User:Doug Bell/Probability derivations for making rank-based hands in Omaha hold 'em

Main article: Poker probability (Omaha)

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

1. The board XXYYY is included in both XX*** and YYY**, so it is subtracted from the total.
2. 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

See also

• Binomial coefficient

Omaha Rank-based hands