Probability derivations for making rank-based hands in Omaha hold 'em

See Poker probability (Omaha)#Making hands based on rank type for the probabilities derived from these equations.

The probability derivations for starting hands making four of a kind, a full house, three of a kind, two pair, one pair and no pair in Omaha hold 'em are separate for each of the starting hand rank types.

The derivations require identifying the individual cases that yield 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.

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 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

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

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

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 YYY**, 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.

Derivations for starting hands with no pair

Starting hands with no pair (XYZR) can improve 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