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

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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 boards on the flop, boards on the turn, and boards at the river. The probabilities for the boards in each table total 1.0.


Derivations for starting hands with four of a kind

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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 48 0.0027752 359.3 : 1
Two pair rrs 3,168 0.1831637 4.5 : 1
One pair rst 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* 2,112 0.0108541 91.1 : 1
rrrr 12 0.0000617 16,214.0 : 1
Total 2,124 0.0109158 90.6 : 1
Two pair rrss 2,376 0.0122109 80.9 : 1
rrst 63,360 0.3256244 2.1 : 1
Total 65,736 0.3378353 2.0 : 1
One pair rstu 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** 45,408 0.0265187 36.7 : 1
rrrr* 528 0.0003084 3,242.0 : 1
Total 45,936 0.0268270 36.3 : 1
Two pair rrsst 95,040 0.0555042 17.0 : 1
rrstu 760,320 0.4440333 1.3 : 1
Total 855,360 0.4995375 1.0 : 1
One pair rstuv 811,008 0.4736355 1.1 : 1

Derivations for starting hands with three of a kind

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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 1 0.0000578 17,295.0 : 1
Full house XYY 3 0.0001735 5,764.3 : 1
Xrr 66 0.0038159 261.1 : 1
rrr 44 0.0025439 392.1 : 1
Total 113 0.0065333 152.1 : 1
Three of a kind XYr 132 0.0076318 130.0 : 1
Xrs 880 0.0508788 18.7 : 1
YYr 132 0.0076318 130.0 : 1
Total 1,144 0.0661425 14.1 : 1
Two pair Yrr 198 0.0114477 86.4 : 1
rrs 2,640 0.1526364 5.6 : 1
Total 2,838 0.1640842 5.1 : 1
One pair Yrs 2,640 0.1526364 5.6 : 1
rst 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* 45 0.0002313 4,323.0 : 1
Full house XYYr 132 0.0006784 1,473.1 : 1
XYrr 198 0.0010176 981.7 : 1
Xrrs 2,640 0.0135677 72.7 : 1
rrr* 1,936 0.0099496 99.5 : 1
rrrr 11 0.0000565 17,688.1 : 1
Total 4,917 0.0252698 38.6 : 1
Three of a kind XYrs 2,640 0.0135677 72.7 : 1
Xrst 10,560 0.0542707 17.4 : 1
YY?? 2,838 0.0145853 67.6 : 1
Total 16,038 0.0824237 11.1 : 1
Two pair Yrrs 7,920 0.0407031 23.6 : 1
rrss 1,980 0.0101758 97.3 : 1
rrst 47,520 0.2442183 3.1 : 1
Total 57,420 0.2950971 2.4 : 1
One pair Yrst 31,680 0.1628122 5.1 : 1
rstu 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** 990 0.0005782 1,728.6 : 1
Full house XYY?? 2,838 0.0016574 602.3 : 1
XYrrs 7,920 0.0046253 215.2 : 1
Xrrss 1,980 0.0011563 863.8 : 1
Xrrst 47,520 0.0277521 35.0 : 1
rrr** 41,624 0.0243088 40.1 : 1
rrrr* 484 0.0002827 3,536.8 : 1
Total 102,366 0.0597826 15.7 : 1
Three of a kind XYrst 31,680 0.0185014 53.1 : 1
Xrstu 84,480 0.0493370 19.3 : 1
YYrrs 7,920 0.0046253 215.2 : 1
YYrst 31,680 0.0185014 53.1 : 1
Total 155,760 0.0909652 10.0 : 1
Two pair Yrrss 5,940 0.0034690 287.3 : 1
Yrrst 142,560 0.0832562 11.0 : 1
rrsst 71,280 0.0416281 23.0 : 1
rrstu 506,880 0.2960222 2.4 : 1
Total 726,660 0.4243756 1.4 : 1
One pair Yrstu 253,440 0.1480111 5.8 : 1
rstuv 473,088 0.2762874 2.6 : 1
Total 726,528 0.4242985 1.4 : 1

Derivations for starting hands with two pair

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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* 92 0.0053191 187.0 : 1
Full house Xrr 264 0.0152636 64.5 : 1
rrr 44 0.0025439 392.1 : 1
Total 308 0.0178076 55.2 : 1
Three of a kind XY? 176 0.0101758 97.3 : 1
Xrs 3,520 0.2035153 3.9 : 1
Total 3,696 0.2136910 3.7 : 1
Two pair rrs 2,640 0.1526364 5.6 : 1
One pair rst 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 1 0.0000051 194,579.0 : 1
XXYr 176 0.0009045 1,104.6 : 1
XX?? 1,892 0.0097235 101.8 : 1
Total 2,069 0.0106332 93.0 : 1
Full house XYrr 264 0.0013568 736.0 : 1
Xrrr 176 0.0009045 1,104.6 : 1
Xrrs 10,560 0.0542707 17.4 : 1
rrrr 11 0.0000565 17,688.1 : 1
rrrs 1,760 0.0090451 109.6 : 1
Total 12,771 0.0656337 14.2 : 1
Three of a kind XYrs 3,520 0.0180902 54.3 : 1
Xrst 42,240 0.2170829 3.6 : 1
Total 45,760 0.2351732 3.3 : 1
Two pair rrss 1,980 0.0101758 97.3 : 1
rrst 47,520 0.2442183 3.1 : 1
Total 49,500 0.2543941 2.9 : 1
One pair rstu 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* 44 0.0000257 38,917.0 : 1
XXY?? 3,784 0.0022099 451.5 : 1
XX??? 26,488 0.0154692 63.6 : 1
Total 30,316 0.0177048 55.5 : 1
Full house XYrrr 176 0.0001028 9,728.0 : 1
XYrrs 10,560 0.0061671 161.2 : 1
Xrrrs 7,040 0.0041114 242.2 : 1
Xrrss 7,922 0.0046253 215.2 : 1
Xrrst 190,080 0.1110083 8.0 : 1
rrr?? 34,320 0.0200432 48.9 : 1
rrrr* 484 0.0002827 3,536.8 : 1
Total 250,580 0.1463408 5.8 : 1
Three of a kind XYrst 42,240 0.0246685 39.5 : 1
Xrstu 337,920 0.1973481 4.1 : 1
Total 380,160 0.2220167 3.5 : 1
Two pair rrsst 71,280 0.0416281 23.0 : 1
rrstu 506,880 0.2960222 2.4 : 1
Total 578,160 0.3376503 2.0 : 1
One pair rstuv 473,088 0.2762874 2.6 : 1

Derivations for starting hands with one pair

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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* 46 0.0026596 375.0 : 1
YYY 2 0.0001156 8,647.0 : 1
Total 48 0.0027752 359.3 : 1
Full house XYY 12 0.0006938 1,440.3 : 1
Xrr 120 0.0069380 143.1 : 1
YYZ 18 0.0010407 959.9 : 1
rrr 40 0.0023127 431.4 : 1
Total 190 0.0109852 90.0 : 1
Three of a kind XYZ 18 0.0010407 959.9 : 1
XYr 480 0.0277521 35.0 : 1
Xrs 1,440 0.0832562 11.0 : 1
YYr 240 0.0138760 71.1 : 1
Total 2,178 0.1259251 6.9 : 1
Two pair YZr 360 0.0208141 47.0 : 1
Yrr 360 0.0208141 47.0 : 1
rrs 2,160 0.1248844 7.0 : 1
Total 2,880 0.1665125 5.0 : 1
One pair Yrs 4,320 0.2497687 3.0 : 1
rst 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** 1,035 0.0053191 187.0 : 1
YYY* 90 0.0004625 2,161.0 : 1
Total 1,125 0.0057817 172.0 : 1
Full house XYYZ 36 0.0001850 5,404.0 : 1
XYYr 480 0.0024669 404.4 : 1
XYrr 720 0.0037003 269.3 : 1
Xrrs 4,320 0.0222017 44.0 : 1
YYZZ 9 0.0000463 21,619.0 : 1
YYZr 720 0.0037003 269.3 : 1
rrr* 1,760 0.0090451 109.6 : 1
rrrr 10 0.0000514 19,457.0 : 1
Total 8,055 0.0413969 23.2 : 1
Three of a kind XYZr 720 0.0037003 269.3 : 1
XYrs 8,640 0.0444033 21.5 : 1
Xrst 15,360 0.0789393 11.7 : 1
YY?? 4,680 0.0240518 40.6 : 1
Total 29,400 0.1510947 5.6 : 1
Two pair YZ?? 7,020 0.0360777 26.7 : 1
Yrrs 12,960 0.0666050 14.0 : 1
rrss 1,620 0.0083256 119.1 : 1
rrst 34,560 0.1776133 4.6 : 1
Total 56,160 0.2886216 2.5 : 1
One pair Yrst 46,080 0.2368178 3.2 : 1
rstu 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*** 15,180 0.0088652 111.8 : 1
YYY** 1,980 0.0011563 863.8 : 1
XXYYY −2 −0.0000012 −856,153 : 1
Total (see #1 below) 17,158 0.0100204 98.8 : 1
Full house XYYZZ 18 0.0000105 95,127.0 : 1
XYYZr 1,440 0.0008410 1,188.1 : 1
XYY?? 9,360 0.0054663 181.9 : 1
XYZrr 1,080 0.0006307 1,584.5 : 1
XYrrs 25,920 0.0151375 65.1 : 1
Xrrss 3,240 0.0018922 527.5 : 1
Xrrst 69,120 0.0403667 23.8 : 1
YYZZr 360 0.0002102 4,755.4 : 1
YYZ?? 14,040 0.0081995 121.0 : 1
rrr** 37,840 0.0220989 44.3 : 1
rrrXX −40 −0.0000234 −42,808.6 : 1
rrrr* 440 0.0002570 3,890.6 : 1
Total (see #2 below) 162,818 0.0950871 9.5 : 1
Three of a kind XYZrs 12,960 0.0075687 131.1 : 1
XYrst 92,160 0.0538222 17.6 : 1
Xrstu 107,520 0.0627926 14.9 : 1
YYrrs 12,960 0.0075687 131.1 : 1
YYrst 46,080 0.0269111 36.2 : 1
Total 271,680 0.1586634 5.3 : 1
Two pair YZrrs 19,440 0.0113531 87.1 : 1
YZrst 69,120 0.0403667 23.8 : 1
Yrrss 9,720 0.0056766 175.2 : 1
Yrrst 207,360 0.1211000 7.3 : 1
rrsst 51,840 0.0302750 32.0 : 1
rrstu 322,560 0.1883778 4.3 : 1
Total 680,040 0.3971491 1.5 : 1
One pair Yrstu 322,560 0.1883778 4.3 : 1
rstuv 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

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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 4 0.0002313 4,323.0 : 1
Full house XXY 108 0.0062442 159.1 : 1
Three of a kind XXs 432 0.0249769 39.0 : 1
sss 36 0.0020814 479.4 : 1
Total 468 0.0270583 36.0 : 1
Two pair XYZ 108 0.0062442 159.1 : 1
XYs 1,944 0.1123959 7.9 : 1
Total 2,052 0.1186401 7.4 : 1
One pair X?? 7,560 0.4370953 1.3 : 1
sst 1,728 0.0999075 9.0 : 1
Total 9,288 0.5370028 0.9 : 1
No pair stu 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* 180 0.0009251 1,080.0 : 1
Full house XXYY 54 0.0002775 3,602.3 : 1
XXYZ 324 0.0016651 599.6 : 1
XXYs 3,888 0.0199815 49.0 : 1
Total 4,266 0.0219241 44.6 : 1
Three of a kind XX?? 7,560 0.0388529 24.7 : 1
Xsss 432 0.0022202 449.4 : 1
ssss 9 0.0000463 21,619.0 : 1
ssst 1,152 0.0059204 167.9 : 1
Total 9,153 0.0470398 20.3 : 1
Two pair XYZR 81 0.0004163 2,401.2 : 1
XYZs 3,888 0.0199815 49.0 : 1
XY?? 34,020 0.1748381 4.7 : 1
Total 37,989 0.1952359 4.1 : 1
One pair Xsst 20,736 0.1065680 8.4 : 1
Xstu 64,512 0.3315449 2.0 : 1
sstt 1,296 0.0066605 149.1 : 1
sstu 24,192 0.1243293 7.0 : 1
Total 110,736 0.5691027 0.8 : 1
No pair stuv 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** 3,960 0.0023127 431.4 : 1
Full house XXYYZ 324 0.0001892 5,283.9 : 1
XXYYs 1,944 0.0011353 879.8 : 1
XXYZR 324 0.0001892 5,283.9 : 1
XXYZs 11,664 0.0068119 145.8 : 1
XXY?? 68,040 0.0397359 24.2 : 1
Total 82,296 0.0480616 19.8 : 1
Three of a kind XX??? 85,680 0.0500378 19.0 : 1
XYsss 1,944 0.0011353 879.8 : 1
Xssst 13,824 0.0080733 122.9 : 1
ssss* 396 0.0002313 4,323.0 : 1
sss?? 17,856 0.0104281 94.9 : 1
Total 119,700 0.0699058 13.3 : 1
Two pair XYZRs 2,916 0.0017030 586.2 : 1
XYZ?? 68,040 0.0397359 24.2 : 1
XYsst 93,312 0.0544950 17.4 : 1
XYstu 290,304 0.1695400 4.9 : 1
Total 454,572 0.2654739 2.8 : 1
One pair Xsstt 15,552 0.0090825 109.1 : 1
Xsstu 290,304 0.1695400 4.9 : 1
Xstuv 387,072 0.2260533 3.4 : 1
ssttu 36,288 0.0211925 46.2 : 1
sstuv 193,536 0.1130267 7.8 : 1
Total 922,752 0.5388950 0.9 : 1
No pair stuvw 129,024 0.0753511 12.3 : 1

See also

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Omaha Rank-based hands