214	0.39 : 1
xxxy	903,504	0.5277	0.90 : 1	284,830	0.1663	5.01 : 1	523,970	0.3060	2.27 : 1	0.7603	0.32 : 1
xxyy	903,504	0.5277	0.90 : 1	289,380	0.1690	4.92 : 1	519,420	0.3033	2.30 : 1	0.7574	0.32 : 1
xyz*	1,134,204	0.6624	0.51 : 1	122,526	0.0716	12.98 : 1	455,574	0.2661	2.76 : 1	0.9025	0.11 : 1
xxyz	1,097,184	0.6408	0.56 : 1	100,368	0.0586	16.06 : 1	514,752	0.3006	2.33 : 1	0.9162	0.09 : 1
xyzr	1,182,004	0.6903	0.45 : 1	20,196	0.0118	83.78 : 1	510,104	0.2979	2.36 : 1	0.9832	0.02 : 1

There are several interesting things to note about low hands. With an 8-high qualifier, the shape xyzr, which has four different low ranks, is the only starting hand with a greater than 50% chance of making a low hand before the flop. With a 9-high qualifier, any hand that can make a low hand has a greater than 50% chance of making a low hand. When playing a hand that has no chance at low, a hand like A-A-Q-J, which has two of the low aces, creates a significantly lower chance of there being a low than a hand like K-K-Q-J which has only high cards. With an 8-high qualifier the probability of a low hand being possible is reduced by about 10% (from 68.6% to 61.5%) and with a 9-high qualifier the probability is reduced by about 7% (from 82.2% to 76.2%). This is an often overlooked additional advantage of pocket aces over pocket kings with a high-only hand in Omaha Hi-Lo.

See Probability derivations for making low hands in Omaha hold 'em for the derivations for the probabilities in the preceding tables of making low hands based on the low hand shape of the starting hand.

[edit] Making the nuts

The nuts is the best possible poker hand that can be made from the community cards. Due to the large number of opportunities to make a hand, it is not uncommon in Omaha for the winning hand to be the nuts, or at least close to it.

On the flop every poker hand uses the three community cards plus two other cards. For each board then, there are $\begin{matrix} {49 \choose 2} = 1,176 \end{matrix}$ ways to make a poker hand and

${52 \choose 3}{3 \choose 3}{49 \choose 2} = 25,989,600$

different possible combinations of poker hands made from the three community cards and two cards from a player's hand. On the turn there are $\begin{matrix} {4 \choose 3} = 4 \end{matrix}$ ways to select three community cards and $\begin{matrix} {48 \choose 2} = 1,128 \end{matrix}$ hand combinations giving 4 × 1,128 = 4,512 ways to make a poker hand for each board. This gives

${52 \choose 4}{4 \choose 3}{48 \choose 2} = 1,221,511,200$

combinations of hands on the turn. Finally, on the river there are $\begin{matrix} {5 \choose 3} = 10 \end{matrix}$ ways to select three community cards and $\begin{matrix} {47 \choose 2} = 1,081 \end{matrix}$ hand combinations giving 10,810 ways to make a poker hand for each board, resulting in

${52 \choose 5}{5 \choose 3}{47 \choose 2} = 28,094,757,600$

poker hand combinations possible on the river. For each board, one or more of the possible poker hands is the nuts.

[edit] Making high nuts

In the case of high hands, when the nuts is a royal flush or straight flush, the winning hand is often a flush; when the nuts is four of a kind, the winning hand is often a full house. The lowest hand that can be the nuts is three of a kind, which occurs when there is no straight or flush possible and no pair on the board. The lowest possible nut hand at the river is Q-Q-Q-8-7 which occurs when the board is one of the 600 possible combinations of Q-8-7-3-2 that doesn't have three or more of the same suit.

The following table shows the probability of the nut hand for the board on the flop, turn and river.

Poker hand	After the flop			After the turn			After the river
Poker hand	Combos	Probability	Odds	Combos	Probability	Odds	Combos	Probability	Odds
Straight flush	256	0.0116	85.33 : 1	11,712	0.0433	22.12 : 1	261,920	0.1008	8.923 : 1
Four of a kind	3,796	0.1718	4.822 : 1	85,368	0.3153	2.171 : 1	1,173,696	0.4516	1.214 : 1
Full house	0	0	∞	13	< 0.0001	20,824 : 1	624	0.0002	4164 : 1
Flush	888	0.0402	23.89 : 1	27,772	0.1026	8.748 : 1	390,520	0.1503	5.655 : 1
Straight	3,840	0.1738	4.755 : 1	88,128	0.3255	2.072 : 1	724,800	0.2789	2.586 : 1
Three of a kind	13,320	0.6027	0.6592 : 1	57,732	0.2132	3.689 : 1	47,400	0.0182	53.83 : 1
TOTAL	22,100	1	0 : 1	270,725	1	0 : 1	2,598,960	1	0 : 1

Notice that while three of a kind is a 60% favorite to be the nuts after the flop, it's less than 2% to still be on top at the river—although the three of a kind has a good chance of improving to a full house or four of a kind, if it doesn't improve, chances are the nut hand at the river is a straight, flush or straight flush. At the river, having the nuts be four of a kind is more likely (45.2%) than all of the hands ranked below four of a kind combined (44.8%). Also, despite the rarity of straight flushes at showdown, 10% of the boards will have one as the nut hand by the river.

[edit] Derivation of the high nut boards

A straight flush is possible whenever the board contains at least three cards of the same suit where the ranks of the suited cards can create a straight with the addition of exactly two ranks. For three ranks, the two lower ranks must be chosen from the up to four next lower ranks, counting the rank of ace as low when trying to make the striaght A-2-3-4-5. There are 10 possible straights (Ace high to 5 high). A straight is also possible when the high rank is 4 combined with two of the three lower ranks A to 3, or when the three ranks are 3-2-A, which gives $\begin{matrix} 10 \times {4 \choose 2} + {3 \choose 2} + {2 \choose 2} = 64 \end{matrix}$ ways. This gives

$64 \times {4 \choose 1} = 256$

combinations for a straight flush after the flop. With four or five cards of different ranks, the determination of the number of rank sets that yield a straight with the addition of exactly two cards is more involved because any enumeration must eliminate rank sets that are counted more than once, but it turns out that there are 432 such rank sets with four ranks and 1,208 with five ranks.^[4] At the turn there are two ways to make a straight flush—there can either be four cards of the same suit with a rank set that allows a straight, or three cards of the same suit that allow a straight combined with any of the 39 cards with a different suit. This gives

$432 \times {4 \choose 1} + 64 \times {4 \choose 1}{39 \choose 1} = 11,712$

combinations after the turn. At the river a straight flush is possible with a suited rank set of either five cards, four cards combined with one of the 39 cards of another suit, or three cards combined with two of the remaining 39 cards, giving

$1,208 \times {4 \choose 1} + 432 \times {4 \choose 1}{39 \choose 1} + 64 \times {4 \choose 1}{39 \choose 2} = 261,920$

combinations.

Four of a kind is the nuts whenever there is a pair or three of a kind on the board and no possibility for a straight flush. After the flop, three of a kind is possible by choosing one of the 13 ranks and three of the four cards in that rank; a pair is possible by choosing one of the 13 ranks and two of the four cards in that rank combined with a card in one of the other 12 ranks in any of the four suits. So on the flop there are

${13 \choose 1}{4 \choose 3} + {13 \choose 1}{4 \choose 2}{12 \choose 1}{4 \choose 1} = 3,796$

boards with a pair or three of a kind. At the turn there can either be three of a kind and another rank, two pair, or one pair and two other ranks. In the case with one pair, any straight flushes made possible by the three different ranks must be subtracted. At the turn, the number of possible straight flushes with a pair on the board is one of the 64 rank sets with three cards that can make a straight in one of the four suits combined with a card that pairs one of the three cards to the straight flush, which is $\begin{matrix} 64 \times {4 \choose 1}{3 \choose 1}{3 \choose 1} = 2,304\end{matrix}$ . So there are

${13 \choose 1}{4 \choose 3}{12 \choose 1}{4 \choose 1} + {13 \choose 2}{4 \choose 2}^2 + \left [ {13 \choose 2}{4 \choose 2}{12 \choose 2}{4 \choose 1}^2 - 2,304 \right ] = 85,368$

combinations that make four of a kind possible at the turn. On the river, four of a kind can be made when there is either a full house on the board, three of a kind and two other ranks, two pair and one other rank, or a pair and three other ranks. With three of a kind or two pair, any straight flushes made possible by the three different ranks must be subtracted and with a pair, any straight flushes made possible by the four different ranks are subtracted. For three of a kind, choose one of the three cards for the straight flush and then choose 2 of the 3 remaining cards of that rank to make three of a kind for $\begin{matrix} 64 \times {4 \choose 1}{3 \choose 1}{3 \choose 2} = 2,304 \end{matrix}$ possible straight flushes. With two pair, choose two of the three cards that make a possible straight flush and then choose one of the three remaining cards for each rank to make one of $\begin{matrix} 64 \times {4 \choose 1}{3 \choose 2}{3 \choose 1}^2 = 6,912 \end{matrix}$ straight flushes. With a pair there are three cases where a straight flush is possible that have a total of 97,536 combinations:

the three non-pair ranks have the same suit as one of the cards in the pair and the four suited cards form one of the 432 rank sets that allows a straight (example 3♠ 9♠ J♠ Q♠ J♦), so choosing one of the four suited ranks and one of the three remaining cards of that rank gives $\begin{matrix} 432 \times {4 \choose 1}{3 \choose 1} = 20,736 \end{matrix}$ possible straight flushes;
two of the non-pair ranks have the same suit as one of the cards in the pair and the three suited cards form one of the 64 rank sets that allows a straight (example 8♠ 9♠ J♠ J♥ K♦), so choosing one of the three suited ranks and one of the three remaining cards of that rank to combine with a card from one of the 10 remaining ranks in one of the three remaining suits gives $\begin{matrix} 64 \times {4 \choose 1}{3 \choose 1}{3 \choose 1}{10 \choose 1}{3 \choose 1} = 69,120 \end{matrix}$ possible straight flushes;
the three non-pair ranks share a suit that is different than either of the suits in the pair and the three suited cards form one of the 64 rank sets that allows a straight (example 4♠ 6♠ 7♠ J♥ J♦), so choose one of the 10 remaining ranks not used by the suited cards and choose two of the three cards of that rank that have a different suit to give $\begin{matrix} 64 \times {4 \choose 1}{10 \choose 1}{3 \choose 2} = 7,680 \end{matrix}$ possible straight flushes.

Altogether there are

${13 \choose 1}{4 \choose 3}{12 \choose 1}{4 \choose 2}$		$3,744\,$	full houses
$+ {13 \choose 1}{4 \choose 3}{12 \choose 2}{4 \choose 1}^2 - 2,304$		$52,608\,$	three of a kind
$+ {13 \choose 2}{4 \choose 2}^2{11 \choose 2}{4 \choose 1} - 6,912$		$116,640\,$	two pair
$+ {13 \choose 1}{4 \choose 2}{12 \choose 3}{4 \choose 1}^3 - 97,536$		$1,000,704\,$	one pair
	for a total of	$1,173,696\,$

combinations on the river that make four of a kind possible without making a straight flush possible.

Oddly enough, a full house can only be the nuts when there is four of a kind on the board. This means that there is no chance of a full house being the nuts on the flop. On the turn and river there are

${13 \choose 1} = 13$ and ${13 \choose 1}{48 \choose 1} = 624$

combinations that make four of a kind, respectively.

A flush is the nuts when no two cards share the same rank (no pairs, trips or quads) and there are three or more cards of the same suit that do not form a rank set that can make a straight. The number of rank sets that can't make a straight is $\begin{matrix}{13 \choose 3} - 64 = 222\end{matrix}$ with three cards, $\begin{matrix}{13 \choose 4} - 432 = 283\end{matrix}$ with four cards, and $\begin{matrix}{13 \choose 5} - 1,208 = 79\end{matrix}$ with five cards. On the flop all three cards must be part of the flush which gives

$222 \times {4 \choose 1} = 888$

combinations where the nuts is a flush. On the turn a nut flush is possible with either four cards of the same suit that form one of the 283 rank sets that doesn't allow a straight or with three cards of the same suit that form one of the 222 rank sets that doesn't allow a straight combined with one of the other 10 ranks in one of the other three suits. This gives

$283 \times {4 \choose 1} + 222 \times {4 \choose 1}{10 \choose 1}{3 \choose 1} = 27,772$

ways for a flush to be the nuts on the turn. On the river there are three ways to make a nut flush—five cards of the same suit that form one of 79 rank sets that can't make a straight; four cards of the same suit that can't make a straight combined with a card in one of the other nine ranks and one of the other three suits; or three cards of the same suit that can't make a straight combinded with two of the remaining 10 ranks, each selected from the three remaining suits. So there are

$79 \times {4 \choose 1} + 283 \times {4 \choose 1}{9 \choose 1}{3 \choose 1} + 222 \times {4 \choose 1}{10 \choose 2}{3 \choose 1}^2 = 390,520$

combinations on the river where the nut hand is a flush.

A straight is the nuts when no two cards share the same rank (no pairs, trips or quads), the ranks form a rank set that makes a straight possible with the addition of two cards, and no more than two cards share the same suit. Given $n$ cards of distinct ranks, there are $4 n$ ways to assign suits to the cards. This includes 4 suit sets that assign the same suit to each card; $\begin{matrix} {n \choose n-1}{4 \choose 1}{3 \choose 1} = 12n \end{matrix}$ suit sets that assign the same suit to $n - 1$ cards, and $\begin{matrix} {n \choose n-x}{4 \choose 1}{3 \choose 1}^x \end{matrix}$ suit sets that assign the same suit to $n - x$ cards. The number of combinations on the flop where a straight is the nuts is then the 64 rank sets that allow a straight multiplied by the $43 - 4 = 60$ suit sets that don't have three of the same suit:

$64 \times 60 = 3,840\,$

For four ranks, there are $\begin{matrix} 4 + {4 \choose 3}{4 \choose 1}{3 \choose 1} = 52 \end{matrix}$ suit sets that have either three or four of the same suit, giving $44 - 52 = 204$ suit sets where no more than two cards share the same suit. For five ranks there are $\begin{matrix} 4 + {5 \choose 4}{4 \choose 1}{3 \choose 1} + {5 \choose 3}{4 \choose 1}{3 \choose 1}^2 = 424 \end{matrix}$ suit sets that have three or more of the same suit and $45 - 424 = 600$ suit sets where no more than two cards share the same suit. This gives

$432 \times 204 = 88,128\,$ and $1,208 \times 600 = 724,800\,$

combinations on the turn and river, respectively, where the nut hand is a straight.

Finally, three of a kind is only the nuts when no two cards share the same rank (no pairs, trips or quads), the ranks form a rank set that can't make a straight with the addition of two cards, and no more than two cards share the same suit. As with a straight, the number of combinations is the number of possible rank sets multiplied by the number of allowed suit sets. On the flop, turn and river, respectively, the number of combinations where three of a kind is the nuts are

$222 \times 60 = 13,320\,$ and $283 \times 204 = 57,732\,$ and $79 \times 600 = 47,400\,$

[edit] Making low nuts

See the section Low starting hands for a description of low starting hand shapes.

In Omaha Hi-Lo, it is often the case that when there is a low hand, the winning hand is the nut low hand. When there are more than two people in the pot at showdown and a low hand is possible, it is not uncommon for two or more players to both have the nut low hand. This makes playing a hand that is only contesting for the low half of the pot risky. For example, if at showdown there are two players in the pot and they each have the nut low hand but different high hands, the player with the better high hand will win 75% of the pot (split the low half and win the high half) and the other player wins just 25% of the pot (called "getting quartered").

A hand must have at least two different ranks from {A, 2, 3, 4, 5} in order to make a nut low hand. As with low starting hands in general, there are seven different shapes of low hands that can make the nut low. The probability of being dealt each of these hands is different. Using * to represent any of the 32 cards higher than 5 and lower case letters to represent card ranks from A–5, the following gives the probability of being dealt the various low hand shapes that can make a nut low.

Nut low shape	Derivations		Starting hands drawing to nut low
	Distinct hands	>5 rank & low suit combinations	Distinct hands	Hand combos	Dealt any hand
	Distinct hands	>5 rank & low suit combinations	Distinct hands	Hand combos	Probability	Odds
xy**	$\begin{matrix}{5 \choose 2}\end{matrix}$	$\begin{matrix}{4 \choose 1}^2{32 \choose 2}\end{matrix}$	10	79,360	0.29314	2.41 : 1
xxy*	$\begin{matrix}{5 \choose 2}{2 \choose 1}\end{matrix}$	$\begin{matrix}{4 \choose 2}{4 \choose 1}{32 \choose 1}\end{matrix}$	20	15,360	0.05674	16.6 : 1
xxxy	$\begin{matrix}{5 \choose 2}{2 \choose 1}\end{matrix}$	$\begin{matrix}{4 \choose 3}{4 \choose 1}\end{matrix}$	20	320	0.00118	845 : 1
xxyy	$\begin{matrix}{5 \choose 2}\end{matrix}$	$\begin{matrix}{4 \choose 2}^2\end{matrix}$	10	360	0.00133	751 : 1
xyz*	$\begin{matrix}{5 \choose 3}\end{matrix}$	$\begin{matrix}{4 \choose 1}^3{32 \choose 1}\end{matrix}$	10	20,048	0.07565	12.2 : 1
xxyz	$\begin{matrix}{5 \choose 3}{3 \choose 1}\end{matrix}$	$\begin{matrix}{4 \choose 2}{4 \choose 1}^2\end{matrix}$	30	2,880	0.01064	93.0 : 1
xyzr	$\begin{matrix}{5 \choose 4}\end{matrix}$	$\begin{matrix}{4 \choose 1}^4\end{matrix}$	5	1,280	0.00473	211 : 1
TOTAL			105	120,040	0.44340	1.26 : 1

As the table shows, there is a better then 44% chance of having a hand that has a possibility of making a nut low hand.

While the table above shows 105 distinct hands that can make a nut low, there are actually 165 different cases to consider. The extra cases come from how the * ranks higher than 5 are assigned. More low ranks in the hand decreases the number of low cards available to make a low hand possible, although they increase the chance of the hand making a non-nut low hand. For shapes xxy* and xyz*, * can either be a qualifying low rank higher than 5, or a non-qualifying high rank. This doubles the distinct hands to consider in these cases, adding 20 and 10 distinct hands for xxy* and xyz*, respectively. For shape xy** there are four ways ** can be assigned: either two different qualifying low ranks, the same qualifying low rank (a low pair), one qualifying low rank and one high rank, or two high ranks. This adds 3 × 10 = 30 more distinct hands for a total of 60 additional distinct hands.

The probability of a hand making the nut low depends on several aspects of the hand:

How many different ranks from A–5 the hand has;
The rank of the second lowest rank in the hand (2–5);
Pairs and trips involving the lowest two ranks—more cards here reduces the chance of the low being counterfeited (shapes xxy*, xxxy, xxyy, and xxyz);
The rank of the third lowest rank in the hand (shapes xyz*, xxyz, and xyzr);
The rank of the fourth lowest rank in the hand (shape xyzr);
Whether ranks above 5 can qualify for a low hand—that is, for the shapes xy**, xxy* and xyz*, if * is one or more cards of rank 6–8 or 6–9 for 8-high and 9-high qualifiers, respectively (more low cards here reduces the chance of a low hand being possible);
Pairs involving the third lowest rank in the hand when the third lowest rank can qualify for a low hand (shapes xy** and xxyz).

Perhaps surprisingly, although it affects the strength of non-nut low hands, the rank of the lowest card has no influence on either making a low hand or making the nut low hand. So the hands A-4-K-K and 3-4-10-K have the same probability of making both a nut-low hand and any low hand, although A-4-K-K is likely to make a better non-nut low hand. The hand A-4-7-K will have a slightly lower chance of making the nut low than either of the previous hands because the 7 in the hand reduces the chance of the board 2-3-7-10-J or A-2-7-10-J, which make nut low hands for A-4-K-K and 3-4-10-K, respectively. However, A-4-7-K will have a better chance of making a non-nut low hand because boards like 4-5-6-10-J still make a non-nut low for it, but make no low hand for either A-4-K-K or 3-4-10-K.

Combining the hands into groups based on the factors that determine the probabilities for making the nut low hand and making low hands, the 165 different cases fall into 56 groupings. The nut low hand shapes are denoted using the lower case letters x and y to represent qualifying low ranks higher than 5 (6–8 or 6–9 for 8- and 9-high qualifiers, respectively), and * to represent any high card. The following tables give the probability for select starting hands to make the nut low hand and make a non-nut low hand on the flop, turn and river. The hands in the table are listed in order of the probability of having the nut hand on the river, from highest probability to lowest. (See Probability of making the nut low hand in Omaha hold 'em for complete tables of all 165 nut low hand shapes.) The tables also give the probability that the hand will make a nut low hand if at least three different low ranks are on the board, making a low hand possible. See the section Making a low hand for the probabilities of a low hand being possible and the probability of making or missing a low hand when one is possible.

**Making a nut low hand with an 8-high qualifier (selected hands)**
Hands	Make on flop						Make on turn						Make on river
	Nut low		Non-nut low		Nuts if low		Nut low		Non-nut low		Nuts if low		Nut low		Non-nut low		Nuts if low
	Prob.	Odds	Prob.	Odds	Prob.	Odds	Prob.	Odds	Prob.	Odds	Prob.	Odds	Prob.	Odds	Prob.	Odds	Prob.	Odds
A-2-3-4	0.1314	6.6 : 1	0	∞	0.9546	<0.1 : 1	0.3246	2.1 : 1	0.0067	149.1 : 1	0.9424	0.1 : 1	0.5087	1.0 : 1	0.0221	44.2 : 1	0.9283	0.1 : 1
A-2-3-5	0.1126	7.9 : 1	0.0187	52.4 : 1	0.8185	0.2 : 1	0.2788	2.6 : 1	0.0525	18.1 : 1	0.8095	0.2 : 1	0.4385	1.3 : 1	0.0923	9.8 : 1	0.8002	0.2 : 1
A-2-3-*	0.1203	7.3 : 1	0	∞	0.7858	0.3 : 1	0.2859	2.5 : 1	0.0200	49.0 : 1	0.7641	0.3 : 1	0.4314	1.3 : 1	0.0638	14.7 : 1	0.7400	0.4 : 1
A-A-2-3, A-2-2-3, A-2-3-3	0.1110	8.0 : 1	0	∞	0.8142	0.2 : 1	0.2723	2.7 : 1	0.0155	63.3 : 1	0.7965	0.3 : 1	0.4229	1.4 : 1	0.0507	18.7 : 1	0.7764	0.3 : 1
A-2-3-x	0.1064	8.4 : 1	0.0250	39.0 : 1	0.7731	0.3 : 1	0.2591	2.9 : 1	0.0722	12.9 : 1	0.7523	0.3 : 1	0.3997	1.5 : 1	0.1311	6.6 : 1	0.7294	0.4 : 1
A-2-4-5	0.0835	11.0 : 1	0.0479	19.9 : 1	0.6067	0.6 : 1	0.2108	3.7 : 1	0.1205	7.3 : 1	0.6120	0.6 : 1	0.3393	1.9 : 1	0.1915	4.2 : 1	0.6192	0.6 : 1
A-A-2-4, A-2-2-4	0.0833	11.0 : 1	0.0278	35.0 : 1	0.6107	0.6 : 1	0.2070	3.8 : 1	0.0808	11.4 : 1	0.6055	0.7 : 1	0.3267	2.1 : 1	0.1468	5.8 : 1	0.5999	0.7 : 1
A-A-A-2, A-2-2-2	0.0740	12.5 : 1	0	∞	0.5536	0.8 : 1	0.1822	4.5 : 1	0.0250	39.0 : 1	0.5416	0.8 : 1	0.2842	2.5 : 1	0.0810	11.3 : 1	0.5283	0.9 : 1
A-A-2-2	0.0740	12.5 : 1	0	∞	0.5479	0.8 : 1	0.1822	4.5 : 1	0.0250	39.0 : 1	0.5372	0.9 : 1	0.2842	2.5 : 1	0.0810	11.3 : 1	0.5250	0.9 : 1
A-A-2-5, A-2-2-5	0.0694	13.4 : 1	0.0416	23.0 : 1	0.5089	1.0 : 1	0.1727	4.8 : 1	0.1151	7.7 : 1	0.5051	1.0 : 1	0.2735	2.7 : 1	0.2000	4.0 : 1	0.5022	1.0 : 1
A-A-2-, A-2-2-	0.0740	12.5 : 1	0	∞	0.4878	1.1 : 1	0.1760	4.7 : 1	0.0312	31.0 : 1	0.4737	1.1 : 1	0.2657	2.8 : 1	0.0995	9.0 : 1	0.4583	1.2 : 1
A-3-4-5, 2-3-4-5	0.0569	16.6 : 1	0.0745	12.4 : 1	0.4134	1.4 : 1	0.1500	5.7 : 1	0.1813	4.5 : 1	0.4355	1.3 : 1	0.2527	3.0 : 1	0.2782	2.6 : 1	0.4611	1.2 : 1
A-2--	0.0740	12.5 : 1	0	∞	0.4360	1.3 : 1	0.1697	4.9 : 1	0.0375	25.7 : 1	0.4191	1.4 : 1	0.2479	3.0 : 1	0.1173	7.5 : 1	0.4013	1.5 : 1
A-A-2-x, A-2-2-x	0.0648	14.4 : 1	0.0463	20.6 : 1	0.4750	1.1 : 1	0.1579	5.3 : 1	0.1299	6.7 : 1	0.4619	1.2 : 1	0.2440	3.1 : 1	0.2295	3.4 : 1	0.4480	1.2 : 1
A-2-x-*	0.0648	14.4 : 1	0.0555	17.0 : 1	0.4231	1.4 : 1	0.1525	5.6 : 1	0.1534	5.5 : 1	0.4075	1.5 : 1	0.2282	3.4 : 1	0.2670	2.7 : 1	0.3914	1.6 : 1
A-2-x-y	0.0564	16.7 : 1	0.0749	12.3 : 1	0.4101	1.4 : 1	0.1363	6.3 : 1	0.1950	4.1 : 1	0.3956	1.5 : 1	0.2090	3.8 : 1	0.3219	2.1 : 1	0.3813	1.6 : 1
A-2-x-x	0.0555	17.0 : 1	0.0555	17.0 : 1	0.4071	1.5 : 1	0.1344	6.4 : 1	0.1534	5.5 : 1	0.3931	1.5 : 1	0.2065	3.8 : 1	0.2670	2.7 : 1	0.3791	1.6 : 1
A-3--, 2-3--	0.0370	26.0 : 1	0.0370	26.0 : 1	0.2180	3.6 : 1	0.0891	10.2 : 1	0.1181	7.5 : 1	0.2201	3.5 : 1	0.1383	6.2 : 1	0.2269	3.4 : 1	0.2238	3.5 : 1
A-4--, 2-4--, 3-4--	0.0148	66.6 : 1	0.0592	15.9 : 1	0.0872	10.5 : 1	0.0395	24.3 : 1	0.1677	5.0 : 1	0.0975	9.3 : 1	0.0687	13.6 : 1	0.2965	2.4 : 1	0.1111	8.0 : 1
A-5--, 2-5--, 3-5--, 4-5--	0.0037	269.3 : 1	0.0703	13.2 : 1	0.0218	44.9 : 1	0.0133	74.1 : 1	0.1939	4.2 : 1	0.0329	29.4 : 1	0.0300	32.4 : 1	0.3352	2.0 : 1	0.0485	19.6 : 1

**Making a nut low hand with a 9-high qualifier (selected hands)**
Hands	Make on flop						Make on turn						Make on river
	Nut low		Non-nut low		Nuts if low		Nut low		Non-nut low		Nuts if low		Nut low		Non-nut low		Nuts if low
	Prob.	Odds	Prob.	Odds	Prob.	Odds	Prob.	Odds	Prob.	Odds	Prob.	Odds	Prob.	Odds	Prob.	Odds	Prob.	Odds
A-2-3-4	0.2105	3.8 : 1	0	∞	0.9712	<0.1 : 1	0.4657	1.1 : 1	0.0089	111.6 : 1	0.9591	<0.1 : 1	0.6628	0.5 : 1	0.0275	35.4 : 1	0.9440	0.1 : 1
A-2-3-5	0.1855	4.4 : 1	0.0250	39.0 : 1	0.8559	0.2 : 1	0.4091	1.4 : 1	0.0655	14.3 : 1	0.8425	0.2 : 1	0.5811	0.7 : 1	0.1092	8.2 : 1	0.8277	0.2 : 1
A-2-3-*	0.1989	4.0 : 1	0	∞	0.8360	0.2 : 1	0.4204	1.4 : 1	0.0311	31.2 : 1	0.8101	0.2 : 1	0.5721	0.7 : 1	0.0903	10.1 : 1	0.7794	0.3 : 1
A-A-2-3, A-2-2-3, A-2-3-3	0.1850	4.4 : 1	0	∞	0.8598	0.2 : 1	0.4051	1.5 : 1	0.0242	40.4 : 1	0.8387	0.2 : 1	0.5689	0.8 : 1	0.0718	12.9 : 1	0.8135	0.2 : 1
A-2-3-x	0.1792	4.6 : 1	0.0312	31.0 : 1	0.8271	0.2 : 1	0.3894	1.6 : 1	0.0852	10.7 : 1	0.8019	0.2 : 1	0.5424	0.8 : 1	0.1479	5.8 : 1	0.7725	0.3 : 1
A-2-4-5	0.1355	6.4 : 1	0.0749	12.3 : 1	0.6254	0.6 : 1	0.3042	2.3 : 1	0.1703	4.9 : 1	0.6266	0.6 : 1	0.4421	1.3 : 1	0.2482	3.0 : 1	0.6296	0.6 : 1
A-A-2-4, A-2-2-4	0.1388	6.2 : 1	0.0463	20.6 : 1	0.6448	0.6 : 1	0.3072	2.3 : 1	0.1220	7.2 : 1	0.6361	0.6 : 1	0.4377	1.3 : 1	0.2030	3.9 : 1	0.6259	0.6 : 1
A-A-A-2, A-2-2-2	0.1295	6.7 : 1	0	∞	0.6107	0.6 : 1	0.2834	2.5 : 1	0.0447	21.4 : 1	0.5932	0.7 : 1	0.3973	1.5 : 1	0.1304	6.7 : 1	0.5725	0.7 : 1
A-A-2-2	0.1295	6.7 : 1	0	∞	0.6061	0.7 : 1	0.2834	2.5 : 1	0.0447	21.4 : 1	0.5899	0.7 : 1	0.3973	1.5 : 1	0.1304	6.7 : 1	0.5703	0.8 : 1
A-A-2-5, A-2-2-5	0.1203	7.3 : 1	0.0648	14.4 : 1	0.5588	0.8 : 1	0.2648	2.8 : 1	0.1645	5.1 : 1	0.5483	0.8 : 1	0.3758	1.7 : 1	0.2650	2.8 : 1	0.5373	0.9 : 1
A-A-2-, A-2-2-	0.1295	6.7 : 1	0	∞	0.5479	0.8 : 1	0.2722	2.7 : 1	0.0559	16.9 : 1	0.5272	0.9 : 1	0.3679	1.7 : 1	0.1598	5.3 : 1	0.5030	1.0 : 1
A-3-4-5, 2-3-4-5	0.1156	7.6 : 1	0.0694	13.4 : 1	0.5373	0.9 : 1	0.2500	3.0 : 1	0.1793	4.6 : 1	0.5176	0.9 : 1	0.3462	1.9 : 1	0.2945	2.4 : 1	0.4951	1.0 : 1
A-2--	0.1295	6.7 : 1	0	∞	0.4969	1.0 : 1	0.2610	2.8 : 1	0.0671	13.9 : 1	0.4721	1.1 : 1	0.3397	1.9 : 1	0.1879	4.3 : 1	0.4443	1.3 : 1
A-A-2-x, A-2-2-x	0.1085	8.2 : 1	0.1020	8.8 : 1	0.5005	1.0 : 1	0.2373	3.2 : 1	0.2372	3.2 : 1	0.4888	1.0 : 1	0.3355	2.0 : 1	0.3548	1.8 : 1	0.4778	1.1 : 1
A-2-x-*	0.1018	8.8 : 1	0.0833	11.0 : 1	0.4729	1.1 : 1	0.2178	3.6 : 1	0.2114	3.7 : 1	0.4510	1.2 : 1	0.2989	2.3 : 1	0.3419	1.9 : 1	0.4273	1.3 : 1
A-2-x-y	0.0728	12.7 : 1	0.1376	6.3 : 1	0.3362	2.0 : 1	0.1703	4.9 : 1	0.3043	2.3 : 1	0.3507	1.9 : 1	0.2580	2.9 : 1	0.4323	1.3 : 1	0.3675	1.7 : 1
A-2-x-x	0.0728	12.7 : 1	0.1376	6.3 : 1	0.3362	2.0 : 1	0.1703	4.9 : 1	0.3043	2.3 : 1	0.3507	1.9 : 1	0.2580	2.9 : 1	0.4323	1.3 : 1	0.3675	1.7 : 1
A-3--, 2-3--	0.0509	18.7 : 1	0.1341	6.5 : 1	0.2364	3.2 : 1	0.1168	7.6 : 1	0.3125	2.2 : 1	0.2418	3.1 : 1	0.1737	4.8 : 1	0.4670	1.1 : 1	0.2484	3.0 : 1
A-4--, 2-4--, 3-4--	0.0185	53.1 : 1	0.1110	8.0 : 1	0.0710	13.1 : 1	0.0456	21.0 : 1	0.2825	2.5 : 1	0.0824	11.1 : 1	0.0748	12.4 : 1	0.4529	1.2 : 1	0.0978	9.2 : 1
A-5--, 2-5--, 3-5--, 4-5--	0.0037	269.3 : 1	0.1258	6.9 : 1	0.0142	69.4 : 1	0.0133	74.1 : 1	0.3148	2.2 : 1	0.0241	40.5 : 1	0.0300	32.4 : 1	0.4977	1.0 : 1	0.0392	24.5 : 1

[edit] Making high hands

The probabilities for making high hands in Omaha hold 'em fall into three categories based on the poker hand:

Rank-based hands that are based solely on the rank type of the starting hand. This includes the poker hands four of a kind, full house, three of a kind, two pair, one pair, and no pair (high card).
Suit-based hands that are based solely on the suit type of the starting hand. The flush is the only poker hand based solely on suit.
Sequence-based hands that are based on the rank sequences in the starting hand. This includes both straights and staight flushes.

[edit] Making hands based on rank type

See the section "Starting hands" for a description of starting hands and rank types.

The probability of making either four of a kind, a full house, three of a kind, two pair, one pair or no pair depends only on the rank type of the starting hand. This ignores when these hands also make straights, flushes and straight flushes—these hands are based on the suit type and rank sequences of the starting hand. Starting hands consisting of four of a kind can only make a full house, two pair or one pair. Starting hands that include at least two cards of the same rank can make no less than one pair. The rank types have the following probabilities of improving on the flop, turn and river.

Rank type	Poker hand	Make on flop		Make by turn		Make by river
Rank type	Poker hand	Probability	Odds	Probability	Odds	Probability	Odds
Four of a kind	Full house	0.0027752	359.33 : 1	0.0109158	90.61 : 1	0.0268270	36.28 : 1
	Two pair	0.1831637	4.46 : 1	0.3378353	1.96 : 1	0.4995375	1.00 : 1
	One pair	0.8140611	0.23 : 1	0.6512488	0.54 : 1	0.4736355	1.11 : 1
Three of a kind	Four of a kind	0.0000578	17,295.00 : 1	0.0002313	4,323.00 : 1	0.0005782	1,728.60 : 1
	Full house	0.0065333	152.06 : 1	0.0252698	38.57 : 1	0.0597826	15.73 : 1
	Three of a kind	0.0661425	14.12 : 1	0.0824237	11.13 : 1	0.0909652	9.99 : 1
	Two pair	0.1640842	5.09 : 1	0.2950971	2.39 : 1	0.4243756	1.36 : 1
	One pair	0.7631822	0.31 : 1	0.5969781	0.68 : 1	0.4242985	1.36 : 1
	Either four of a kind or a full house	0.0065911	150.72 : 1	0.0255011	38.21 : 1	0.0603608	15.57 : 1
	Four of a kind, a full house, or three of a kind	0.0727336	12.75 : 1	0.1079248	8.27 : 1	0.1513259	5.61 : 1
Two pair	Four of a kind	0.0053191	187.00 : 1	0.0106332	93.05 : 1	0.0177048	55.48 : 1
	Full house	0.0178076	55.16 : 1	0.0656337	14.24 : 1	0.1463408	5.83 : 1
	Three of a kind	0.2136910	3.68 : 1	0.2351732	3.25 : 1	0.2220167	3.50 : 1
	Two pair	0.1526364	5.55 : 1	0.2543941	2.93 : 1	0.3376503	1.96 : 1
	One pair	0.6105458	0.64 : 1	0.4341659	1.30 : 1	0.2762874	2.62 : 1
	Either four of a kind or a full house	0.0231267	42.24 : 1	0.0762668	12.11 : 1	0.1640456	5.10 : 1
	Four of a kind, a full house, or three of a kind	0.2368178	3.22 : 1	0.3114400	2.21 : 1	0.3860623	1.59 : 1
One pair	Four of a kind	0.0027752	359.33 : 1	0.0057817	171.96 : 1	0.0100204	98.80 : 1
	Full house	0.0109852	90.03 : 1	0.0413969	23.16 : 1	0.0950871	9.52 : 1
	Three of a kind	0.1259251	6.94 : 1	0.1510947	5.62 : 1	0.1586634	5.30 : 1
	Two pair	0.1665125	5.01 : 1	0.2886216	2.46 : 1	0.3971491	1.52 : 1
	One pair	0.6938020	0.44 : 1	0.5131051	0.95 : 1	0.3390800	1.95 : 1
	Either four of a kind or a full house	0.0137604	71.67 : 1	0.0471785	20.20 : 1	0.1051075	8.51 : 1
	Four of a kind, a full house, or three of a kind	0.1396855	6.16 : 1	0.1982732	4.04 : 1	0.2637709	2.79 : 1
No pair	Four of a kind	0.0002313	4,323.00 : 1	0.0009251	1,080.00 : 1	0.0023127	431.40 : 1
	Full house	0.0062442	159.15 : 1	0.0219241	44.61 : 1	0.0480616	19.81 : 1
	Three of a kind	0.0270583	35.96 : 1	0.0470398	20.26 : 1	0.0699058	13.30 : 1
	Two pair	0.1186401	7.43 : 1	0.1952359	4.12 : 1	0.2654739	2.77 : 1
	One pair	0.5370028	0.86 : 1	0.5691027	0.76 : 1	0.5388950	0.86 : 1
	No pair	0.3108233	2.22 : 1	0.1657724	5.03 : 1	0.0753511	12.27 : 1
	Either four of a kind or a full house	0.0064755	153.43 : 1	0.0228492	42.77 : 1	0.0503742	18.85 : 1
	Four of a kind, a full house, or three of a kind	0.0335338	28.82 : 1	0.0698890	13.31 : 1	0.1202800	7.31 : 1

Not surprisingly, starting with two pair gives the best overall chance of making four of a kind, a full house or three of a kind; one pair has the next best chance for each of these hands. Two pair will improve to at least three of a kind by the river more than one in three times and will make a full house or four of a kind almost one in six times. However, starting with three of a kind is only marginally better than starting with no pair, and starting with three of a kind actually yields the lowest probability of making four of a kind. Starting with four of a kind has very few possibilities to improve—there is almost never a reason to play these hands.

See Probability derivations for making rank-based hands in Omaha hold 'em for the derivations for the probabilities in the preceding tables of making hands based on the rank type of the hand.

[edit] Making a flush

See the section "Starting hands" for a description of starting hands and suit types.

The probability of making a flush depends only on the suit type of the starting hand. This ignores when these hands also make four of a kind and full houses—these hands are based on the rank type of the starting hand. Starting hands consisting of all four suits (suit type abcd) can't make a flush. The starting hands that can make straight flushes are a subset of the hands that can make flushes and the boards that make straight flushes are a subset of the boards that make flushes. The subset of both starting hands and boards that can make straight flushes are based on the rank sequences of their respective suited cards.

To make a flush on the flop, all three cards must be the same suit. If $s$ is the number of cards with the same suit in the hand (s = 2, 3, 4) then the probability $P f$ of making a flush on the flop with that suit is

$P_f = \frac{{13 - s \choose 3}}{{48 \choose 3}}$

On the turn a flush is possible with either four cards of the same suit, or three cards of the same suit combined with one of the $39 - (4 - s) = 35 + s$ cards from one of the other suits, which gives the probability

$P_t = \frac{{13 - s \choose 4} + {13 - s \choose 3}{35 + s \choose 1}}{{48 \choose 4}}$

for completing a flush in a suit by the turn. On the river there are three ways to fill a flush—five cards of the same suit; four cards of the same suit combined with one of the 35 + s cards in one of the other three suits; or three cards of the same suit combinded with two of the remaining 35 + s cards in one of the other three suits. This gives the probability

$P_r = \frac{{13 - s \choose 5} + {13 - s \choose 4}{35 + s \choose 1} + {13 - s \choose 3}{35 + s \choose 2}} {{48 \choose 5}}$

of making the flush by the river. For hands with flush draws in two suits (suit type aabb), multiply the probability of making a flush with two suited cards (s = 2) by the two suits to give probabilities of $2 P f$ , $2 P t$ and $2 P r$ .

The suit types with at least two of the same suit have the following probabilities of making a flush on the flop, turn and river.

Suit type	Make on flop		Make by turn		Make by river
Suit type	Probability	Odds	Probability	Odds	Probability	Odds
aaaa	0.0048566	204.90 : 1	0.01748381	56.20 : 1	0.0392944	24.45 : 1
aaab	0.0069380	143.13 : 1	0.02451434	39.79 : 1	0.0540745	17.49 : 1
aabb	0.0190796	51.41 : 1	0.06614246	14.12 : 1	0.1431545	5.99 : 1
aabc	0.0095398	103.82 : 1	0.03307123	29.24 : 1	0.0715772	12.97 : 1

[edit] Making a straight flush

See the section "Starting hands for straight flushes" for a description of straight flush starting hand types.

The probability of making a straight flush depends primarily on the number of different sets of three cards that can fill a straight flush in the hand. For convenience, the term straight flush sequence means a three-card set that can make a straight flush when combined with the starting hand. A secondary factor to the number of straight flush sequences, although much less significant, is the amount of overlap (shared cards) in the straight flush sequences—the more overlap, the lower the probability for a straight flush on the turn and river. More overlap reduces the probability because some of the board combinations make more than one straight flush and are thus counted multiple times.

For example, the hand 5♥ 6♥ 7♥ A♣ makes a straight flush with either of the two straight flush sequences {2♥ 3♥ 4♥} or {8♥ 9♥ 10♥}, which have no overlap. The hand 5♥ 8♥ A♣ A♠ makes a straight flush with either of the two straight flush sequences {4♥ 6♥ 7♥} or {6♥ 7♥ 9♥}, which have an overlap of two cards—the 6♥ and 7♥. On the turn, the board 4♥ 6♥ 7♥ 9♥ satisfies both sets and is only counted once, making the frequency of boards that make a straight flush on the turn for 5♥ 8♥ A♣ A♠ one less than for 5♥ 6♥ 7♥ A♣. On the river, any of the 44 remaining cards combined with 4♥ 6♥ 7♥ 9♥ makes two straight flushes, making the frequency of boards 44 less.

To make a straight flush on the flop, the three cards on the board must exactly match one of the straight flush sequences for the hand. If $s$ is the number of straight flush sequences for a hand, then the frequency $F f$ of boards that make a straight flush on the flop is

$F_f = {s \choose 1}$

On the turn, one of the $s$ straight flush sequences can be combined with any of the remaining 45 cards. Enumerating the frequencies this way ends up counting any board that can form two different straight flushes twice. On the turn, for every pair of three card sets that share two cards (for example {A♠ 3♠ 4♠} and {3♠ 4♠ 6♠}) there is exactly one board that makes two straight flushes ({A♠ 3♠ 4♠ 6♠} in the example). Where $n 42$ is the number of boards containing four cards that make exactly two straight flushes, then the frequency $F t$ of boards that make a straight flush on the turn is

$F_t = {s \choose 1}{45 \choose 1} - n_{42}$

On the river, one of the $s$ straight flush sequences can be combined with any two of the remaining 45 cards. Now all boards that make exactly two straight flushes are counted twice, and all boards the make exactly three straight flushes are counted three time. Where $n 52$ is the number of boards containing five cards that make exactly two straight flushes and $n 53$ is the number of boards containing five cards that make exactly three straight flushes, then the frequency $F r$ of boards that make a straight flush on the river is

$F_r = {s \choose 1}{45 \choose 2} - n_{52} - 2n_{53}$

The probabilities of making a straight flush are the same for any two starting hands that can make a straight flush with exactly two straight flush sequences that contain no overlap. So A♥ 2♥ 10♥ A♣ (which makes a straight flush with either {3♥ 4♥ 5♥} or {J♥ Q♥ K♥}) and 5♥ 6♥ 7♥ A♣ have the same probabilities on the flop, turn and river. Likewise, 2♥ 3♥ A♣ A♠ (which makes a straight flush with either {A♥ 4♥ 5♥} or {4♥ 5♥ 6♥}) and 5♥ 8♥ A♣ A♠ each make a straight flush with one of two straight flush sequences, with two cards overlapping.

A complete straight flush hand pattern is then the number of straight flush sequences for the hand combined with the overlaps between all of the straight flush sequences. The following rules can be used to derive a notation for describing complete straight flush hand patterns:

Label each element in each straight flush sequence according to the number of straight flush sequences that contain it.
Sort the labelled elements of each straight flush sequence to derive the labeled straight flush sequence. Arbitrarily, choose to sort the elements in decending order.
Sort the labelled straight flush sequences. Arbitrarily, choose to sort the straight flush sequences in decending order.

Each element can be either the low, middle, or high rank of a straight flush sequence. Using numbers to label the straight flush sequence elements, each element in a straight flush sequence is assigned a label from 1–3 depending on whether is appears in 1, 2 or 3 straight flush sequences. For example, the two straight flush sequences for a hand {2♥ 3♥ 4♥} and {8♥ 9♥ 10♥} are labelled as 111 and 111 since each element appears in only a single straight flush sequence, and the straight flush hand pattern is 111+111. The two straight flush sequences {4♥ 6♥ 7♥} and {6♥ 7♥ 9♥} are labelled as 221 and 221, and the straight flush hand pattern is 221+221.

To determine the probability of making a straight flush from any starting hand, first identify all of the straight flush hand patterns, and then determine the probabilities for each hand pattern. It turns out that there are 32 hand patterns possible using a single suit to make the straight flush, with either 2, 3, or 4 cards from the suit being used to make straight flushes. The following table shows each of the single-suit straight flush hand patterns, listed in order of probability of making a straight flush on the river, from highest to lowest probability.

Straight flush hand pattern	Make on flop			Make by turn			Make by river
Straight flush hand pattern	Combos	Probability	Odds	Combos	Probability	Odds	Combos	Probability	Odds
333+333+333+333+332+332+322+322	8	0.0004625	2,161 : 1	353	0.0018142	550 : 1	7,611	0.0044449	224 : 1
332+332+322+322+322+322+322	7	0.0004047	2,470 : 1	310	0.0015932	627 : 1	6,708	0.0039175	254 : 1
333+333+333+332+332+321+321	7	0.0004047	2,470 : 1	309	0.0015880	629 : 1	6,666	0.0038930	256 : 1
332+332+322+321+222+221	6	0.0003469	2,882 : 1	266	0.0013670	731 : 1	5,763	0.0033656	296 : 1
322+322+322+322+321+321	6	0.0003469	2,882 : 1	266	0.0013670	731 : 1	5,763	0.0033656	296 : 1
333+332+332+322+321+211	6	0.0003469	2,882 : 1	266	0.0013670	731 : 1	5,763	0.0033656	296 : 1
333+333+332+332+321+321	6	0.0003469	2,882 : 1	265	0.0013619	733 : 1	5,720	0.0033405	298 : 1
222+222+221+221+221	5	0.0002891	3,458 : 1	223	0.0011461	872 : 1	4,860	0.0028383	351 : 1
222+222+222+221+211	5	0.0002891	3,458 : 1	223	0.0011461	872 : 1	4,860	0.0028383	351 : 1
322+322+321+221+211	5	0.0002891	3,458 : 1	223	0.0011461	872 : 1	4,860	0.0028383	351 : 1
322+321+321+221+221	5	0.0002891	3,458 : 1	222	0.0011409	875 : 1	4,818	0.0028138	354 : 1
332+332+321+321+111	5	0.0002891	3,458 : 1	222	0.0011409	875 : 1	4,818	0.0028138	354 : 1
322+322+321+222+221	5	0.0002891	3,458 : 1	222	0.0011409	875 : 1	4,817	0.0028132	354 : 1
332+332+322+321+211	5	0.0002891	3,458 : 1	222	0.0011409	875 : 1	4,817	0.0028132	354 : 1
333+332+332+321+321	5	0.0002891	3,458 : 1	221	0.0011358	879 : 1	4,774	0.0027881	358 : 1
222+221+211+111	4	0.0002313	4,323 : 1	179	0.0009199	1,086 : 1	3,915	0.0022864	436 : 1
221+221+211+211	4	0.0002313	4,323 : 1	179	0.0009199	1,086 : 1	3,915	0.0022864	436 : 1
222+222+211+211	4	0.0002313	4,323 : 1	179	0.0009199	1,086 : 1	3,914	0.0022858	436 : 1
221+221+221+221	4	0.0002313	4,323 : 1	178	0.0009148	1,092 : 1	3,872	0.0022613	441 : 1
322+321+321+111	4	0.0002313	4,323 : 1	178	0.0009148	1,092 : 1	3,872	0.0022613	441 : 1
222+222+221+221	4	0.0002313	4,323 : 1	178	0.0009148	1,092 : 1	3,871	0.0022607	441 : 1
322+322+321+211	4	0.0002313	4,323 : 1	178	0.0009148	1,092 : 1	3,871	0.0022607	441 : 1
332+332+321+321	4	0.0002313	4,323 : 1	177	0.0009097	1,098 : 1	3,828	0.0022356	446 : 1
111+111+111	3	0.0001735	5,764 : 1	135	0.0006938	1,440 : 1	2,970	0.0017345	576 : 1
211+211+111	3	0.0001735	5,764 : 1	135	0.0006938	1,440 : 1	2,969	0.0017339	576 : 1
221+221+111	3	0.0001735	5,764 : 1	134	0.0006887	1,451 : 1	2,926	0.0017088	584 : 1
222+221+211	3	0.0001735	5,764 : 1	134	0.0006887	1,451 : 1	2,925	0.0017082	584 : 1
322+321+321	3	0.0001735	5,764 : 1	133	0.0006835	1,462 : 1	2,882	0.0016831	593 : 1
111+111	2	0.0001156	8,647 : 1	90	0.0004625	2,161 : 1	1,980	0.0011563	864 : 1
211+211	2	0.0001156	8,647 : 1	90	0.0004625	2,161 : 1	1,979	0.0011558	864 : 1
221+221	2	0.0001156	8,647 : 1	89	0.0004574	2,185 : 1	1,936	0.0011306	883 : 1
111	1	0.0000578	17,295 : 1	45	0.0002313	4,323 : 1	990	0.0005782	1,729 : 1

Of the 32 hand patterns that can make a straight flush in one suit, four of the hand patterns (111, 211+211, 322+321+321, and 332+332+321+321) use exactly two cards in the suit. For hands that can make a straight flush in two suits, each of these hand patterns can be used by one of the two suits. This gives $\begin{matrix} {4 + 2 - 1 \choose 2} \end{matrix} = 10$ different combinations of single suit hand patterns for making a straight flush in one of two suits. There is no overlap in the straight flush sequences between suits and it is not possible to make a straight flush in more than one suit. Therefore, where $P 1$ and $P 2$ are the probabilities of making a flush in each of the two suits, then the probability $P$ of making a straight flush in either suit is $P = P 1 + P 2$ . The following table gives the double-suit straight flush hand patterns, listed in order of probability of making a straight flush on the river, from highest to lowest probability.

Straight flush hand pattern		Make on flop			Make by turn			Make by river
Suit 1	Suit 2	Combos	Probability	Odds	Combos	Probability	Odds	Combos	Probability	Odds
332+332+321+321	332+332+321+321	8	0.0004625	2,161 : 1	354	0.0018193	549 : 1	7,656	0.0044712	223 : 1
322+321+321	332+332+321+321	7	0.0004047	2,470 : 1	310	0.0015932	627 : 1	6,710	0.0039187	254 : 1
211+211	332+332+321+321	6	0.0003469	2,882 : 1	267	0.0013722	728 : 1	5,807	0.0033913	294 : 1
322+321+321	322+321+321	6	0.0003469	2,882 : 1	266	0.0013670	731 : 1	5,764	0.0033662	296 : 1
211+211	322+321+321	5	0.0002891	3,458 : 1	223	0.0011461	872 : 1	4,861	0.0028389	351 : 1
111	332+332+321+321	5	0.0002891	3,458 : 1	222	0.0011409	875 : 1	4,818	0.0028138	354 : 1
211+211	211+211	4	0.0002313	4,323 : 1	180	0.0009251	1,080 : 1	3,958	0.0023115	432 : 1
111	322+321+321	4	0.0002313	4,323 : 1	178	0.0009148	1,092 : 1	3,872	0.0022613	441 : 1
111	211+211	3	0.0001735	5,764 : 1	135	0.0006938	1,440 : 1	2,969	0.0017339	576 : 1
111	111	2	0.0001156	8,647 : 1	90	0.0004625	2,161 : 1	1,980	0.0011563	864 : 1

[edit] See also

Derivations and probability tables for Omaha:

Poker topics:

Math and probability topics:

[edit] Notes

^{^} The odds presented in this article use the notation x : 1 which translates to x to 1 odds against the event happening. The odds are calculated from the probability p of the event happening using the formula: odds = [(1 − p) ÷ p] : 1, or odds = [(1 ÷ p) − 1] : 1. Another way of expressing the odds x : 1 is to state that there is a 1 in x+1 chance of the event occurring or the probability of the event occurring is 1 ÷ (x + 1). So for example, the odds of a role of a fair six-sided die coming up three is 5 : 1 against because there are 5 chances for a number other than three and 1 chance for a three; alternatively, this could be described as a 1 in 6 chance or $\begin{matrix}\frac{1}{6}\end{matrix}$ probability of a three being rolled because the three is 1 of 6 equally-likely possible outcomes.

[edit] References

^ Edward Hutchison. Hutchison Omaha Point System (HTML). Retrieved on 2006-11-02.
^ Edward Hutchison (December, 1997). Hutchison Point Count System for Omaha High-Low Poker (HTML). Canadian Poker Monthly. Retrieved on 2006-11-02.
^ Ian Berry. Mr Hutchison, You Suck.... (HTML). bet-the-pot. Retrieved on 2006-11-02.
^ Brian Alspach (2003). Rank Sets and Straights. Retrieved on 2006-10-30.

[edit] External links

Poker Computations by Brian Alspach
Omaha odds calculator on www.cardplayer.com

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