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 T-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-T-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-T-J that can make a straight with 20 different combinations of three ranks (5-6-7, 6-7-8, 6-7-9, 6-7-T, 7-8-9, 7-8-T, 7-8-J, 7-9-T, 7-9-J, 7-T-J, 8-9-Q, 8-T-Q, 8-J-Q, 9-T-Q, 9-J-Q, 9-Q-K, T-J-Q, T-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 T-Q-K or 3-4-5), and four ranks (e.g. 3-9-K-A makes a straight with 2-4-5 or T-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 ranks ({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 preceeding 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. AaAbKaKb is the strongest starting hand in Omaha, 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 the vast majority of the time when a qualifying low hand is possible.

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 choosen 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 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. 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.

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.

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
Royal flush	40	0.0018	551.5 : 1	1,880	0.0069	143.0 : 1	43,240	0.0166	59.11 : 1
Straight flush	216	0.0098	101.3 : 1	9,832	0.0363	26.54 : 1	218,680	0.0841	10.88 : 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 and royal flushs at showdown, 10% of the boards will have one as the nut hand by the river.

[edit] Derivation of the nut boards

A royal flush is possible whenever the board contains exactly three cards from 10 to A of the same suit. For the flop there are

${5 \choose 3}{4 \choose 1} = 40$

combinations. At the turn and river the three cards to a royal flush can be combined with any of the 47 remaining cards that aren't part of the royal flush, giving

${5 \choose 3}{4 \choose 1}{47 \choose 1} = 1,880$ and ${5 \choose 3}{4 \choose 1}{47 \choose 2} = 43,240$

combinations, respectively.

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. The royal flushes have already been accounted for and are subtracted from the straight flush combinations. This gives

$64 \times {4 \choose 1} - 40 = 216$

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 yeild 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 ranks 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

$432 \times {4 \choose 1} + 64 \times {4 \choose 1}{39 \choose 1} - 1,880 = 9,832$

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

$1,208 \times {4 \choose 1} + 432 \times {4 \choose 1}{39 \choose 1} + 64 \times {4 \choose 1}{39 \choose 2} - 43,240 = 218,680$

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 ranks 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 ranks sets that doesn't allow a straight or with three cards of the same suit that form one of the 222 ranks 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 ranks 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 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 soley 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 soley on the suit type of the starting hand. The flush is the only poker hand based soley 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 yeilds 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.

[edit] Derivations for making rank-based hands

The derivations for starting hands making four of a kind, a full house, three of a kind, two pair, one pair and no pair are separate for each of the starting hand rank types. The derivations require identifying the individual cases that yeild each possible hand and are sometimes rather detailed, so it is useful to use a notation to indicate the shape of the board for each case. The rank type of the hand is shown using upper case letters to indicate ranks. The ranks on the board are indicated using upper case letters for matches with the starting hand and lower case letters to indicate ranks that don't match the starting hand. So the rank type XXYZ is any hand with a pair of X with two additional ranks Y and Z and the board XYr represents a flop that contains one X, one of the non-paired ranks Y and one other rank r. Note that since Y and Z have an identical relationship to the starting hand—each represents an unpaired rank—XYr and XZr represent the same set of boards and are interchangeable, so derivations for this hand choose one of the two choices represented by Y. In addition to the upper and lower case letters, * is used to represent any rank not already represented on the board, and ? is used to represent any rank not already represented on the board and not included in the starting hand. So for the rank type XXYZ, the board XX* represents a flop that contains two Xs and any other rank (including Y and Z), but X?? is any flop that contains an X and any two cards of a rank other than X, Y or Z, and rrr?? is any board on the river that contains three cards of rank r and any two cards of ranks other than X, Y, Z or r.

Each table shows all of the boards that can make each hand and the derivation for the combinations for that board. Probabilities are determined by dividing the number of combinations for each hand by the $\begin{matrix} {48 \choose 3} = 17,296 \end{matrix}$ boards on the flop, $\begin{matrix} {48 \choose 4} = 194,580 \end{matrix}$ boards on the turn, and $\begin{matrix} {48 \choose 5} = 1,712,304 \end{matrix}$ boards at the river. The probabilities for the boards in each table total 1.0.

[edit] Derivations for starting hands with four of a kind

Starting hands with four of a kind (XXXX) can only improve to a full house or two pair. To make a full house, this hand needs to have two or three cards of the same rank appear on the board. To make two pair, another pair on the board is needed. Of course, any other hand holding a pair also makes at least a full house or two with either of these boards. The following table shows the derivations for making a full house, two pair or one pair when holding four of a kind.

**Derivations for rank type XXXX (four of a kind) on the flop**
Hand to make	Board	Derivation	Combos	Probability	Odds
Full house	rrr	$\begin{matrix} {12 \choose 1}{4 \choose 3} \end{matrix}$	48	0.0027752	359.3 : 1
Two pair	rrs	$\begin{matrix} {12 \choose 1}{4 \choose 2}{44 \choose 1} \end{matrix}$	3,168	0.1831637	4.5 : 1
One pair	rst	$\begin{matrix} {12 \choose 3}{4 \choose 1}^3 \end{matrix}$	14,080	0.8140611	0.2 : 1

**Derivations for rank type XXXX (four of a kind) on the turn**
Hand to make	Board	Derivation	Combos	Probability	Odds
Full house	rrr*	$\begin{matrix} {12 \choose 1}{4 \choose 3}{44 \choose 1} \end{matrix}$	2,112	0.0108541	91.1 : 1
	rrrr	$\begin{matrix} {12 \choose 1}{4 \choose 4} \end{matrix}$	12	0.0000617	16,214.0 : 1
	Total		2,124	0.0109158	90.6 : 1
Two pair	rrss	$\begin{matrix} {12 \choose 2}{4 \choose 2}^2 \end{matrix}$	2,376	0.0122109	80.9 : 1
	rrst	$\begin{matrix} {12 \choose 1}{4 \choose 2}{11 \choose 2}{4 \choose 1}^2 \end{matrix}$	63,360	0.3256244	2.1 : 1
	Total		65,736	0.3378353	2.0 : 1
One pair	rstu	$\begin{matrix} {12 \choose 4}{4 \choose 1}^4 \end{matrix}$	126,720	0.6512488	0.5 : 1

**Derivations for rank type XXXX (four of a kind) on the river**
Hand to make	Board	Derivation	Combos	Probability	Odds
Full house	rrr**	$\begin{matrix} {12 \choose 1}{4 \choose 3}{44 \choose 2} \end{matrix}$	45,408	0.0265187	36.7 : 1
	rrrr*	$\begin{matrix} {12 \choose 1}{4 \choose 4}{44 \choose 1} \end{matrix}$	528	0.0003084	3,242.0 : 1
	Total		45,936	0.0268270	36.3 : 1
Two pair	rrsst	$\begin{matrix} {12 \choose 2}{4 \choose 2}^2{40 \choose 1} \end{matrix}$	95,040	0.0555042	17.0 : 1
	rrstu	$\begin{matrix} {12 \choose 1}{4 \choose 2}{11 \choose 3}{4 \choose 1}^3 \end{matrix}$	760,320	0.4440333	1.3 : 1
	Total		855,360	0.4995375	1.0 : 1
One pair	rstuv	$\begin{matrix} {12 \choose 5}{4 \choose 1}^5 \end{matrix}$	811,008	0.4736355	1.1 : 1

[edit] Derivations for starting hands with three of a kind

To make a full house or three or four of a kind, starting hands with three of a kind (XXXY) need to either catch the case (last) X or catch two or three of the remaining Y cards (YY or YYY). They also improve to a full house if three or more of another rank appears on the board (rrr or rrrr), although any other hand holding a pair also makes a full house with this board. Three of a kind makes two pair if either a Y card or another pair appears on the board. The following tables show all the ways for XXXY to make four of a kind, a full house, three of a kind, two pair or one pair on the flop, turn and river.

**Derivations for rank type XXXY (three of a kind) on the flop**
Hand to make	Board	Derivation	Combos	Probability	Odds
Four of a kind	YYY	$\begin{matrix} {3 \choose 3} \end{matrix}$	1	0.0000578	17,295.0 : 1
Full house	XYY	$\begin{matrix} {1 \choose 1}{3 \choose 2} \end{matrix}$	3	0.0001735	5,764.3 : 1
	Xrr	$\begin{matrix} {1 \choose 1}{11 \choose 1}{4 \choose 2} \end{matrix}$	66	0.0038159	261.1 : 1
	rrr	$\begin{matrix} {11 \choose 1}{4 \choose 3} \end{matrix}$	44	0.0025439	392.1 : 1
	Total		113	0.0065333	152.1 : 1
Three of a kind	XYr	$\begin{matrix} {1 \choose 1}{3 \choose 1}{44 \choose 1} \end{matrix}$	132	0.0076318	130.0 : 1
	Xrs	$\begin{matrix} {1 \choose 1}{11 \choose 2}{4 \choose 1}^2 \end{matrix}$	880	0.0508788	18.7 : 1
	YYr	$\begin{matrix} {3 \choose 2}{44 \choose 1} \end{matrix}$	132	0.0076318	130.0 : 1
	Total		1,144	0.0661425	14.1 : 1
Two pair	Yrr	$\begin{matrix} {3 \choose 1}{11 \choose 1}{4 \choose 2} \end{matrix}$	198	0.0114477	86.4 : 1
	rrs	$\begin{matrix} {11 \choose 1}{4 \choose 2}{40 \choose 1} \end{matrix}$	2,640	0.1526364	5.6 : 1
	Total		2,838	0.1640842	5.1 : 1
One pair	Yrs	$\begin{matrix} {3 \choose 1}{11 \choose 2}{4 \choose 1}^2 \end{matrix}$	2,640	0.1526364	5.6 : 1
	rst	$\begin{matrix} {11 \choose 3}{4 \choose 1}^3 \end{matrix}$	10,560	0.6105458	0.6 : 1
	Total		13,200	0.7631822	0.3 : 1

**Derivations for rank type XXXY (three of a kind) on the turn**
Hand to make	Board	Derivation	Combos	Probability	Odds
Four of a kind	YYY*	$\begin{matrix} {3 \choose 3}{45 \choose 1} \end{matrix}$	45	0.0002313	4,323.0 : 1
Full house	XYYr	$\begin{matrix} {1 \choose 1}{3 \choose 2}{44 \choose 1} \end{matrix}$	132	0.0006784	1,473.1 : 1
	XYrr	$\begin{matrix} {1 \choose 1}{3 \choose 1}{11 \choose 1}{4 \choose 2} \end{matrix}$	198	0.0010176	981.7 : 1
	Xrrs	$\begin{matrix} {1 \choose 1}{11 \choose 1}{4 \choose 2}{40 \choose 1} \end{matrix}$	2,640	0.0135677	72.7 : 1
	rrr*	$\begin{matrix} {11 \choose 1}{4 \choose 3}{44 \choose 1} \end{matrix}$	1,936	0.0099496	99.5 : 1
	rrrr	$\begin{matrix} {11 \choose 1}{4 \choose 4} \end{matrix}$	11	0.0000565	17,688.1 : 1
	Total		4,917	0.0252698	38.6 : 1
Three of a kind	XYrs	$\begin{matrix} {1 \choose 1}{3 \choose 1}{11 \choose 2}{4 \choose 1}^2 \end{matrix}$	2,640	0.0135677	72.7 : 1
	Xrst	$\begin{matrix} {1 \choose 1}{11 \choose 3}{4 \choose 1}^3 \end{matrix}$	10,560	0.0542707	17.4 : 1
	YY??	$\begin{matrix} {3 \choose 2}{44 \choose 2} \end{matrix}$	2,838	0.0145853	67.6 : 1
	Total		16,038	0.0824237	11.1 : 1
Two pair	Yrrs	$\begin{matrix} {3 \choose 1}{11 \choose 1}{4 \choose 2}{40 \choose 1} \end{matrix}$	7,920	0.0407031	23.6 : 1
	rrss	$\begin{matrix} {11 \choose 2}{4 \choose 2}^2 \end{matrix}$	1,980	0.0101758	97.3 : 1
	rrst	$\begin{matrix} {11 \choose 1}{4 \choose 2}{10 \choose 2}{4 \choose 1}^2 \end{matrix}$	47,520	0.2442183	3.1 : 1
	Total		57,420	0.2950971	2.4 : 1
One pair	Yrst	$\begin{matrix} {3 \choose 1}{11 \choose 3}{4 \choose 1}^3 \end{matrix}$	31,680	0.1628122	5.1 : 1
	rstu	$\begin{matrix} {11 \choose 4}{4 \choose 1}^4 \end{matrix}$	84,480	0.4341659	1.3 : 1
	Total		116,160	0.5969781	0.7 : 1

**Derivations for rank type XXXY (three of a kind) on the river**
Hand to make	Board	Derivation	Combos	Probability	Odds
Four of a kind	YYY**	$\begin{matrix} {3 \choose 3}{45 \choose 2} \end{matrix}$	990	0.0005782	1,728.6 : 1
Full house	XYY??	$\begin{matrix} {1 \choose 1}{3 \choose 2}{44 \choose 2} \end{matrix}$	2,838	0.0016574	602.3 : 1
	XYrrs	$\begin{matrix} {1 \choose 1}{3 \choose 1}{11 \choose 1}{4 \choose 2}{40 \choose 1} \end{matrix}$	7,920	0.0046253	215.2 : 1
	Xrrss	$\begin{matrix} {1 \choose 1}{11 \choose 2}{4 \choose 2}^2 \end{matrix}$	1,980	0.0011563	863.8 : 1
	Xrrst	$\begin{matrix} {1 \choose 1}{11 \choose 1}{4 \choose 2}{10 \choose 2}{4 \choose 1}^2 \end{matrix}$	47,520	0.0277521	35.0 : 1
	rrr**	$\begin{matrix} {11 \choose 1}{4 \choose 3}{44 \choose 2} \end{matrix}$	41,624	0.0243088	40.1 : 1
	rrrr*	$\begin{matrix} {11 \choose 1}{4 \choose 4}{44 \choose 1} \end{matrix}$	484	0.0002827	3,536.8 : 1
	Total		102,366	0.0597826	15.7 : 1
Three of a kind	XYrst	$\begin{matrix} {1 \choose 1}{3 \choose 1}{11 \choose 3}{4 \choose 1}^3 \end{matrix}$	31,680	0.0185014	53.1 : 1
	Xrstu	$\begin{matrix} {1 \choose 1}{11 \choose 4}{4 \choose 1}^4 \end{matrix}$	84,480	0.0493370	19.3 : 1
	YYrrs	$\begin{matrix} {3 \choose 2}{11 \choose 1}{4 \choose 2}{40 \choose 1} \end{matrix}$	7,920	0.0046253	215.2 : 1
	YYrst	$\begin{matrix} {3 \choose 2}{11 \choose 3}{4 \choose 2}^3 \end{matrix}$	31,680	0.0185014	53.1 : 1
	Total		155,760	0.0909652	10.0 : 1
Two pair	Yrrss	$\begin{matrix} {3 \choose 1}{11 \choose 2}{4 \choose 2}^2 \end{matrix}$	5,940	0.0034690	287.3 : 1
	Yrrst	$\begin{matrix} {3 \choose 1}{11 \choose 1}{4 \choose 2}{10 \choose 2}{4 \choose 1}^2 \end{matrix}$	142,560	0.0832562	11.0 : 1
	rrsst	$\begin{matrix} {11 \choose 2}{4 \choose 2}^2{36 \choose 1} \end{matrix}$	71,280	0.0416281	23.0 : 1
	rrstu	$\begin{matrix} {11 \choose 1}{4 \choose 2}{10 \choose 3}{4 \choose 1}^3 \end{matrix}$	506,880	0.2960222	2.4 : 1
	Total		726,660	0.4243756	1.4 : 1
One pair	Yrstu	$\begin{matrix} {3 \choose 1}{11 \choose 4}{4 \choose 1}^4 \end{matrix}$	253,440	0.1480111	5.8 : 1
	rstuv	$\begin{matrix} {11 \choose 5}{4 \choose 1}^5 \end{matrix}$	473,088	0.2762874	2.6 : 1
	Total		726,528	0.4242985	1.4 : 1

[edit] Derivations for starting hands with two pair

Starting hands with two pair (XXYY) can improve to three of a kind, a full house or four of a kind when one or more of the four remaining X or Y cards appears (X, XX or XY). They also improve to a full house if three or more of another rank appears on the board (rrr or rrrr), although any other hand holding a pair also makes at least a full house with this board. If another pair appears the hand makes two pair, although any other hand holding a pair also makes at least two pair. The following tables show all the ways for XXYY to make four of a kind, a full house, three of a kind, two pair or one pair on the flop, turn and river.

**Derivations for rank type XXYY (two pair) on the flop**
Hand to make	Board	Derivation	Combos	Probability	Odds
Four of a kind	XX*	$\begin{matrix} {2 \choose 2}{2 \choose 1}{46 \choose 1} \end{matrix}$	92	0.0053191	187.0 : 1
Full house	Xrr	$\begin{matrix} {2 \choose 1}{2 \choose 1}{11 \choose 1}{4 \choose 2} \end{matrix}$	264	0.0152636	64.5 : 1
	rrr	$\begin{matrix} {11 \choose 1}{4 \choose 3} \end{matrix}$	44	0.0025439	392.1 : 1
	Total		308	0.0178076	55.2 : 1
Three of a kind	XY?	$\begin{matrix} {2 \choose 1}^2{44 \choose 1} \end{matrix}$	176	0.0101758	97.3 : 1
	Xrs	$\begin{matrix} {2 \choose 1}{2 \choose 1}{11 \choose 2}{4 \choose 1}^2 \end{matrix}$	3,520	0.2035153	3.9 : 1
	Total		3,696	0.2136910	3.7 : 1
Two pair	rrs	$\begin{matrix} {11 \choose 1}{4 \choose 2}{40 \choose 1} \end{matrix}$	2,640	0.1526364	5.6 : 1
One pair	rst	$\begin{matrix} {11 \choose 3}{4 \choose 1}^3 \end{matrix}$	10,560	0.6105458	0.6 : 1

**Derivations for rank type XXYY (two pair) on the turn**
Hand to make	Board	Derivation	Combos	Probability	Odds
Four of a kind	XXYY	$\begin{matrix} {2 \choose 2}{2 \choose 2} \end{matrix}$	1	0.0000051	194,579.0 : 1
	XXYr	$\begin{matrix} {2 \choose 2}{2 \choose 1}{2 \choose 1}{44 \choose 1} \end{matrix}$	176	0.0009045	1,104.6 : 1
	XX??	$\begin{matrix} {2 \choose 2}{2 \choose 1}{44 \choose 2} \end{matrix}$	1,892	0.0097235	101.8 : 1
	Total		2,069	0.0106332	93.0 : 1
Full house	XYrr	$\begin{matrix} {2 \choose 1}^2{11 \choose 1}{4 \choose 2} \end{matrix}$	264	0.0013568	736.0 : 1
	Xrrr	$\begin{matrix} {2 \choose 1}{2 \choose 1}{11 \choose 1}{4 \choose 3} \end{matrix}$	176	0.0009045	1,104.6 : 1
	Xrrs	$\begin{matrix} {2 \choose 1}{2 \choose 1}{11 \choose 1}{4 \choose 2}{40 \choose 1} \end{matrix}$	10,560	0.0542707	17.4 : 1
	rrrr	$\begin{matrix} {11 \choose 1}{4 \choose 4} \end{matrix}$	11	0.0000565	17,688.1 : 1
	rrrs	$\begin{matrix} {11 \choose 1}{4 \choose 3}{40 \choose 1} \end{matrix}$	1,760	0.0090451	109.6 : 1
	Total		12,771	0.0656337	14.2 : 1
Three of a kind	XYrs	$\begin{matrix} {2 \choose 1}^2{11 \choose 2}{4 \choose 1}^2 \end{matrix}$	3,520	0.0180902	54.3 : 1
	Xrst	$\begin{matrix} {2 \choose 1}{2 \choose 1}{11 \choose 3}{4 \choose 1}^3 \end{matrix}$	42,240	0.2170829	3.6 : 1
	Total		45,760	0.2351732	3.3 : 1
Two pair	rrss	$\begin{matrix} {11 \choose 2}{4 \choose 2}^2 \end{matrix}$	1,980	0.0101758	97.3 : 1
	rrst	$\begin{matrix} {11 \choose 1}{4 \choose 2}{10 \choose 2}{4 \choose 1}^2 \end{matrix}$	47,520	0.2442183	3.1 : 1
	Total		49,500	0.2543941	2.9 : 1
One pair	rstu	$\begin{matrix} {11 \choose 4}{4 \choose 1}^4 \end{matrix}$	84,480	0.4341659	1.3 : 1

**Derivations for rank type XXYY (two pair) on the river**
Hand to make	Board	Derivation	Combos	Probability	Odds
Four of a kind	XXYY*	$\begin{matrix} {2 \choose 2}{2 \choose 2}{44 \choose 1} \end{matrix}$	44	0.0000257	38,917.0 : 1
	XXY??	$\begin{matrix} {2 \choose 1}{2 \choose 2}{2 \choose 1}{44 \choose 2} \end{matrix}$	3,784	0.0022099	451.5 : 1
	XX???	$\begin{matrix} {2 \choose 1}{2 \choose 2}{44 \choose 3} \end{matrix}$	26,488	0.0154692	63.6 : 1
	Total		30,316	0.0177048	55.5 : 1
Full house	XYrrr	$\begin{matrix} {2 \choose 1}^2{11 \choose 1}{4 \choose 3} \end{matrix}$	176	0.0001028	9,728.0 : 1
	XYrrs	$\begin{matrix} {2 \choose 1}^2{11 \choose 1}{4 \choose 2}{40 \choose 1} \end{matrix}$	10,560	0.0061671	161.2 : 1
	Xrrrs	$\begin{matrix} {2 \choose 1}{2 \choose 1}{11 \choose 1}{4 \choose 3}{40 \choose 1} \end{matrix}$	7,040	0.0041114	242.2 : 1
	Xrrss	$\begin{matrix} {2 \choose 1}{2 \choose 1}{11 \choose 2}{4 \choose 2}^2 \end{matrix}$	7,922	0.0046253	215.2 : 1
	Xrrst	$\begin{matrix} {2 \choose 1}{2 \choose 1}{11 \choose 1}{4 \choose 2}{10 \choose 2}{4 \choose 1}^2 \end{matrix}$	190,080	0.1110083	8.0 : 1
	rrr??	$\begin{matrix} {11 \choose 1}{4 \choose 3}{40 \choose 2} \end{matrix}$	34,320	0.0200432	48.9 : 1
	rrrr*	$\begin{matrix} {11 \choose 1}{4 \choose 4}{44 \choose 1} \end{matrix}$	484	0.0002827	3,536.8 : 1
	Total		250,580	0.1463408	5.8 : 1
Three of a kind	XYrst	$\begin{matrix} {2 \choose 1}^2{11 \choose 3}{4 \choose 1}^3 \end{matrix}$	42,240	0.0246685	39.5 : 1
	Xrstu	$\begin{matrix} {2 \choose 1}{2 \choose 1}{11 \choose 4}{4 \choose 1}^4 \end{matrix}$	337,920	0.1973481	4.1 : 1
	Total		380,160	0.2220167	3.5 : 1
Two pair	rrsst	$\begin{matrix} {11 \choose 2}{4 \choose 2}^2{36 \choose 1} \end{matrix}$	71,280	0.0416281	23.0 : 1
	rrstu	$\begin{matrix} {11 \choose 1}{4 \choose 2}{10 \choose 3}{4 \choose 1}^3 \end{matrix}$	506,880	0.2960222	2.4 : 1
	Total		578,160	0.3376503	2.0 : 1
One pair	rstuv	$\begin{matrix} {11 \choose 5}{4 \choose 1}^5 \end{matrix}$	473,088	0.2762874	2.6 : 1

[edit] Derivations for starting hands with one pair

Starting hands with one pair (XXYZ) can improve to three of a kind, a full house or four of a kind when either an X card is on the board or when two or three of the remaining Y or Z cards (YY or YYY) is on the board. They also improve to a full house if three or more of another rank is on the board (rrr or rrrr), although any other hand holding a pair also makes a full house with this board. These hands make two pair if another pair (rr) appears on the board. The following tables show all the ways for XXYZ to make four of a kind, a full house, three of a kind, two pair or one pair on the flop, turn and river.

**Derivations for rank type XXYZ (one pair) on the flop**
Hand to make	Board	Derivation	Combos	Probability	Odds
Four of a kind	XX*	$\begin{matrix} {2 \choose 2}{46 \choose 1} \end{matrix}$	46	0.0026596	375.0 : 1
	YYY	$\begin{matrix} {2 \choose 1}{3 \choose 3} \end{matrix}$	2	0.0001156	8,647.0 : 1
	Total		48	0.0027752	359.3 : 1
Full house	XYY	$\begin{matrix} {2 \choose 1}{2 \choose 1}{3 \choose 2} \end{matrix}$	12	0.0006938	1,440.3 : 1
	Xrr	$\begin{matrix} {2 \choose 1}{10 \choose 1}{4 \choose 2} \end{matrix}$	120	0.0069380	143.1 : 1
	YYZ	$\begin{matrix} {2 \choose 1}{3 \choose 2}{3 \choose 1} \end{matrix}$	18	0.0010407	959.9 : 1
	rrr	$\begin{matrix} {10 \choose 1}{4 \choose 3} \end{matrix}$	40	0.0023127	431.4 : 1
	Total		190	0.0109852	90.0 : 1
Three of a kind	XYZ	$\begin{matrix} {2 \choose 1}{3 \choose 1}^2 \end{matrix}$	18	0.0010407	959.9 : 1
	XYr	$\begin{matrix} {2 \choose 1}{2 \choose 1}{3 \choose 1}{40 \choose 1} \end{matrix}$	480	0.0277521	35.0 : 1
	Xrs	$\begin{matrix} {2 \choose 1}{10 \choose 2}{4 \choose 1}^2 \end{matrix}$	1,440	0.0832562	11.0 : 1
	YYr	$\begin{matrix} {2 \choose 1}{3 \choose 2}{40 \choose 1} \end{matrix}$	240	0.0138760	71.1 : 1
	Total		2,178	0.1259251	6.9 : 1
Two pair	YZr	$\begin{matrix} {3 \choose 1}^2{40 \choose 1} \end{matrix}$	360	0.0208141	47.0 : 1
	Yrr	$\begin{matrix} {2 \choose 1}{3 \choose 1}{10 \choose 1}{4 \choose 2} \end{matrix}$	360	0.0208141	47.0 : 1
	rrs	$\begin{matrix} {10 \choose 1}{4 \choose 2}{36 \choose 1} \end{matrix}$	2,160	0.1248844	7.0 : 1
	Total		2,880	0.1665125	5.0 : 1
One pair	Yrs	$\begin{matrix} {2 \choose 1}{3 \choose 1}{10 \choose 2}{4 \choose 1}^2 \end{matrix}$	4,320	0.2497687	3.0 : 1
	rst	$\begin{matrix} {10 \choose 3}{4 \choose 1}^3 \end{matrix}$	7,680	0.4440333	1.3 : 1
	Total		12,000	0.6938020	0.4 : 1

**Derivations for rank type XXYZ (one pair) on the turn**
Hand to make	Board	Derivation	Combos	Probability	Odds
Four of a kind	XX**	$\begin{matrix} {2 \choose 2}{46 \choose 2} \end{matrix}$	1,035	0.0053191	187.0 : 1
	YYY*	$\begin{matrix} {2 \choose 1}{3 \choose 3}{45 \choose 1} \end{matrix}$	90	0.0004625	2,161.0 : 1
	Total		1,125	0.0057817	172.0 : 1
Full house	XYYZ	$\begin{matrix} {2 \choose 1}{2 \choose 1}{3 \choose 2}{3 \choose 1} \end{matrix}$	36	0.0001850	5,404.0 : 1
	XYYr	$\begin{matrix} {2 \choose 1}{2 \choose 1}{3 \choose 2}{40 \choose 1} \end{matrix}$	480	0.0024669	404.4 : 1
	XYrr	$\begin{matrix} {2 \choose 1}{2 \choose 1}{3 \choose 1}{10 \choose 1}{4 \choose 2} \end{matrix}$	720	0.0037003	269.3 : 1
	Xrrs	$\begin{matrix} {2 \choose 1}{10 \choose 1}{4 \choose 2}{36 \choose 1} \end{matrix}$	4,320	0.0222017	44.0 : 1
	YYZZ	$\begin{matrix} {2 \choose 2}{3 \choose 2}^2 \end{matrix}$	9	0.0000463	21,619.0 : 1
	YYZr	$\begin{matrix} {2 \choose 1}{2 \choose 2}{3 \choose 2}{3 \choose 1}{40 \choose 1} \end{matrix}$	720	0.0037003	269.3 : 1
	rrr*	$\begin{matrix} {10 \choose 1}{4 \choose 3}{44 \choose 1} \end{matrix}$	1,760	0.0090451	109.6 : 1
	rrrr	$\begin{matrix} {10 \choose 1}{4 \choose 4} \end{matrix}$	10	0.0000514	19,457.0 : 1
	Total		8,055	0.0413969	23.2 : 1
Three of a kind	XYZr	$\begin{matrix} {2 \choose 1}{2 \choose 2}{3 \choose 1}^2{40 \choose 1} \end{matrix}$	720	0.0037003	269.3 : 1
	XYrs	$\begin{matrix} {2 \choose 1}{2 \choose 1}{3 \choose 1}{10 \choose 2}{4 \choose 1}^2 \end{matrix}$	8,640	0.0444033	21.5 : 1
	Xrst	$\begin{matrix} {2 \choose 1}{10 \choose 3}{4 \choose 1}^3 \end{matrix}$	15,360	0.0789393	11.7 : 1
	YY??	$\begin{matrix} {2 \choose 1}{3 \choose 2}{40 \choose 2} \end{matrix}$	4,680	0.0240518	40.6 : 1
	Total		29,400	0.1510947	5.6 : 1
Two pair	YZ??	$\begin{matrix} {3 \choose 1}^2{40 \choose 2} \end{matrix}$	7,020	0.0360777	26.7 : 1
	Yrrs	$\begin{matrix} {2 \choose 1}{3 \choose 1}{10 \choose 1}{4 \choose 2}{36 \choose 1} \end{matrix}$	12,960	0.0666050	14.0 : 1
	rrss	$\begin{matrix} {10 \choose 2}{4 \choose 2}^2 \end{matrix}$	1,620	0.0083256	119.1 : 1
	rrst	$\begin{matrix} {10 \choose 1}{4 \choose 2}{9 \choose 2}{4 \choose 1}^2 \end{matrix}$	34,560	0.1776133	4.6 : 1
	Total		56,160	0.2886216	2.5 : 1
One pair	Yrst	$\begin{matrix} {2 \choose 1}{3 \choose 1}{10 \choose 3}{4 \choose 1}^3 \end{matrix}$	46,080	0.2368178	3.2 : 1
	rstu	$\begin{matrix} {10 \choose 4}{4 \choose 1}^4 \end{matrix}$	53,760	0.2762874	2.6 : 1
	Total		99,840	0.5131051	0.9 : 1

**Derivations for rank type XXYZ (one pair) on the river**
Hand to make	Board	Derivation	Combos	Probability	Odds
Four of a kind	XX***	$\begin{matrix} {2 \choose 2}{46 \choose 3} \end{matrix}$	15,180	0.0088652	111.8 : 1
	YYY**	$\begin{matrix} {2 \choose 1}{3 \choose 3}{45 \choose 2} \end{matrix}$	1,980	0.0011563	863.8 : 1
	XXYYY	$\begin{matrix} {2 \choose 2}{2 \choose 1}{3 \choose 3} \end{matrix}$	−2	−0.0000012	−856,153 : 1
	Total ^{(see #1 below)}		17,158	0.0100204	98.8 : 1
Full house	XYYZZ	$\begin{matrix} {2 \choose 1}{2 \choose 2}{3 \choose 2}{3 \choose 2} \end{matrix}$	18	0.0000105	95,127.0 : 1
	XYYZr	$\begin{matrix} {2 \choose 1}{2 \choose 1}{3 \choose 2}{3 \choose 1}{40 \choose 1} \end{matrix}$	1,440	0.0008410	1,188.1 : 1
	XYY??	$\begin{matrix} {2 \choose 1}{2 \choose 1}{3 \choose 2}{40 \choose 2} \end{matrix}$	9,360	0.0054663	181.9 : 1
	XYZrr	$\begin{matrix} {2 \choose 1}{2 \choose 2}{3 \choose 1}^2{10 \choose 1}{4 \choose 2} \end{matrix}$	1,080	0.0006307	1,584.5 : 1
	XYrrs	$\begin{matrix} {2 \choose 1}{2 \choose 1}{3 \choose 1}{10 \choose 1}{4 \choose 2}{36 \choose 1} \end{matrix}$	25,920	0.0151375	65.1 : 1
	Xrrss	$\begin{matrix} {2 \choose 1}{10 \choose 1}{4 \choose 2}{36 \choose 1} \end{matrix}$	3,240	0.0018922	527.5 : 1
	Xrrst	$\begin{matrix} {2 \choose 1}{10 \choose 1}{4 \choose 2}{36 \choose 1} \end{matrix}$	69,120	0.0403667	23.8 : 1
	YYZZr	$\begin{matrix} {2 \choose 2}{3 \choose 2}^2{40 \choose 1} \end{matrix}$	360	0.0002102	4,755.4 : 1
	YYZ??	$\begin{matrix} {2 \choose 1}{2 \choose 2}{3 \choose 2}{3 \choose 1}{40 \choose 2} \end{matrix}$	14,040	0.0081995	121.0 : 1
	rrr**	$\begin{matrix} {10 \choose 1}{4 \choose 3}{44 \choose 2} \end{matrix}$	37,840	0.0220989	44.3 : 1
	rrrXX	$\begin{matrix} {10 \choose 1}{4 \choose 3}{2 \choose 2} \end{matrix}$	−40	−0.0000234	−42,808.6 : 1
	rrrr*	$\begin{matrix} {10 \choose 1}{4 \choose 4}{44 \choose 1} \end{matrix}$	440	0.0002570	3,890.6 : 1
	Total ^{(see #2 below)}		162,818	0.0950871	9.5 : 1
Three of a kind	XYZrs	$\begin{matrix} {2 \choose 1}{2 \choose 2}{3 \choose 1}^2{10 \choose 2}{4 \choose 1}^2 \end{matrix}$	12,960	0.0075687	131.1 : 1
	XYrst	$\begin{matrix} {2 \choose 1}{2 \choose 1}{3 \choose 1}{10 \choose 3}{4 \choose 1}^3 \end{matrix}$	92,160	0.0538222	17.6 : 1
	Xrstu	$\begin{matrix} {2 \choose 1}{10 \choose 4}{4 \choose 1}^4 \end{matrix}$	107,520	0.0627926	14.9 : 1
	YYrrs	$\begin{matrix} {2 \choose 1}{3 \choose 2}{10 \choose 1}{4 \choose 2}{36 \choose 1} \end{matrix}$	12,960	0.0075687	131.1 : 1
	YYrst	$\begin{matrix} {2 \choose 1}{3 \choose 2}{10 \choose 3}{4 \choose 1}^3 \end{matrix}$	46,080	0.0269111	36.2 : 1
	Total		271,680	0.1586634	5.3 : 1
Two pair	YZrrs	$\begin{matrix} {3 \choose 1}^2{10 \choose 1}{4 \choose 2}{36 \choose 1} \end{matrix}$	19,440	0.0113531	87.1 : 1
	YZrst	$\begin{matrix} {3 \choose 1}^2{10 \choose 3}{4 \choose 1}^3 \end{matrix}$	69,120	0.0403667	23.8 : 1
	Yrrss	$\begin{matrix} {2 \choose 1}{3 \choose 1}{10 \choose 2}{4 \choose 2}^2 \end{matrix}$	9,720	0.0056766	175.2 : 1
	Yrrst	$\begin{matrix} {2 \choose 1}{3 \choose 1}{10 \choose 1}{4 \choose 2}{9 \choose 2}{4 \choose 1}^2 \end{matrix}$	207,360	0.1211000	7.3 : 1
	rrsst	$\begin{matrix} {10 \choose 2}{4 \choose 2}^2{32 \choose 1} \end{matrix}$	51,840	0.0302750	32.0 : 1
	rrstu	$\begin{matrix} {10 \choose 1}{4 \choose 2}{9 \choose 3}{4 \choose 1}^3 \end{matrix}$	322,560	0.1883778	4.3 : 1
	Total		680,040	0.3971491	1.5 : 1
One pair	Yrstu	$\begin{matrix} {2 \choose 1}{3 \choose 1}{10 \choose 4}{4 \choose 1}^4 \end{matrix}$	322,560	0.1883778	4.3 : 1
	rstuv	$\begin{matrix} {10 \choose 5}{4 \choose 1}^5 \end{matrix}$	258,048	0.1507022	5.6 : 1
	Total		580,608	0.3390800	1.9 : 1

The board XXYYY is included in both XX*** and YY***, so it is subtracted from the total.
The board rrrXX makes four of a kind X and is included in rrr**, so it is subtracted from the total.

[edit] Derivations for starting hands with no pair

Starting hands with no pair (XYZR) can improve when when two or three of the remaining X, Y, Z or R cards (XX or XXX) appears on the board. These hands can make two pair or a full house when two of more ranks from the hand appear (XY or XXY). They also can make three of a kind or a pair if two or three other ranks (ss or sss) appear, although these boards are likely to improve other hands at least as much. The following tables show all the ways for XYZR to make four of a kind, a full house, three of a kind, two pair, one pair or no pair (high card) on the flop, turn and river.

**Derivations for rank type XYZR (no pair) on the flop**
Hand to make	Board	Derivation	Combos	Probability	Odds
Four of a kind	XXX	$\begin{matrix} {4 \choose 1}{3 \choose 3} \end{matrix}$	4	0.0002313	4,323.0 : 1
Full house	XXY	$\begin{matrix} {4 \choose 1}{3 \choose 2}{3 \choose 1}{3 \choose 1} \end{matrix}$	108	0.0062442	159.1 : 1
Three of a kind	XXs	$\begin{matrix} {4 \choose 1}{3 \choose 2}{36 \choose 1} \end{matrix}$	432	0.0249769	39.0 : 1
	sss	$\begin{matrix} {9 \choose 1}{4 \choose 3} \end{matrix}$	36	0.0020814	479.4 : 1
	Total		468	0.0270583	36.0 : 1
Two pair	XYZ	$\begin{matrix} {4 \choose 3}{3 \choose 1}^3 \end{matrix}$	108	0.0062442	159.1 : 1
	XYs	$\begin{matrix} {4 \choose 2}{3 \choose 1}^2{36 \choose 1} \end{matrix}$	1,944	0.1123959	7.9 : 1
	Total		2,052	0.1186401	7.4 : 1
One pair	X??	$\begin{matrix} {4 \choose 1}{3 \choose 1}{36 \choose 2} \end{matrix}$	7,560	0.4370953	1.3 : 1
	sst	$\begin{matrix} {9 \choose 1}{4 \choose 2}{32 \choose 1} \end{matrix}$	1,728	0.0999075	9.0 : 1
	Total		9,288	0.5370028	0.9 : 1
No pair	stu	$\begin{matrix} {9 \choose 3}{4 \choose 1}^3 \end{matrix}$	5,376	0.3108233	2.2 : 1

**Derivations for rank type XYZR (no pair) on the turn**
Hand to make	Board	Derivation	Combos	Probability	Odds
Four of a kind	XXX*	$\begin{matrix} {4 \choose 1}{3 \choose 3}{45 \choose 1} \end{matrix}$	180	0.0009251	1,080.0 : 1
Full house	XXYY	$\begin{matrix} {4 \choose 2}{3 \choose 2}^2 \end{matrix}$	54	0.0002775	3,602.3 : 1
	XXYZ	$\begin{matrix} {4 \choose 1}{3 \choose 2}{3 \choose 2}{3 \choose 1}^2 \end{matrix}$	324	0.0016651	599.6 : 1
	XXYs	$\begin{matrix} {4 \choose 1}{3 \choose 2}{3 \choose 1}{3 \choose 1}{36 \choose 1} \end{matrix}$	3,888	0.0199815	49.0 : 1
	Total		4,266	0.0219241	44.6 : 1
Three of a kind	XX??	$\begin{matrix} {4 \choose 1}{3 \choose 2}{36 \choose 2} \end{matrix}$	7,560	0.0388529	24.7 : 1
	Xsss	$\begin{matrix} {4 \choose 1}{3 \choose 1}{9 \choose 1}{4 \choose 3} \end{matrix}$	432	0.0022202	449.4 : 1
	ssss	$\begin{matrix} {9 \choose 1}{4 \choose 4} \end{matrix}$	9	0.0000463	21,619.0 : 1
	ssst	$\begin{matrix} {9 \choose 1}{4 \choose 3}{32 \choose 1} \end{matrix}$	1,152	0.0059204	167.9 : 1
	Total		9,153	0.0470398	20.3 : 1
Two pair	XYZR	$\begin{matrix} {4 \choose 4}{3 \choose 1}^4 \end{matrix}$	81	0.0004163	2,401.2 : 1
	XYZs	$\begin{matrix} {4 \choose 3}{3 \choose 1}^3{36 \choose 1} \end{matrix}$	3,888	0.0199815	49.0 : 1
	XY??	$\begin{matrix} {4 \choose 2}{3 \choose 1}^2{36 \choose 2} \end{matrix}$	34,020	0.1748381	4.7 : 1
	Total		37,989	0.1952359	4.1 : 1
One pair	Xsst	$\begin{matrix} {4 \choose 1}{3 \choose 1}{9 \choose 1}{4 \choose 2}{32 \choose 1} \end{matrix}$	20,736	0.1065680	8.4 : 1
	Xstu	$\begin{matrix} {4 \choose 1}{3 \choose 1}{9 \choose 3}{4 \choose 1}^3 \end{matrix}$	64,512	0.3315449	2.0 : 1
	sstt	$\begin{matrix} {9 \choose 2}{4 \choose 2}^2 \end{matrix}$	1,296	0.0066605	149.1 : 1
	sstu	$\begin{matrix} {9 \choose 1}{4 \choose 2}{8 \choose 2}{4 \choose 1}^2 \end{matrix}$	24,192	0.1243293	7.0 : 1
	Total		110,736	0.5691027	0.8 : 1
No pair	stuv	$\begin{matrix} {9 \choose 4}{4 \choose 1}^4 \end{matrix}$	32,256	0.1657724	5.0 : 1

**Derivations for rank type XYZR (no pair) on the river**
Hand to make	Board	Derivation	Combos	Probability	Odds
Four of a kind	XXX**	$\begin{matrix} {4 \choose 1}{3 \choose 3}{45 \choose 2} \end{matrix}$	3,960	0.0023127	431.4 : 1
Full house	XXYYZ	$\begin{matrix} {4 \choose 2}{3 \choose 2}^2{2 \choose 1}{3 \choose 1} \end{matrix}$	324	0.0001892	5,283.9 : 1
	XXYYs	$\begin{matrix} {4 \choose 2}{3 \choose 2}^2{36 \choose 1} \end{matrix}$	1,944	0.0011353	879.8 : 1
	XXYZR	$\begin{matrix} {4 \choose 1}{3 \choose 2}{3 \choose 3}{3 \choose 1}^3 \end{matrix}$	324	0.0001892	5,283.9 : 1
	XXYZs	$\begin{matrix} {4 \choose 1}{3 \choose 2}{3 \choose 2}{3 \choose 1}^2{36 \choose 1} \end{matrix}$	11,664	0.0068119	145.8 : 1
	XXY??	$\begin{matrix} {4 \choose 1}{3 \choose 2}{3 \choose 1}{3 \choose 1}{36 \choose 2} \end{matrix}$	68,040	0.0397359	24.2 : 1
	Total		82,296	0.0480616	19.8 : 1
Three of a kind	XX???	$\begin{matrix} {4 \choose 1}{3 \choose 2}{36 \choose 3} \end{matrix}$	85,680	0.0500378	19.0 : 1
	XYsss	$\begin{matrix} {4 \choose 2}{3 \choose 1}^2{9 \choose 1}{4 \choose 3} \end{matrix}$	1,944	0.0011353	879.8 : 1
	Xssst	$\begin{matrix} {4 \choose 1}{3 \choose 1}{9 \choose 1}{4 \choose 3}{32 \choose 1} \end{matrix}$	13,824	0.0080733	122.9 : 1
	ssss*	$\begin{matrix} {9 \choose 1}{4 \choose 4}{44 \choose 1} \end{matrix}$	396	0.0002313	4,323.0 : 1
	sss??	$\begin{matrix} {9 \choose 1}{4 \choose 3}{32 \choose 2} \end{matrix}$	17,856	0.0104281	94.9 : 1
	Total		119,700	0.0699058	13.3 : 1
Two pair	XYZRs	$\begin{matrix} {4 \choose 4}{3 \choose 1}^4{36 \choose 1} \end{matrix}$	2,916	0.0017030	586.2 : 1
	XYZ??	$\begin{matrix} {4 \choose 3}{3 \choose 1}^3{36 \choose 2} \end{matrix}$	68,040	0.0397359	24.2 : 1
	XYsst	$\begin{matrix} {4 \choose 2}{3 \choose 1}^2{9 \choose 1}{4 \choose 2}{32 \choose 1} \end{matrix}$	93,312	0.0544950	17.4 : 1
	XYstu	$\begin{matrix} {4 \choose 2}{3 \choose 1}^2{9 \choose 3}{4 \choose 1}^3 \end{matrix}$	290,304	0.1695400	4.9 : 1
	Total		454,572	0.2654739	2.8 : 1
One pair	Xsstt	$\begin{matrix} {4 \choose 1}{3 \choose 1}{9 \choose 2}{4 \choose 2}^2 \end{matrix}$	15,552	0.0090825	109.1 : 1
	Xsstu	$\begin{matrix} {4 \choose 1}{3 \choose 1}{9 \choose 1}{4 \choose 2}{8 \choose 2}{4 \choose 1}^2 \end{matrix}$	290,304	0.1695400	4.9 : 1
	Xstuv	$\begin{matrix} {4 \choose 1}{3 \choose 1}{9 \choose 4}{4 \choose 1}^4 \end{matrix}$	387,072	0.2260533	3.4 : 1
	ssttu	$\begin{matrix} {9 \choose 2}{4 \choose 2}^2{28 \choose 1} \end{matrix}$	36,288	0.0211925	46.2 : 1
	sstuv	$\begin{matrix} {9 \choose 1}{4 \choose 2}{8 \choose 3}{4 \choose 1}^3 \end{matrix}$	193,536	0.1130267	7.8 : 1
	Total		922,752	0.5388950	0.9 : 1
No pair	stuvw	$\begin{matrix} {9 \choose 5}{4 \choose 1}^5 \end{matrix}$	129,024	0.0753511	12.3 : 1

[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] See also

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

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Categories: Poker gameplay and terminology | Probability and statistics