# Poker/Omaha/Probability derivations for making low hands in Omaha hold 'em

The probability derivations for starting hands making a qualifying low hand in Omaha hold 'em Hi-Lo are separate for each of the starting hand low shapes. The derivations require identifying the individual cases that complete a low hand; make a low hand possible on the board, but don't make a low hand; and that create a board with fewer than three low ranks, making no low hand possible.
The ranks on the board are indicated using lower case letters for matches with the starting hand and upper case letters to indicate qualifying low ranks that don't match the starting hand. So the low hand shape xxyz is any hand with a low pair of x with two additional low ranks y and z and the board xyA represents a flop that contains one x, one of the non-paired ranks y and one other low rank A. Note that since y and z have an identical relationship to the starting hand—each represents an unpaired rank—xyA and xzA 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 higher than the qualifying low. So for the low hand shape xxy*, the board xx* represents a flop that contains two xs and a high rank (either the same high rank as xxy* or a different high rank).
One common calculation regardless of the hand is that no low is possible when there are two or fewer low cards on the board. If $h$ is the number of high cards in the hand ($h$ = 0, 1, 2, 3, 4) and $r$ is the low qualifier ($r$ = 8, 9), then $4r-4+h$ is the number of low cards remaining in the deck and $52-4r-h$ is the number of high cards remaining in the deck. Using ? to represent any low card, the number of boards with 0, 1 or 2 low cards on the flop, turn and river are calculated as follows:
    FlopTurnRiver BoardDerivationBoardDerivationBoardDerivation ??* ${\begin{matrix}{4r-4+h \choose 2}{52-4r-h \choose 1}\end{matrix}}$ ??** ${\begin{matrix}{4r-4+h \choose 2}{52-4r-h \choose 2}\end{matrix}}$ ??*** ${\begin{matrix}{4r-4+h \choose 2}{52-4r-h \choose 3}\end{matrix}}$ ?** ${\begin{matrix}{4r-4+h \choose 1}{52-4r-h \choose 2}\end{matrix}}$ ?*** ${\begin{matrix}{4r-4+h \choose 1}{52-4r-h \choose 3}\end{matrix}}$ ?**** ${\begin{matrix}{4r-4+h \choose 1}{52-4r-h \choose 4}\end{matrix}}$ *** ${\begin{matrix}{52-4r-h \choose 3}\end{matrix}}$ **** ${\begin{matrix}{52-4r-h \choose 4}\end{matrix}}$ ***** ${\begin{matrix}{52-4r-h \choose 5}\end{matrix}}$
Each table in the following subsections 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 making low hands with four high cards

Starting hands with four high cards (****) cannot make a low hand, so the only possibilities are having a low hand possible on the board or not. With four high cards in the hand, there are $52-4r-4=48-4r$ high cards and $4r-4+4=4r$ low cards left in the deck. The following tables show the derivations when holding four high cards.
    Derivations for low hand shape **** (four high cards) on the flop Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Miss low ABC ${\begin{matrix}{r \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 3,5840.20721553.8 : 1 5,3760.31082332.2 : 1 No low AAA ${\begin{matrix}{r \choose 1}{4 \choose 3}\end{matrix}}$ 320.0018501539.5 : 1 360.0020814479.4 : 1 AAB ${\begin{matrix}{r \choose 1}{4 \choose 2}{r-1 \choose 1}{4 \choose 1}\end{matrix}}$ 1,3440.077705811.9 : 1 1,7280.09990759.0 : 1  ??* ${\begin{matrix}{4r \choose 2}{48-4r \choose 1}\end{matrix}}$ 7,1680.41443111.4 : 1 6,9120.39963001.5 : 1  ?** ${\begin{matrix}{4r \choose 1}{48-4r \choose 2}\end{matrix}}$ 7,9360.45883441.2 : 1 7,5600.43709531.3 : 1 *** ${\begin{matrix}{48-4r \choose 3}\end{matrix}}$ 5600.032377429.9 : 1 2200.012719777.6 : 1 Total 13,7120.79278450.3 : 1 11,9200.68917670.5 : 1
    Derivations for low hand shape **** (four high cards) on the turn Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Miss low ABCD ${\begin{matrix}{r \choose 4}{4 \choose 1}^{4}\end{matrix}}$ 17,9200.09209589.9 : 1 32,2560.16577245.0 : 1 AABC ${\begin{matrix}{r \choose 1}{4 \choose 2}{r-1 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 16,1280.082886211.1 : 1 24,1920.12432937.0 : 1 ABC* ${\begin{matrix}{r \choose 3}{4 \choose 1}^{3}{48-4r \choose 1}\end{matrix}}$ 57,3440.29470652.4 : 1 64,5120.33154492.0 : 1 Total 91,3920.79278451.1 : 1 120,9600.62164660.6 : 1 No low AAAA ${\begin{matrix}{r \choose 1}{4 \choose 4}\end{matrix}}$ 80.000041124,321.5 : 1 90.000046321,619.0 : 1 AAAB ${\begin{matrix}{r \choose 1}{4 \choose 3}{r-1 \choose 1}{4 \choose 1}\end{matrix}}$ 8960.0046048216.2 : 1 1,1520.0059204167.9 : 1 AABB ${\begin{matrix}{r \choose 2}{4 \choose 2}^{2}\end{matrix}}$ 1,0080.0051804192.0 : 1 1,2960.0066605149.1 : 1 AAA* ${\begin{matrix}{r \choose 1}{4 \choose 3}{48-4r \choose 1}\end{matrix}}$ 5120.0026313379.0 : 1 4320.0022202449.4 : 1 AAB* ${\begin{matrix}{r \choose 1}{4 \choose 2}{r-1 \choose 1}{4 \choose 1}{48-4r \choose 1}\end{matrix}}$ 21,5040.11051508.0 : 1 20,7360.10656808.4 : 1  ??** ${\begin{matrix}{4r \choose 2}{48-4r \choose 2}\end{matrix}}$ 59,5200.30588962.3 : 1 41,5800.21369103.7 : 1  ?*** ${\begin{matrix}{4r \choose 1}{48-4r \choose 3}\end{matrix}}$ 17,9200.09209589.9 : 1 7,9200.040703123.6 : 1 **** ${\begin{matrix}{48-4r \choose 4}\end{matrix}}$ 1,8200.0093535105.9 : 1 4950.0025439392.1 : 1 Total 103,1880.53031140.9 : 1 73,6200.37835341.6 : 1
    Derivations for low hand shape **** (four high cards) on the river Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Miss low ABCDE ${\begin{matrix}{r \choose 5}{4 \choose 1}^{5}\end{matrix}}$ 57,3440.033489428.9 : 1 129,0240.075351112.3 : 1 AABCD ${\begin{matrix}{r \choose 1}{4 \choose 2}{r-1 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 107,5200.062792614.9 : 1 193,5360.11302677.8 : 1 ABCD* ${\begin{matrix}{r \choose 4}{4 \choose 1}^{4}{48-4r \choose 1}\end{matrix}}$ 286,7200.16744695.0 : 1 387,0720.22605333.4 : 1 AABBC ${\begin{matrix}{r \choose 2}{4 \choose 2}^{2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 24,1920.014128369.8 : 1 36,2880.021192546.2 : 1 AAABC ${\begin{matrix}{r \choose 1}{4 \choose 3}{r-1 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 10,7520.0062793158.3 : 1 16,1280.0094189105.2 : 1 AABC* ${\begin{matrix}{r \choose 1}{4 \choose 2}{r-1 \choose 2}{4 \choose 1}^{2}{48-4r \choose 1}\end{matrix}}$ 258,0480.15070225.6 : 1 290,3040.16954004.9 : 1 ABC** ${\begin{matrix}{r \choose 3}{4 \choose 1}^{3}{48-4r \choose 2}\end{matrix}}$ 430,0800.25117043.0 : 1 354,8160.20721553.8 : 1 Total 1,174,6560.68600900.5 : 1 1,407,1680.82179800.2 : 1 No low AAAAB ${\begin{matrix}{r \choose 1}{4 \choose 4}{r-1 \choose 1}{4 \choose 1}\end{matrix}}$ 2240.00013087,643.2 : 1 2880.00016825,944.5 : 1 AAAA* ${\begin{matrix}{r \choose 1}{4 \choose 4}{48-4r \choose 1}\end{matrix}}$ 1280.000074813,376.4 : 1 1080.000063115,853.7 : 1 AAABB ${\begin{matrix}{r \choose 1}{4 \choose 3}{r-1 \choose 1}{4 \choose 2}\end{matrix}}$ 1,3440.00078491,273.0 : 1 1,7280.0010092989.9 : 1 AAAB* ${\begin{matrix}{r \choose 1}{4 \choose 3}{r-1 \choose 1}{4 \choose 1}{48-4r \choose 1}\end{matrix}}$ 14,3360.0083723118.4 : 1 13,8240.0080733122.9 : 1 AABB* ${\begin{matrix}{r \choose 2}{4 \choose 2}^{2}{48-4r \choose 1}\end{matrix}}$ 16,1280.0094189105.2 : 1 15,5520.0090825109.1 : 1 AAA** ${\begin{matrix}{r \choose 1}{4 \choose 3}{48-4r \choose 2}\end{matrix}}$ 3,8400.0022426444.9 : 1 2,3760.0013876719.7 : 1 AAB** ${\begin{matrix}{r \choose 1}{4 \choose 2}{r-1 \choose 1}{4 \choose 1}{48-4r \choose 2}\end{matrix}}$ 161,2800.09418899.6 : 1 114,0480.066605014.0 : 1  ??*** ${\begin{matrix}{4r \choose 2}{48-4r \choose 3}\end{matrix}}$ 277,7600.16221425.2 : 1 138,6000.080943611.4 : 1  ?**** ${\begin{matrix}{4r \choose 1}{48-4r \choose 4}\end{matrix}}$ 58,2400.034012728.4 : 1 17,8200.010407095.1 : 1 ***** ${\begin{matrix}{48-4r \choose 5}\end{matrix}}$ 4,3680.0025509391.0 : 1 7920.00046252,161.0 : 1 Total 537,6480.31399102.2 : 1 305,1360.17820204.6 : 1

## Derivations for making low hands with one low card and three high cards

Starting hands with one low card and three high cards (x***) cannot make a low hand, so the only possibilities are having a low hand possible on the board or not. With three high cards in the hand, there are $52-4r-3=49-4r$ high cards and $4r-4+3=4r-1$ low cards left in the deck. The following tables show the derivations when holding one low card and three high cards.
    Derivations for low hand shape x*** (one low card and three high cards) on the flop Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Miss low ABC ${\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 2,2400.12950976.7 : 1 3,5840.20721553.8 : 1 xAB ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 1,0080.058279416.2 : 1 1,3440.077705811.9 : 1 Total 3,2480.18778914.3 : 1 4,9280.28492142.5 : 1 No low AAA ${\begin{matrix}{r-1 \choose 1}{4 \choose 3}\end{matrix}}$ 280.0016189616.7 : 1 320.0018501539.5 : 1 AAB ${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 1,0080.058279416.2 : 1 1,3440.077705811.9 : 1 xxx ${\begin{matrix}{1 \choose 1}{3 \choose 3}\end{matrix}}$ 10.000057817,295.0 : 1 10.000057817,295.0 : 1 xxA ${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 1}\end{matrix}}$ 840.0048566204.9 : 1 960.0055504179.2 : 1 xAA ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}\end{matrix}}$ 1260.0072849136.3 : 1 1440.0083256119.1 : 1  ??* ${\begin{matrix}{4r-1 \choose 2}{49-4r \choose 1}\end{matrix}}$ 7,9050.45704211.2 : 1 7,7350.44721321.2 : 1  ?** ${\begin{matrix}{4r-1 \choose 1}{49-4r \choose 2}\end{matrix}}$ 4,2160.24375583.1 : 1 2,7300.15784005.3 : 1 *** ${\begin{matrix}{49-4r \choose 3}\end{matrix}}$ 6800.039315424.4 : 1 2860.016535659.5 : 1 Total 14,0480.81221090.2 : 1 12,3680.71507860.4 : 1
    Derivations for low hand shape x*** (one low card and three high cards) on the turn Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Miss low ABCD ${\begin{matrix}{r-1 \choose 4}{4 \choose 1}^{4}\end{matrix}}$ 8,9600.046047920.7 : 1 17,9200.09209589.9 : 1 AABC ${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 10,0800.051803918.3 : 1 16,1280.082886211.1 : 1 ABC* ${\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}{49-4r \choose 1}\end{matrix}}$ 38,0800.19570364.1 : 1 46,5920.23944913.2 : 1 xxAB ${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-1 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 1,0080.0051804192.0 : 1 1,3440.0069072143.8 : 1 xABC ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 6,7200.034535928.0 : 1 10,7520.055257517.1 : 1 xAAB ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 3,0240.015541263.3 : 1 4,0320.020721647.3 : 1 xAB* ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}{49-4r \choose 1}\end{matrix}}$ 17,1360.088066610.4 : 1 17,4720.089793410.1 : 1 Total 85,0080.43687941.3 : 1 114,2400.58711070.7 : 1 No low AAAA ${\begin{matrix}{r-1 \choose 1}{4 \choose 4}\end{matrix}}$ 70.000036027,796.1 : 1 80.000041124,321.5 : 1 AAAB ${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 6720.0034536288.6 : 1 8960.0046048216.2 : 1 AABB ${\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}\end{matrix}}$ 7560.0038853256.4 : 1 1,0080.0051804192.0 : 1 AAA* ${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{49-4r \choose 1}\end{matrix}}$ 4760.0024463407.8 : 1 4160.0021379466.7 : 1 AAB* ${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{49-4r \choose 1}\end{matrix}}$ 17,1360.088066610.4 : 1 17,4720.089793410.1 : 1 xxx* ${\begin{matrix}{1 \choose 1}{3 \choose 3}{49-4r \choose 1}\end{matrix}}$ 170.000087411,444.9 : 1 130.000066814,966.7 : 1 xxxA ${\begin{matrix}{1 \choose 1}{3 \choose 3}{r-1 \choose 1}{4 \choose 1}\end{matrix}}$ 280.00014396,948.3 : 1 320.00016456,079.6 : 1 xxAA ${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 2}\end{matrix}}$ 1260.00064751,543.3 : 1 1440.00074011,350.3 : 1 xxA* ${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 1}{49-4r \choose 1}\end{matrix}}$ 1,4280.0073389135.3 : 1 1,2480.0064138154.9 : 1 xAAA ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 3}\end{matrix}}$ 840.00043172,315.4 : 1 960.00049342,025.9 : 1 xAA* ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{49-4r \choose 1}\end{matrix}}$ 2,1420.011008389.8 : 1 1,8720.0096207102.9 : 1  ??** ${\begin{matrix}{4r-1 \choose 2}{49-4r \choose 2}\end{matrix}}$ 63,2400.32500772.1 : 1 46,4100.23851373.2 : 1  ?*** ${\begin{matrix}{4r-1 \choose 1}{49-4r \choose 3}\end{matrix}}$ 21,0800.10833598.2 : 1 10,0100.051444118.4 : 1 **** ${\begin{matrix}{49-4r \choose 4}\end{matrix}}$ 2,3800.012231580.8 : 1 7150.0036746271.1 : 1 Total 109,5720.56312060.8 : 1 80,3400.41288931.4 : 1
    Derivations for low hand shape x*** (one low card and three high cards) on the river Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Miss low ABCDE ${\begin{matrix}{r-1 \choose 5}{4 \choose 1}^{5}\end{matrix}}$ 21,5040.012558578.6 : 1 57,3440.033489428.9 : 1 AABCD ${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 53,7600.031396330.9 : 1 107,5200.062792614.9 : 1 ABCD* ${\begin{matrix}{r-1 \choose 4}{4 \choose 1}^{4}{49-4r \choose 1}\end{matrix}}$ 152,3200.088956210.2 : 1 232,9600.13605066.4 : 1 AABBC ${\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 15,1200.0088302112.2 : 1 24,1920.014128369.8 : 1 AAABC ${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 6,7200.0039245253.8 : 1 10,7520.0062793158.3 : 1 AABC* ${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}{49-4r \choose 1}\end{matrix}}$ 171,3600.10007579.0 : 1 209,6640.12244557.2 : 1 ABC** ${\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}{49-4r \choose 2}\end{matrix}}$ 304,6400.17791234.6 : 1 279,5520.16326075.1 : 1 xABCD ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 4}{4 \choose 1}^{4}\end{matrix}}$ 26,8800.015698162.7 : 1 53,7600.031396330.9 : 1 xABC* ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 3}{4 \choose 1}^{3}{49-4r \choose 1}\end{matrix}}$ 114,2400.066717114.0 : 1 139,7760.081630411.3 : 1 xAABC ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 30,2400.017660455.6 : 1 48,3840.028256734.4 : 1 xAAB* ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{49-4r \choose 1}\end{matrix}}$ 51,4080.030022732.3 : 1 52,4160.030611431.7 : 1 xAAAB ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 2,0160.0011774848.4 : 1 2,6880.0015698636.0 : 1 xAABB ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 2}^{2}\end{matrix}}$ 2,2680.0013245754.0 : 1 3,0240.0017660565.2 : 1 xAB** ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}{49-4r \choose 2}\end{matrix}}$ 137,0880.080060511.5 : 1 104,8320.061222815.3 : 1 xxABC ${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-1 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 6,7200.0039245253.8 : 1 10,7520.0062793158.3 : 1 xxAB* ${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-1 \choose 2}{4 \choose 1}^{2}{49-4r \choose 1}\end{matrix}}$ 17,1360.010007698.9 : 1 17,4720.010203897.0 : 1 xxAAB ${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 3,0240.0017660565.2 : 1 4,0320.0023547423.7 : 1 xxxAB ${\begin{matrix}{1 \choose 1}{3 \choose 3}{r-1 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 3360.00019625,095.1 : 1 4480.00026163,821.1 : 1 Total 1,116,7800.65220900.5 : 1 1,359,5680.79399920.3 : 1 No low AAAAB ${\begin{matrix}{r-1 \choose 1}{4 \choose 4}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 1680.000098110,191.3 : 1 2240.00013087,643.2 : 1 AAAA* ${\begin{matrix}{r-1 \choose 1}{4 \choose 4}{49-4r \choose 1}\end{matrix}}$ 1190.000069514,388.1 : 1 1040.000060716,463.5 : 1 AAABB ${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 2}\end{matrix}}$ 1,0080.00058871,697.7 : 1 1,3440.00078491,273.0 : 1 AAAB* ${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}{49-4r \choose 1}\end{matrix}}$ 11,4240.0066717148.9 : 1 11,6480.0068025146.0 : 1 AABB* ${\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}{49-4r \choose 1}\end{matrix}}$ 12,8520.0075057132.2 : 1 13,1040.0076528129.7 : 1 AAA** ${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{49-4r \choose 2}\end{matrix}}$ 3,8080.0022239448.7 : 1 2,4960.0014577685.0 : 1 AAB** ${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{49-4r \choose 2}\end{matrix}}$ 137,0880.080060511.5 : 1 104,8320.061222815.3 : 1 xxxAA ${\begin{matrix}{1 \choose 1}{3 \choose 3}{r-1 \choose 1}{4 \choose 2}\end{matrix}}$ 420.000024540,768.1 : 1 480.000028035,672.0 : 1 xxxA* ${\begin{matrix}{1 \choose 1}{3 \choose 3}{r-1 \choose 1}{4 \choose 1}{49-4r \choose 1}\end{matrix}}$ 4760.00027803,596.3 : 1 4160.00024294,115.1 : 1 xxx** ${\begin{matrix}{1 \choose 1}{3 \choose 3}{49-4r \choose 2}\end{matrix}}$ 1360.000079412,589.5 : 1 780.000045621,951.6 : 1 xxAAA ${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 3}\end{matrix}}$ 840.000049120,383.6 : 1 960.000056117,835.5 : 1 xxAA* ${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 2}{49-4r \choose 1}\end{matrix}}$ 2,1420.0012509798.4 : 1 1,8720.0010933913.7 : 1 xxA** ${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 1}{49-4r \choose 2}\end{matrix}}$ 11,4240.0066717148.9 : 1 7,4880.0043731227.7 : 1 xAAAA ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 4}\end{matrix}}$ 210.000012381,537.3 : 1 240.000014071,345.0 : 1 xAAA* ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 3}{49-4r \choose 1}\end{matrix}}$ 1,4280.00083401,198.1 : 1 1,2480.00072881,371.0 : 1 xAA** ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{49-4r \choose 2}\end{matrix}}$ 17,1360.010007698.9 : 1 11,2320.0065596151.4 : 1  ??*** ${\begin{matrix}{4r-1 \choose 2}{49-4r \choose 3}\end{matrix}}$ 316,2000.18466354.4 : 1 170,1700.09938079.1 : 1  ?**** ${\begin{matrix}{4r-1 \choose 1}{49-4r \choose 4}\end{matrix}}$ 73,7800.043088122.2 : 1 25,0250.014614867.4 : 1 ***** ${\begin{matrix}{49-4r \choose 5}\end{matrix}}$ 6,1880.0036138275.7 : 1 1,2870.00075161,329.5 : 1 Total 595,5240.34779101.9 : 1 352,7360.20600083.9 : 1

## Derivations for making low hands with a low pair and two high cards

Starting hands with a low pair and two high cards (xx**) cannot make a low hand, so the only possibilities are having a low hand possible on the board or not. With two high cards in the hand, there are $52-4r-2=50-4r$ high cards and $4r-4+2=4r-2$ low cards left in the deck. The following tables show the derivations when holding a low pair and two high cards.
    Derivations for low hand shape xx** (a low pair and two high cards) on the flop Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Miss low ABC ${\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 2,2400.12950976.7 : 1 3,5840.20721553.8 : 1 xAB ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 6720.038852924.7 : 1 8960.051803918.3 : 1 Total 2,9120.16836264.9 : 1 4,4800.25901942.9 : 1 No low AAA ${\begin{matrix}{r-1 \choose 1}{4 \choose 3}\end{matrix}}$ 280.0016189616.7 : 1 320.0018501539.5 : 1 AAB ${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 1,0080.058279416.2 : 1 1,3440.077705811.9 : 1 xxA ${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 1}\end{matrix}}$ 280.0016189616.7 : 1 320.0018501539.5 : 1 xAA ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}\end{matrix}}$ 840.0048566204.9 : 1 960.0055504179.2 : 1  ??* ${\begin{matrix}{4r-2 \choose 2}{50-4r \choose 1}\end{matrix}}$ 7,8300.45270581.2 : 1 7,8540.45409341.2 : 1  ?** ${\begin{matrix}{4r-2 \choose 1}{50-4r \choose 2}\end{matrix}}$ 4,5900.26537932.8 : 1 3,0940.17888534.6 : 1 *** ${\begin{matrix}{50-4r \choose 3}\end{matrix}}$ 8160.047178520.2 : 1 3640.021045346.5 : 1 Total 14,3840.83163740.2 : 1 12,8160.74098060.3 : 1
    Derivations for low hand shape xx** (a low pair and two high cards) on the turn Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Miss low ABCD ${\begin{matrix}{r-1 \choose 4}{4 \choose 1}^{4}\end{matrix}}$ 8,9600.046047920.7 : 1 17,9200.09209589.9 : 1 AABC ${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 10,0800.051803918.3 : 1 16,1280.082886211.1 : 1 ABC* ${\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}{50-4r \choose 1}\end{matrix}}$ 40,3200.20721553.8 : 1 50,1760.25786822.9 : 1 xxAB ${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-1 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 3360.0017268578.1 : 1 4480.0023024433.3 : 1 xABC ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 4,4800.023023942.4 : 1 7,1680.036838326.1 : 1 xAAB ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 2,0160.010360895.5 : 1 2,6880.013814471.4 : 1 xAB* ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}{50-4r \choose 1}\end{matrix}}$ 12,0960.062164715.1 : 1 12,5440.064467114.5 : 1 Total 78,2880.40234351.5 : 1 107,0720.55027240.8 : 1 No low AAAA ${\begin{matrix}{r-1 \choose 1}{4 \choose 4}\end{matrix}}$ 70.000036027,796.1 : 1 80.000041124,321.5 : 1 AAAB ${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 6720.0034536288.6 : 1 8960.0046048216.2 : 1 AABB ${\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}\end{matrix}}$ 7560.0038853256.4 : 1 1,0080.0051804192.0 : 1 AAA* ${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{50-4r \choose 1}\end{matrix}}$ 5040.0025902385.1 : 1 4480.0023024433.3 : 1 AAB* ${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1}\end{matrix}}$ 18,1440.09324709.7 : 1 18,8160.09670069.3 : 1 xxAA ${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 2}\end{matrix}}$ 420.00021584,631.9 : 1 480.00024674,052.8 : 1 xxA* ${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 1}{50-4r \choose 1}\end{matrix}}$ 5040.0025902385.1 : 1 4480.0023024433.3 : 1 xAAA ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 3}\end{matrix}}$ 560.00028783,473.6 : 1 640.00032893,039.3 : 1 xAA* ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}{50-4r \choose 1}\end{matrix}}$ 1,5120.0077706127.7 : 1 1,3440.0069072143.8 : 1  ??** ${\begin{matrix}{4r-2 \choose 2}{50-4r \choose 2}\end{matrix}}$ 66,5550.34204441.9 : 1 51,0510.26236512.8 : 1  ?*** ${\begin{matrix}{4r-2 \choose 1}{50-4r \choose 3}\end{matrix}}$ 24,4800.12580946.9 : 1 12,3760.063603714.7 : 1 **** ${\begin{matrix}{50-4r \choose 4}\end{matrix}}$ 3,0600.015726262.6 : 1 1,0010.0051444193.4 : 1 Total 116,2920.59765650.7 : 1 87,5080.44972761.2 : 1
    Derivations for low hand shape xx** (a low pair and two high cards) on the river Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Miss low ABCDE ${\begin{matrix}{r-1 \choose 5}{4 \choose 1}^{5}\end{matrix}}$ 21,5040.012558578.6 : 1 57,3440.033489428.9 : 1 AABCD ${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 53,7600.031396330.9 : 1 107,5200.062792614.9 : 1 ABCD* ${\begin{matrix}{r-1 \choose 4}{4 \choose 1}^{4}{50-4r \choose 1}\end{matrix}}$ 161,2800.09418899.6 : 1 250,8800.14651605.8 : 1 AABBC ${\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 15,1200.0088302112.2 : 1 24,1920.014128369.8 : 1 AAABC ${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 6,7200.0039245253.8 : 1 10,7520.0062793158.3 : 1 AABC* ${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}{50-4r \choose 1}\end{matrix}}$ 181,4400.10596258.4 : 1 225,7920.13186446.6 : 1 ABC** ${\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}{50-4r \choose 2}\end{matrix}}$ 342,7200.20015144.0 : 1 326,1440.19047094.3 : 1 xABCD ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 4}{4 \choose 1}^{4}\end{matrix}}$ 17,9200.010465494.6 : 1 35,8400.020930946.8 : 1 xABC* ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 3}{4 \choose 1}^{3}{50-4r \choose 1}\end{matrix}}$ 80,6400.047094420.2 : 1 100,3520.058606416.1 : 1 xAABC ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 20,1600.011773683.9 : 1 32,2560.018837852.1 : 1 xAAB* ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1}\end{matrix}}$ 36,2880.021192546.2 : 1 37,6320.021977444.5 : 1 xAAAB ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 1,3440.00078491,273.0 : 1 1,7920.0010465954.5 : 1 xAABB ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 2}^{2}\end{matrix}}$ 1,5120.00088301,131.5 : 1 2,0160.0011774848.4 : 1 xAB** ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}{50-4r \choose 2}\end{matrix}}$ 102,8160.060045415.7 : 1 81,5360.047617720.0 : 1 xxABC ${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-1 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 2,2400.0013082763.4 : 1 3,5840.0020931476.8 : 1 xxAB* ${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-1 \choose 2}{4 \choose 1}^{2}{50-4r \choose 1}\end{matrix}}$ 6,0480.0035321282.1 : 1 6,2720.0036629272.0 : 1 xxAAB ${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 1,0080.00058871,697.7 : 1 1,3440.00078491,273.0 : 1 Total 1,052,5200.61468060.6 : 1 1,305,2480.76227590.3 : 1 No low AAAAB ${\begin{matrix}{r-1 \choose 1}{4 \choose 4}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 1680.000098110,191.3 : 1 2240.00013087,643.2 : 1 AAAA* ${\begin{matrix}{r-1 \choose 1}{4 \choose 4}{50-4r \choose 1}\end{matrix}}$ 1260.000073613,588.7 : 1 1120.000065415,287.4 : 1 AAABB ${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 2}\end{matrix}}$ 1,0080.00058871,697.7 : 1 1,3440.00078491,273.0 : 1 AAAB* ${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1}\end{matrix}}$ 12,0960.0070642140.6 : 1 12,5440.0073258135.5 : 1 AABB* ${\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}{50-4r \choose 1}\end{matrix}}$ 13,6080.0079472124.8 : 1 14,1120.0082415120.3 : 1 AAA** ${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{50-4r \choose 2}\end{matrix}}$ 4,2840.0025019398.7 : 1 2,9120.0017006587.0 : 1 AAB** ${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 2}\end{matrix}}$ 154,2240.090068110.1 : 1 122,3040.071426613.0 : 1 xxAAA ${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 3}\end{matrix}}$ 280.000016461,152.7 : 1 320.000018753,508.5 : 1 xxAA* ${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 2}{50-4r \choose 1}\end{matrix}}$ 7560.00044152,264.0 : 1 6720.00039252,547.1 : 1 xxA** ${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 1}{50-4r \choose 2}\end{matrix}}$ 4,2840.0025019398.7 : 1 2,9120.0017006587.0 : 1 xAAAA ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 4}\end{matrix}}$ 140.0000082122,306.4 : 1 160.0000093107,018.0 : 1 xAAA* ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 3}{50-4r \choose 1}\end{matrix}}$ 1,0080.00058871,697.7 : 1 8960.00052331,910.1 : 1 xAA** ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}{50-4r \choose 2}\end{matrix}}$ 12,8520.0075057132.2 : 1 8,7360.0051019195.0 : 1  ??*** ${\begin{matrix}{4r-2 \choose 2}{50-4r \choose 3}\end{matrix}}$ 354,9600.20729963.8 : 1 204,2040.11925697.4 : 1  ?**** ${\begin{matrix}{4r-2 \choose 1}{50-4r \choose 4}\end{matrix}}$ 91,8000.053612017.7 : 1 34,0340.019876149.3 : 1 ***** ${\begin{matrix}{50-4r \choose 5}\end{matrix}}$ 8,5680.0050038198.8 : 1 2,0020.0011692854.3 : 1 Total 659,7840.38531941.6 : 1 407,0560.23772413.2 : 1

## Derivations for making low hands with a low three of a kind and a high card

Starting hands with a low three of a kind and a high card (xxx*) cannot make a low hand, so the only possibilities are having a low hand possible on the board or not. With one high card in the hand, there are $52-4r-1=51-4r$ high cards and $4r-4+1=4r-3$ low cards left in the deck. The following tables show the derivations when holding a low three of a kind and a high card.
    Derivations for low hand shape xxx* (a low three of a kind and a high card) on the flop Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Miss low ABC ${\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 2,2400.12950976.7 : 1 3,5840.20721553.8 : 1 xAB ${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 3360.019426550.5 : 1 4480.025901937.6 : 1 Total 2,5760.14893625.7 : 1 4,0320.23311753.3 : 1 No low AAA ${\begin{matrix}{r-1 \choose 1}{4 \choose 3}\end{matrix}}$ 280.0016189616.7 : 1 320.0018501539.5 : 1 AAB ${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 1,0080.058279416.2 : 1 1,3440.077705811.9 : 1 xAA ${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 2}\end{matrix}}$ 420.0024283410.8 : 1 480.0027752359.3 : 1  ??* ${\begin{matrix}{4r-3 \choose 2}{51-4r \choose 1}\end{matrix}}$ 7,7140.44599911.2 : 1 7,9200.45790931.2 : 1  ?** ${\begin{matrix}{4r-3 \choose 1}{51-4r \choose 2}\end{matrix}}$ 4,9590.28671372.5 : 1 3,4650.20033534.0 : 1 *** ${\begin{matrix}{51-4r \choose 3}\end{matrix}}$ 9690.056024516.8 : 1 4550.026306737.0 : 1 Total 14,7200.85106380.2 : 1 13,2640.76688250.3 : 1
    Derivations for low hand shape xxx* (a low three of a kind and a high card) on the turn Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Miss low ABCD ${\begin{matrix}{r-1 \choose 4}{4 \choose 1}^{4}\end{matrix}}$ 8,9600.046047920.7 : 1 17,9200.09209589.9 : 1 AABC ${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 10,0800.051803918.3 : 1 16,1280.082886211.1 : 1 ABC* ${\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}{51-4r \choose 1}\end{matrix}}$ 42,5600.21872753.6 : 1 53,7600.27628742.6 : 1 xABC ${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 2,2400.011512085.9 : 1 3,5840.018419253.3 : 1 xAAB ${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 1,0080.0051804192.0 : 1 1,3440.0069072143.8 : 1 xAB* ${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}{51-4r \choose 1}\end{matrix}}$ 6,3840.032809129.5 : 1 6,7200.034535928.0 : 1 Total 71,2320.36608081.7 : 1 99,4560.51113171.0 : 1 No low AAAA ${\begin{matrix}{r-1 \choose 1}{4 \choose 4}\end{matrix}}$ 70.000036027,796.1 : 1 80.000041124,321.5 : 1 AAAB ${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 6720.0034536288.6 : 1 8960.0046048216.2 : 1 AABB ${\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}\end{matrix}}$ 7560.0038853256.4 : 1 1,0080.0051804192.0 : 1 AAA* ${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{51-4r \choose 1}\end{matrix}}$ 5320.0027341364.8 : 1 4800.0024669404.4 : 1 AAB* ${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 1}\end{matrix}}$ 19,1520.09842749.2 : 1 20,1600.10360788.7 : 1 xAAA ${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 3}\end{matrix}}$ 280.00014396,948.3 : 1 320.00016456,079.6 : 1 xAA* ${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 2}{51-4r \choose 1}\end{matrix}}$ 7980.0041011242.8 : 1 7200.0037003269.3 : 1  ??** ${\begin{matrix}{4r-3 \choose 2}{51-4r \choose 2}\end{matrix}}$ 69,4260.35679931.8 : 1 55,4400.28492142.5 : 1  ?*** ${\begin{matrix}{4r-3 \choose 1}{51-4r \choose 3}\end{matrix}}$ 28,1010.14441875.9 : 1 15,0150.077166212.0 : 1 **** ${\begin{matrix}{51-4r \choose 4}\end{matrix}}$ 3,8760.019919849.2 : 1 1,3650.0070151141.5 : 1 Total 123,3480.63391920.6 : 1 95,1240.48886831.0 : 1
    Derivations for low hand shape xxx* (a low three of a kind and a high card) on the river Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Miss low ABCDE ${\begin{matrix}{r-1 \choose 5}{4 \choose 1}^{5}\end{matrix}}$ 21,5040.012558578.6 : 1 57,3440.033489428.9 : 1 AABCD ${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 53,7600.031396330.9 : 1 107,5200.062792614.9 : 1 ABCD* ${\begin{matrix}{r-1 \choose 4}{4 \choose 1}^{4}{51-4r \choose 1}\end{matrix}}$ 170,2400.09942169.1 : 1 268,8000.15698155.4 : 1 AABBC ${\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 15,1200.0088302112.2 : 1 24,1920.014128369.8 : 1 AAABC ${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 6,7200.0039245253.8 : 1 10,7520.0062793158.3 : 1 AABC* ${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}{51-4r \choose 1}\end{matrix}}$ 191,5200.11184937.9 : 1 241,9200.14128336.1 : 1 ABC** ${\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}{51-4r \choose 2}\end{matrix}}$ 383,0400.22369863.5 : 1 376,3200.21977413.6 : 1 xABCD ${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 4}{4 \choose 1}^{4}\end{matrix}}$ 8,9600.0052327190.1 : 1 17,9200.010465494.6 : 1 xABC* ${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 3}{4 \choose 1}^{3}{51-4r \choose 1}\end{matrix}}$ 42,5600.024855439.2 : 1 53,7600.031396330.9 : 1 xAABC ${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 10,0800.0058868168.9 : 1 16,1280.0094189105.2 : 1 xAAB* ${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 1}\end{matrix}}$ 19,1520.011184988.4 : 1 20,1600.011773683.9 : 1 xAAAB ${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 6720.00039252,547.1 : 1 8960.00052331,910.1 : 1 xAABB ${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 2}{4 \choose 2}^{2}\end{matrix}}$ 7560.00044152,264.0 : 1 1,0080.00058871,697.7 : 1 xAB** ${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}{51-4r \choose 2}\end{matrix}}$ 57,4560.033554828.8 : 1 47,0400.027471835.4 : 1 Total 981,5400.57322770.7 : 1 1,243,7600.72636630.4 : 1 No low AAAAB ${\begin{matrix}{r-1 \choose 1}{4 \choose 4}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 1680.000098110,191.3 : 1 2240.00013087,643.2 : 1 AAAA* ${\begin{matrix}{r-1 \choose 1}{4 \choose 4}{51-4r \choose 1}\end{matrix}}$ 1330.000077712,873.5 : 1 1200.000070114,268.2 : 1 AAABB ${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 2}\end{matrix}}$ 1,0080.00058871,697.7 : 1 1,3440.00078491,273.0 : 1 AAAB* ${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 1}\end{matrix}}$ 12,7680.0074566133.1 : 1 13,4400.0078491126.4 : 1 AABB* ${\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}{51-4r \choose 1}\end{matrix}}$ 14,3640.0083887118.2 : 1 15,1200.0088302112.2 : 1 AAA** ${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{51-4r \choose 2}\end{matrix}}$ 4,7880.0027962356.6 : 1 3,3600.0019623508.6 : 1 AAB** ${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 2}\end{matrix}}$ 172,3680.10066448.9 : 1 141,1200.082415311.1 : 1 xAAAA ${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 4}\end{matrix}}$ 70.0000041244,613.9 : 1 80.0000047214,037.0 : 1 xAAA* ${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 3}{51-4r \choose 1}\end{matrix}}$ 5320.00031073,217.6 : 1 4800.00028033,566.3 : 1 xAA** ${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 2}{51-4r \choose 2}\end{matrix}}$ 7,1820.0041943237.4 : 1 5,0400.0029434338.7 : 1  ??*** ${\begin{matrix}{4r-3 \choose 2}{51-4r \choose 3}\end{matrix}}$ 393,4140.22975713.4 : 1 240,2400.14030226.1 : 1  ?**** ${\begin{matrix}{4r-3 \choose 1}{51-4r \choose 4}\end{matrix}}$ 112,4040.065644914.2 : 1 45,0450.026306737.0 : 1 ***** ${\begin{matrix}{51-4r \choose 5}\end{matrix}}$ 11,6280.0067909146.3 : 1 3,0030.0017538569.2 : 1 Total 730,7640.42677231.3 : 1 468,5440.27363372.7 : 1

## Derivations for making low hands with a low four of a kind

Starting hands with a low four of a kind (xxxx) cannot make a low hand, so the only possibilities are having a low hand possible on the board or not. With no high cards in the hand, there are $52-4r-0=52-4r$ high cards and $4r-4+0=4r-4$ low cards left in the deck. The following tables show the derivations when holding a low four of a kind.
    Derivations for low hand shape xxxx (a low four of a kind) on the flop Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Miss low ABC ${\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 2,2400.12950976.7 : 1 3,5840.20721553.8 : 1 No low AAA ${\begin{matrix}{r-1 \choose 1}{4 \choose 3}\end{matrix}}$ 280.0016189616.7 : 1 320.0018501539.5 : 1 AAB ${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 1,0080.058279416.2 : 1 1,3440.077705811.9 : 1  ??* ${\begin{matrix}{4r-4 \choose 2}{52-4r \choose 1}\end{matrix}}$ 7,5600.43709531.3 : 1 7,9360.45883441.2 : 1  ?** ${\begin{matrix}{4r-4 \choose 1}{52-4r \choose 2}\end{matrix}}$ 5,3200.30758562.3 : 1 3,8400.22201673.5 : 1 *** ${\begin{matrix}{52-4r \choose 3}\end{matrix}}$ 1,1400.065911214.2 : 1 5600.032377429.9 : 1 Total 15,0560.87049030.1 : 1 13,7120.79278450.3 : 1
    Derivations for low hand shape xxxx (a low four of a kind) on the turn Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Miss low ABCD ${\begin{matrix}{r-1 \choose 4}{4 \choose 1}^{4}\end{matrix}}$ 8,9600.046047920.7 : 1 17,9200.09209589.9 : 1 AABC ${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 10,0800.051803918.3 : 1 16,1280.082886211.1 : 1 ABC* ${\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}{52-4r \choose 1}\end{matrix}}$ 44,8000.23023953.3 : 1 57,3440.29470652.4 : 1 Total 63,8400.32809132.0 : 1 91,3920.46968861.1 : 1 No low AAAA ${\begin{matrix}{r-1 \choose 1}{4 \choose 4}\end{matrix}}$ 70.000036027,796.1 : 1 80.000041124,321.5 : 1 AAAB ${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 6720.0034536288.6 : 1 8960.0046048216.2 : 1 AABB ${\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}\end{matrix}}$ 7560.0038853256.4 : 1 1,0080.0051804192.0 : 1 AAA* ${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{52-4r \choose 1}\end{matrix}}$ 5600.0028780346.5 : 1 5120.0026313379.0 : 1 AAB* ${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 20,1600.10360788.7 : 1 21,5040.11051508.0 : 1  ??** ${\begin{matrix}{4r-4 \choose 2}{52-4r \choose 2}\end{matrix}}$ 71,8200.36910271.7 : 1 59,5200.30588962.3 : 1  ?*** ${\begin{matrix}{4r-4 \choose 1}{52-4r \choose 3}\end{matrix}}$ 31,9200.16404565.1 : 1 17,9200.09209589.9 : 1 **** ${\begin{matrix}{52-4r \choose 4}\end{matrix}}$ 4,8450.024899839.2 : 1 1,8200.0093535105.9 : 1 Total 130,7400.67190870.5 : 1 103,1880.53031140.9 : 1
    Derivations for low hand shape xxxx (a low four of a kind) on the river Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Miss low ABCDE ${\begin{matrix}{r-1 \choose 5}{4 \choose 1}^{5}\end{matrix}}$ 21,5040.012558578.6 : 1 57,3440.033489428.9 : 1 AABCD ${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 53,7600.031396330.9 : 1 107,5200.062792614.9 : 1 ABCD* ${\begin{matrix}{r-1 \choose 4}{4 \choose 1}^{4}{52-4r \choose 1}\end{matrix}}$ 179,2000.10465438.6 : 1 286,7200.16744695.0 : 1 AABBC ${\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 15,1200.0088302112.2 : 1 24,1920.014128369.8 : 1 AAABC ${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 6,7200.0039245253.8 : 1 10,7520.0062793158.3 : 1 AABC* ${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}{52-4r \choose 1}\end{matrix}}$ 201,6000.11773617.5 : 1 258,0480.15070225.6 : 1 ABC** ${\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}{52-4r \choose 2}\end{matrix}}$ 425,6000.24855403.0 : 1 430,0800.25117043.0 : 1 Total 903,5040.52765400.9 : 1 1,174,6560.68600900.5 : 1 No low AAAAB ${\begin{matrix}{r-1 \choose 1}{4 \choose 4}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 1680.000098110,191.3 : 1 2240.00013087,643.2 : 1 AAAA* ${\begin{matrix}{r-1 \choose 1}{4 \choose 4}{52-4r \choose 1}\end{matrix}}$ 1400.000081812,229.7 : 1 1280.000074813,376.4 : 1 AAABB ${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 2}\end{matrix}}$ 1,0080.00058871,697.7 : 1 1,3440.00078491,273.0 : 1 AAAB* ${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 13,4400.0078491126.4 : 1 14,3360.0083723118.4 : 1 AABB* ${\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}{52-4r \choose 1}\end{matrix}}$ 15,1200.0088302112.2 : 1 16,1280.0094189105.2 : 1 AAA** ${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{52-4r \choose 2}\end{matrix}}$ 5,3200.0031069320.9 : 1 3,8400.0022426444.9 : 1 AAB** ${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 2}\end{matrix}}$ 191,5200.11184937.9 : 1 161,2800.09418899.6 : 1  ??*** ${\begin{matrix}{4r-4 \choose 2}{52-4r \choose 3}\end{matrix}}$ 430,9200.25166093.0 : 1 277,7600.16221425.2 : 1  ?**** ${\begin{matrix}{4r-4 \choose 1}{52-4r \choose 4}\end{matrix}}$ 135,6600.079226611.6 : 1 58,2400.034012728.4 : 1 ***** ${\begin{matrix}{52-4r \choose 5}\end{matrix}}$ 15,5040.0090545109.4 : 1 4,3680.0025509391.0 : 1 Total 808,8000.47234601.1 : 1 537,6480.31399102.2 : 1

## Derivations for making low hands with two low ranks and two high cards

Starting hands with two low ranks and two high cards (xy**) can make a low hand when at least three other low ranks beside x and y appear on the board. With two high cards in the hand, there are $52-4r-2=50-4r$ high cards and $4r-4+2=4r-2$ low cards left in the deck. The following tables show the derivations when holding two low ranks and two high cards.
    Derivations for low hand shape xy** (two low ranks and two high cards) on the flop Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Make low ABC ${\begin{matrix}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 1,2800.074005612.5 : 1 2,2400.12950976.7 : 1 Miss low xAB ${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 1,4400.083256211.0 : 1 2,0160.11655877.6 : 1 xyA ${\begin{matrix}{2 \choose 2}{3 \choose 1}^{2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 2160.012488479.1 : 1 2520.014569867.6 : 1 Total 1,6560.09574479.4 : 1 2,2680.13112866.6 : 1 No low AAA ${\begin{matrix}{r-2 \choose 1}{4 \choose 3}\end{matrix}}$ 240.0013876719.7 : 1 280.0016189616.7 : 1 AAB ${\begin{matrix}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 7200.041628123.0 : 1 1,0080.058279416.2 : 1 xxx ${\begin{matrix}{2 \choose 1}{3 \choose 3}\end{matrix}}$ 20.00011568,647.0 : 1 20.00011568,647.0 : 1 xxy ${\begin{matrix}{2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}\end{matrix}}$ 180.0010407959.9 : 1 180.0010407959.9 : 1 xxA ${\begin{matrix}{2 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 1440.0083256119.1 : 1 1680.0097132102.0 : 1 xAA ${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}\end{matrix}}$ 2160.012488479.1 : 1 2520.014569867.6 : 1  ??* ${\begin{matrix}{4r-2 \choose 2}{50-4r \choose 1}\end{matrix}}$ 7,8300.45270581.2 : 1 7,8540.45409341.2 : 1  ?** ${\begin{matrix}{4r-2 \choose 1}{50-4r \choose 2}\end{matrix}}$ 4,5900.26537932.8 : 1 3,0940.17888534.6 : 1 *** ${\begin{matrix}{50-4r \choose 3}\end{matrix}}$ 8160.047178520.2 : 1 3640.021045346.5 : 1 Total 14,3600.83024980.2 : 1 12,7880.73936170.4 : 1
    Derivations for low hand shape xy** (two low ranks and two high cards) on the turn Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Make low ABCD ${\begin{matrix}{r-2 \choose 4}{4 \choose 1}^{4}\end{matrix}}$ 3,8400.019734849.7 : 1 8,9600.046047920.7 : 1 AABC ${\begin{matrix}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 5,7600.029602232.8 : 1 10,0800.051803918.3 : 1 ABC* ${\begin{matrix}{r-2 \choose 3}{4 \choose 1}^{3}{50-4r \choose 1}\end{matrix}}$ 23,0400.11840897.4 : 1 31,3600.16116765.2 : 1 xABC ${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 7,6800.039469624.3 : 1 13,4400.069071813.5 : 1 Total 40,3200.20721553.8 : 1 63,8400.32809132.0 : 1 Miss low xxyA ${\begin{matrix}{2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 4320.0022202449.4 : 1 5040.0025902385.1 : 1 xxAB ${\begin{matrix}{2 \choose 1}{3 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 1,4400.0074006134.1 : 1 2,0160.010360895.5 : 1 xyAA ${\begin{matrix}{2 \choose 2}{3 \choose 1}^{2}{r-2 \choose 1}{4 \choose 2}\end{matrix}}$ 3240.0016651599.6 : 1 3780.0019426513.8 : 1 xAAB ${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 4,3200.022201744.0 : 1 6,0480.031082331.2 : 1 xyAB ${\begin{matrix}{2 \choose 2}{3 \choose 1}^{2}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 2,1600.011100889.1 : 1 3,0240.015541263.3 : 1 xyA* ${\begin{matrix}{2 \choose 2}{3 \choose 1}^{2}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1}\end{matrix}}$ 3,8880.019981549.0 : 1 3,5280.018131454.2 : 1 xAB* ${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^{2}{50-4r \choose 1}\end{matrix}}$ 25,9200.13321006.5 : 1 28,2240.14505095.9 : 1 Total 38,4840.19777984.1 : 1 43,7220.22469943.5 : 1 No low AAAA ${\begin{matrix}{r-2 \choose 1}{4 \choose 4}\end{matrix}}$ 60.000030832,429.0 : 1 70.000036027,796.1 : 1 AAAB ${\begin{matrix}{r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 4800.0024669404.4 : 1 6720.0034536288.6 : 1 AABB ${\begin{matrix}{r-2 \choose 2}{4 \choose 2}^{2}\end{matrix}}$ 5400.0027752359.3 : 1 7560.0038853256.4 : 1 AAA* ${\begin{matrix}{r-2 \choose 1}{4 \choose 3}{50-4r \choose 1}\end{matrix}}$ 4320.0022202449.4 : 1 3920.0020146495.4 : 1 AAB* ${\begin{matrix}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}{50-4r \choose 1}\end{matrix}}$ 14,1120.072525412.8 : 1 12,9600.066605014.0 : 1 xxxy ${\begin{matrix}{2 \choose 1}{3 \choose 3}{1 \choose 1}{3 \choose 1}\end{matrix}}$ 60.000030832,429.0 : 1 60.000030832,429.0 : 1 xxxA ${\begin{matrix}{2 \choose 1}{3 \choose 3}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 560.00028783,473.6 : 1 480.00024674,052.8 : 1 xxx* ${\begin{matrix}{2 \choose 1}{3 \choose 3}{50-4r \choose 1}\end{matrix}}$ 280.00014396,948.3 : 1 360.00018505,404.0 : 1 xxyy ${\begin{matrix}{2 \choose 2}{3 \choose 2}^{2}\end{matrix}}$ 90.000046321,619.0 : 1 90.000046321,619.0 : 1 xxy* ${\begin{matrix}{2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{50-4r \choose 1}\end{matrix}}$ 3240.0016651599.6 : 1 2520.0012951771.1 : 1 xxAA ${\begin{matrix}{2 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2}\end{matrix}}$ 2160.0011101899.8 : 1 2520.0012951771.1 : 1 xxA* ${\begin{matrix}{2 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1}\end{matrix}}$ 2,5920.013321074.1 : 1 2,3520.012087681.7 : 1 xAAA ${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 3}\end{matrix}}$ 1440.00074011,350.3 : 1 1680.00086341,157.2 : 1 xAA* ${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{50-4r \choose 1}\end{matrix}}$ 3,8880.019981549.0 : 1 3,5280.018131454.2 : 1  ??** ${\begin{matrix}{4r-2 \choose 2}{50-4r \choose 2}\end{matrix}}$ 66,5550.34204441.9 : 1 51,0510.26236512.8 : 1  ?*** ${\begin{matrix}{4r-2 \choose 1}{50-4r \choose 3}\end{matrix}}$ 24,4800.12580946.9 : 1 12,3760.063603714.7 : 1 **** ${\begin{matrix}{50-4r \choose 4}\end{matrix}}$ 3,0600.015726262.6 : 1 1,0010.0051444193.4 : 1 Total 115,7760.59500460.7 : 1 87,0180.44720941.2 : 1
    Derivations for low hand shape xy** (two low ranks and two high cards) on the river Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Make low ABCDE ${\begin{matrix}{r-2 \choose 5}{4 \choose 1}^{5}\end{matrix}}$ 6,1440.0035881277.7 : 1 21,5040.012558578.6 : 1 AABCD ${\begin{matrix}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 23,0400.013455673.3 : 1 53,7600.031396330.9 : 1 ABCD* ${\begin{matrix}{r-2 \choose 4}{4 \choose 1}^{4}{50-4r \choose 1}\end{matrix}}$ 69,1200.040366723.8 : 1 125,4400.073258012.7 : 1 AABBC ${\begin{matrix}{r-2 \choose 2}{4 \choose 2}^{2}{r-4 \choose 1}{4 \choose 1}\end{matrix}}$ 8,6400.0050458197.2 : 1 15,1200.0088302112.2 : 1 AABC* ${\begin{matrix}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^{2}{50-4r \choose 1}\end{matrix}}$ 103,6800.060550015.5 : 1 141,1200.082415311.1 : 1 AAABC ${\begin{matrix}{r-2 \choose 1}{4 \choose 3}{r-3 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 3,8400.0022426444.9 : 1 6,7200.0039245253.8 : 1 ABC** ${\begin{matrix}{r-2 \choose 3}{4 \choose 1}^{3}{50-4r \choose 2}\end{matrix}}$ 195,8400.11437227.7 : 1 203,8400.11904437.4 : 1 xABCD ${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-2 \choose 4}{4 \choose 1}^{4}\end{matrix}}$ 23,0400.013455673.3 : 1 53,7600.031396330.9 : 1 xABC* ${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-2 \choose 3}{4 \choose 1}^{3}{50-4r \choose 1}\end{matrix}}$ 138,2400.080733311.4 : 1 188,1600.10988708.1 : 1 xAABC ${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 34,5600.020183348.5 : 1 60,4800.035320827.3 : 1 xxABC ${\begin{matrix}{2 \choose 1}{3 \choose 2}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 7,6800.0044852222.0 : 1 13,4400.0078491126.4 : 1 xyABC ${\begin{matrix}{2 \choose 2}{3 \choose 1}^{2}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 11,5200.0067278147.6 : 1 20,1600.011773683.9 : 1 Total 625,3440.36520621.7 : 1 903,5040.52765400.9 : 1 Miss low xxxyA ${\begin{matrix}{2 \choose 1}{3 \choose 3}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 1440.000084111,890.0 : 1 1680.000098110,191.3 : 1 xxxAB ${\begin{matrix}{2 \choose 1}{3 \choose 3}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 4800.00028033,566.3 : 1 6720.00039252,547.1 : 1 xxyyA ${\begin{matrix}{2 \choose 2}{3 \choose 2}^{2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 2160.00012617,926.3 : 1 2520.00014726,793.9 : 1 xxyAB ${\begin{matrix}{2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 4,3200.0025229395.4 : 1 6,0480.0035321282.1 : 1 xxyAA ${\begin{matrix}{2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}\end{matrix}}$ 6480.00037842,641.4 : 1 7560.00044152,264.0 : 1 xxyA* ${\begin{matrix}{2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1}\end{matrix}}$ 7,7760.0045412219.2 : 1 7,0560.0041208241.7 : 1 xxAAB ${\begin{matrix}{2 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 4,3200.0025229395.4 : 1 6,0480.0035321282.1 : 1 xxAB* ${\begin{matrix}{2 \choose 1}{3 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}{50-4r \choose 1}\end{matrix}}$ 25,9200.015137565.1 : 1 28,2240.016483159.7 : 1 xyAAA ${\begin{matrix}{2 \choose 2}{3 \choose 1}^{2}{r-2 \choose 1}{4 \choose 3}\end{matrix}}$ 2160.00012617,926.3 : 1 2520.00014726,793.9 : 1 xyAAB ${\begin{matrix}{2 \choose 2}{3 \choose 1}^{2}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 6,4800.0037844263.2 : 1 9,0720.0052981187.7 : 1 xyAA* ${\begin{matrix}{2 \choose 2}{3 \choose 1}^{2}{r-2 \choose 1}{4 \choose 2}{50-4r \choose 1}\end{matrix}}$ 5,8320.0034059292.6 : 1 5,2920.0030906322.6 : 1 xyAB* ${\begin{matrix}{2 \choose 2}{3 \choose 1}^{2}{r-2 \choose 2}{4 \choose 1}^{2}{50-4r \choose 1}\end{matrix}}$ 38,8800.022706243.0 : 1 42,3360.024724639.4 : 1 xyA** ${\begin{matrix}{2 \choose 2}{3 \choose 1}^{2}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 2}\end{matrix}}$ 33,0480.019300350.8 : 1 22,9320.013392573.7 : 1 xAAAB ${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 2,8800.0016819593.6 : 1 4,0320.0023547423.7 : 1 xAABB ${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 2}^{2}\end{matrix}}$ 3,2400.0018922527.5 : 1 4,5360.0026491376.5 : 1 xAAB* ${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1}\end{matrix}}$ 77,7600.045412521.0 : 1 84,6720.049449219.2 : 1 xAB** ${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}{50-4r \choose 2}\end{matrix}}$ 220,3200.12866876.8 : 1 183,4560.10713998.3 : 1 Total 432,4800.25257203.0 : 1 405,8040.23699303.2 : 1 No low AAAAB ${\begin{matrix}{r-2 \choose 1}{4 \choose 4}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 1200.000070114,268.2 : 1 1680.000098110,191.3 : 1 AAAA* ${\begin{matrix}{r-2 \choose 1}{4 \choose 4}{50-4r \choose 1}\end{matrix}}$ 1080.000063115,853.7 : 1 980.000057217,471.5 : 1 AAABB ${\begin{matrix}{r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 2}\end{matrix}}$ 7200.00042052,377.2 : 1 1,0080.00058871,697.7 : 1 AAAB* ${\begin{matrix}{r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 1}{50-4r \choose 1}\end{matrix}}$ 8,6400.0050458197.2 : 1 9,4080.0054944181.0 : 1 AABB* ${\begin{matrix}{r-2 \choose 2}{4 \choose 2}^{2}{50-4r \choose 1}\end{matrix}}$ 9,7200.0056766175.2 : 1 10,5840.0061811160.8 : 1 AAA** ${\begin{matrix}{r-2 \choose 1}{4 \choose 3}{50-4r \choose 2}\end{matrix}}$ 3,6720.0021445465.3 : 1 2,5480.0014881671.0 : 1 AAB** ${\begin{matrix}{r-2 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 2}\end{matrix}}$ 110,1600.064334414.5 : 1 91,7280.053569917.7 : 1 xxxyy ${\begin{matrix}{2 \choose 1}{3 \choose 3}{1 \choose 1}{3 \choose 2}\end{matrix}}$ 60.0000035285,383.0 : 1 60.0000035285,383.0 : 1 xxxy* ${\begin{matrix}{2 \choose 1}{3 \choose 3}{1 \choose 1}{3 \choose 1}{50-4r \choose 1}\end{matrix}}$ 1080.000063115,853.7 : 1 840.000049120,383.6 : 1 xxxAA ${\begin{matrix}{2 \choose 1}{3 \choose 3}{r-2 \choose 1}{4 \choose 2}\end{matrix}}$ 720.000042023,781.0 : 1 840.000049120,383.6 : 1 xxxA* ${\begin{matrix}{2 \choose 1}{3 \choose 3}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1}\end{matrix}}$ 8640.00050461,980.8 : 1 7840.00045792,183.1 : 1 xxx** ${\begin{matrix}{2 \choose 1}{3 \choose 3}{50-4r \choose 2}\end{matrix}}$ 3060.00017875,594.8 : 1 1820.00010639,407.3 : 1 xxyy* ${\begin{matrix}{2 \choose 2}{3 \choose 2}^{2}{50-4r \choose 1}\end{matrix}}$ 1620.000094610,568.8 : 1 1260.000073613,588.7 : 1 xxy** ${\begin{matrix}{2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{50-4r \choose 2}\end{matrix}}$ 2,7540.0016084620.8 : 1 1,6380.00095661,044.4 : 1 xxAAA ${\begin{matrix}{2 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 3}\end{matrix}}$ 1440.000084111,890.0 : 1 1680.000098110,191.3 : 1 xxAA* ${\begin{matrix}{2 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2}{50-4r \choose 1}\end{matrix}}$ 3,8880.0022706439.4 : 1 3,5280.0020604484.3 : 1 xxA** ${\begin{matrix}{2 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 2}\end{matrix}}$ 22,0320.012866976.7 : 1 15,2880.0089283111.0 : 1 xAAAA ${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 4}\end{matrix}}$ 360.000021047,563.0 : 1 420.000024540,768.1 : 1 xAAA* ${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 3}{50-4r \choose 1}\end{matrix}}$ 2,5920.0015137659.6 : 1 2,3520.0013736727.0 : 1 xAA** ${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{50-4r \choose 2}\end{matrix}}$ 33,0480.019300350.8 : 1 22,9320.013392573.7 : 1  ??*** ${\begin{matrix}{4r-2 \choose 2}{50-4r \choose 3}\end{matrix}}$ 354,9600.20729963.8 : 1 204,2040.11925697.4 : 1  ?**** ${\begin{matrix}{4r-2 \choose 1}{50-4r \choose 4}\end{matrix}}$ 91,8000.053612017.7 : 1 34,0340.019876149.3 : 1 ***** ${\begin{matrix}{50-4r \choose 5}\end{matrix}}$ 8,5680.0050038198.8 : 1 2,0020.0011692854.3 : 1 Total 654,4800.38222181.6 : 1 402,9960.23535313.2 : 1

## Derivations for making low hands with a low pair, a low card and a high card

Starting hands with a low pair, a low card and a high card (xxy*) can make a low hand when at least three other low ranks beside x and y appear on the board. With one high card in the hand, there are $52-4r-1=51-4r$ high cards and $4r-4+1=4r-3$ low cards left in the deck. The following tables show the derivations when holding a low pair, a low card and a high card.
    Derivations for low hand shape xxy* (a low pair, a low card and a high card) on the flop Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Make low ABC ${\begin{matrix}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 1,2800.074005612.5 : 1 2,2400.12950976.7 : 1 Miss low xAB ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 4800.027752135.0 : 1 6720.038852924.7 : 1 yAB ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 7200.041628123.0 : 1 1,0080.058279416.2 : 1 xyA ${\begin{matrix}{1 \choose 1}{2 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 1440.0083256119.1 : 1 1680.0097132102.0 : 1 Total 1,3440.077705811.9 : 1 1,8480.10684558.4 : 1 No low AAA ${\begin{matrix}{r-2 \choose 1}{4 \choose 3}\end{matrix}}$ 240.0013876719.7 : 1 280.0016189616.7 : 1 AAB ${\begin{matrix}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 7200.041628123.0 : 1 1,0080.058279416.2 : 1 xxy ${\begin{matrix}{1 \choose 1}{2 \choose 2}{1 \choose 1}{3 \choose 1}\end{matrix}}$ 30.00017355,764.3 : 1 30.00017355,764.3 : 1 xxA ${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 240.0013876719.7 : 1 280.0016189616.7 : 1 xyy ${\begin{matrix}{1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 2}\end{matrix}}$ 60.00034692,881.7 : 1 60.00034692,881.7 : 1 xAA ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 2}\end{matrix}}$ 720.0041628239.2 : 1 840.0048566204.9 : 1 yyy ${\begin{matrix}{1 \choose 1}{3 \choose 3}\end{matrix}}$ 10.000057817,295.0 : 1 10.000057817,295.0 : 1 yyA ${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 1080.0062442159.1 : 1 1260.0072849136.3 : 1 yAA ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}\end{matrix}}$ 2160.012488479.1 : 1 2520.014569867.6 : 1  ??* ${\begin{matrix}{4r-3 \choose 2}{51-4r \choose 1}\end{matrix}}$ 7,7140.44599911.2 : 1 7,9200.45790931.2 : 1  ?** ${\begin{matrix}{4r-3 \choose 1}{51-4r \choose 2}\end{matrix}}$ 4,9590.28671372.5 : 1 3,4650.20033534.0 : 1 *** ${\begin{matrix}{51-4r \choose 3}\end{matrix}}$ 9690.056024516.8 : 1 4550.026306737.0 : 1 Total 14,6720.84828860.2 : 1 13,2080.76364480.3 : 1
    Derivations for low hand shape xxy* (a low pair, a low card and a high card) on the turn Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Make low ABCD ${\begin{matrix}{r-2 \choose 4}{4 \choose 1}^{4}\end{matrix}}$ 3,8400.019734849.7 : 1 8,9600.046047920.7 : 1 AABC ${\begin{matrix}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 5,7600.029602232.8 : 1 10,0800.051803918.3 : 1 ABC* ${\begin{matrix}{r-2 \choose 3}{4 \choose 1}^{3}{51-4r \choose 1}\end{matrix}}$ 24,3200.12498727.0 : 1 33,6000.17267964.8 : 1 xABC ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 2,5600.013156575.0 : 1 4,4800.023023942.4 : 1 yABC ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 3,8400.019734849.7 : 1 6,7200.034535928.0 : 1 Total 40,3200.20721553.8 : 1 63,8400.32809132.0 : 1 Miss low xxyA ${\begin{matrix}{1 \choose 1}{2 \choose 2}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 720.00037002,701.5 : 1 840.00043172,315.4 : 1 xxAB ${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 2400.0012334809.8 : 1 3360.0017268578.1 : 1 xyyA ${\begin{matrix}{1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 1440.00074011,350.3 : 1 1680.00086341,157.2 : 1 xyAA ${\begin{matrix}{1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}\end{matrix}}$ 2160.012488479.1 : 1 2520.014569867.6 : 1 xAAB ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 1,4400.0074006134.1 : 1 2,0160.010360895.5 : 1 xyAB ${\begin{matrix}{1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 1,4400.0074006134.1 : 1 2,0160.010360895.5 : 1 xyA* ${\begin{matrix}{1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 1}\end{matrix}}$ 2,7360.014061170.1 : 1 2,5200.012951076.2 : 1 xAB* ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-2 \choose 2}{4 \choose 1}^{2}{51-4r \choose 1}\end{matrix}}$ 9,1200.046870220.3 : 1 10,0800.051803918.3 : 1 yyAB ${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 7200.0037003269.3 : 1 1,0080.0051804192.0 : 1 yAAB ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 2,1600.011100889.1 : 1 3,0240.015541263.3 : 1 yAB* ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^{2}{51-4r \choose 1}\end{matrix}}$ 13,6800.070305313.2 : 1 15,1200.077705811.9 : 1 Total 31,9680.16429235.1 : 1 36,6240.18822084.3 : 1 No low AAAA ${\begin{matrix}{r-2 \choose 1}{4 \choose 4}\end{matrix}}$ 60.000030832,429.0 : 1 70.000036027,796.1 : 1 AAAB ${\begin{matrix}{r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 4800.0024669404.4 : 1 6720.0034536288.6 : 1 AABB ${\begin{matrix}{r-2 \choose 2}{4 \choose 2}^{2}\end{matrix}}$ 5400.0027752359.3 : 1 7560.0038853256.4 : 1 AAA* ${\begin{matrix}{r-2 \choose 1}{4 \choose 3}{51-4r \choose 1}\end{matrix}}$ 4560.0023435425.7 : 1 4200.0021585462.3 : 1 AAB* ${\begin{matrix}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}{51-4r \choose 1}\end{matrix}}$ 13,6800.070305313.2 : 1 15,1200.077705811.9 : 1 xxyy ${\begin{matrix}{1 \choose 1}{2 \choose 2}{1 \choose 1}{3 \choose 2}\end{matrix}}$ 30.000015464,859.0 : 1 30.000015464,859.0 : 1 xxy* ${\begin{matrix}{1 \choose 1}{2 \choose 2}{1 \choose 1}{3 \choose 1}{51-4r \choose 1}\end{matrix}}$ 570.00029293,412.7 : 1 450.00023134,323.0 : 1 xxAA ${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 2}\end{matrix}}$ 360.00018505,404.0 : 1 420.00021584,631.9 : 1 xxA* ${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1}\end{matrix}}$ 4560.0023435425.7 : 1 4200.0021585462.3 : 1 xyyy ${\begin{matrix}{1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 3}\end{matrix}}$ 20.000010397,289.0 : 1 20.000010397,289.0 : 1 xyy* ${\begin{matrix}{1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 2}{51-4r \choose 1}\end{matrix}}$ 1140.00058591,705.8 : 1 900.00046252,161.0 : 1 xAAA ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 3}\end{matrix}}$ 480.00024674,052.8 : 1 560.00028783,473.6 : 1 xAA* ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 2}{51-4r \choose 1}\end{matrix}}$ 1,3680.0070305141.2 : 1 1,2600.0064755153.4 : 1 yyyA ${\begin{matrix}{1 \choose 1}{3 \choose 3}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 240.00012338,106.5 : 1 280.00014396,948.3 : 1 yyy* ${\begin{matrix}{1 \choose 1}{3 \choose 3}{51-4r \choose 1}\end{matrix}}$ 190.000097610,240.1 : 1 150.000077112,971.0 : 1 yyAA ${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2}\end{matrix}}$ 1080.00055501,800.7 : 1 1260.00064751,543.3 : 1 yyA* ${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 1}\end{matrix}}$ 1,3680.0070305141.2 : 1 1,2600.0064755153.4 : 1 yAAA ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 3}\end{matrix}}$ 720.00037002,701.5 : 1 840.00043172,315.4 : 1 yAA* ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{51-4r \choose 1}\end{matrix}}$ 2,0520.010545893.8 : 1 1,8900.0097132102.0 : 1  ??** ${\begin{matrix}{4r-3 \choose 2}{51-4r \choose 2}\end{matrix}}$ 69,4260.35679931.8 : 1 55,4400.28492142.5 : 1  ?*** ${\begin{matrix}{4r-3 \choose 1}{51-4r \choose 3}\end{matrix}}$ 28,1010.14441875.9 : 1 15,0150.077166212.0 : 1 **** ${\begin{matrix}{51-4r \choose 4}\end{matrix}}$ 3,8760.019919849.2 : 1 1,3650.0070151141.5 : 1 Total 122,2920.62849210.6 : 1 94,1160.48368791.1 : 1
    Derivations for low hand shape xxy* (a low pair, a low card and a high card) on the river Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Make low ABCDE ${\begin{matrix}{r-2 \choose 5}{4 \choose 1}^{5}\end{matrix}}$ 6,1440.0035881277.7 : 1 21,5040.012558578.6 : 1 AABCD ${\begin{matrix}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 23,0400.013455673.3 : 1 53,7600.031396330.9 : 1 ABCD* ${\begin{matrix}{r-2 \choose 4}{4 \choose 1}^{4}{51-4r \choose 1}\end{matrix}}$ 72,9600.042609322.5 : 1 134,4000.078490711.7 : 1 AABBC ${\begin{matrix}{r-2 \choose 2}{4 \choose 2}^{2}{r-4 \choose 1}{4 \choose 1}\end{matrix}}$ 8,6400.0050458197.2 : 1 15,1200.0088302112.2 : 1 AABC* ${\begin{matrix}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^{2}{51-4r \choose 1}\end{matrix}}$ 109,4400.063913914.6 : 1 151,2000.088302110.3 : 1 AAABC ${\begin{matrix}{r-2 \choose 1}{4 \choose 3}{r-3 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 3,8400.0022426444.9 : 1 6,7200.0039245253.8 : 1 ABC** ${\begin{matrix}{r-2 \choose 3}{4 \choose 1}^{3}{51-4r \choose 2}\end{matrix}}$ 218,8800.12782786.8 : 1 235,2000.13735886.3 : 1 xABCD ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-2 \choose 4}{4 \choose 1}^{4}\end{matrix}}$ 7,6800.0044852222.0 : 1 17,9200.010465494.6 : 1 xABC* ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-2 \choose 3}{4 \choose 1}^{3}{51-4r \choose 1}\end{matrix}}$ 48,6400.028406234.2 : 1 67,2000.039245424.5 : 1 xAABC ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 11,5200.0067278147.6 : 1 20,1600.011773683.9 : 1 xxABC ${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 1,2800.00074751,336.7 : 1 2,2400.0013082763.4 : 1 xyABC ${\begin{matrix}{1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 7,6800.0044852222.0 : 1 13,4400.0078491126.4 : 1 yyABC ${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 3,8400.0022426444.9 : 1 6,7200.0039245253.8 : 1 yABCD ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-2 \choose 4}{4 \choose 1}^{4}\end{matrix}}$ 11,5200.0067278147.6 : 1 26,8800.015698162.7 : 1 yABC* ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-2 \choose 3}{4 \choose 1}^{3}{51-4r \choose 1}\end{matrix}}$ 72,9600.042609322.5 : 1 100,8000.058868116.0 : 1 yAABC ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 17,2800.010091798.1 : 1 30,2400.017660455.6 : 1 Total 625,3440.36520621.7 : 1 903,5040.52765400.9 : 1 Miss low xxyyA ${\begin{matrix}{1 \choose 1}{2 \choose 2}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 720.000042023,781.0 : 1 840.000049120,383.6 : 1 xxyAB ${\begin{matrix}{1 \choose 1}{2 \choose 2}{1 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 7200.00042052,377.2 : 1 1,0080.00058871,697.7 : 1 xxyAA ${\begin{matrix}{1 \choose 1}{2 \choose 2}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}\end{matrix}}$ 1080.000063115,853.7 : 1 1260.000073613,588.7 : 1 xxyA* ${\begin{matrix}{1 \choose 1}{2 \choose 2}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1}\end{matrix}}$ 1,3680.00079891,250.7 : 1 1,2600.00073591,358.0 : 1 xxAAB ${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 7200.00042052,377.2 : 1 1,0080.00058871,697.7 : 1 xxAB* ${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}{51-4r \choose 1}\end{matrix}}$ 4,5600.0026631374.5 : 1 5,0400.0029434338.7 : 1 xyyyA ${\begin{matrix}{1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 3}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 480.000028035,672.0 : 1 560.000032730,575.9 : 1 xyyAA ${\begin{matrix}{1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 3}\end{matrix}}$ 2160.00012617,926.3 : 1 2520.00014726,793.9 : 1 xyyAB ${\begin{matrix}{1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 1,4400.00084101,188.1 : 1 2,0160.0011774848.4 : 1 xyyA* ${\begin{matrix}{1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2}{51-4r \choose 1}\end{matrix}}$ 2,7360.0015978624.8 : 1 2,5200.0014717678.5 : 1 xyAAA ${\begin{matrix}{1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 3}\end{matrix}}$ 1440.000084111,890.0 : 1 1680.000098110,191.3 : 1 xyAAB ${\begin{matrix}{1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 4,3200.0025229395.4 : 1 6,0480.0035321282.1 : 1 xyAA* ${\begin{matrix}{1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{51-4r \choose 1}\end{matrix}}$ 4,1040.0023968416.2 : 1 3,7800.0022076452.0 : 1 xyAB* ${\begin{matrix}{1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^{2}{51-4r \choose 1}\end{matrix}}$ 27,3600.015978561.6 : 1 30,2400.017660455.6 : 1 xyA** ${\begin{matrix}{1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 2}\end{matrix}}$ 24,6240.014380668.5 : 1 17,6400.010301996.1 : 1 xAAAB ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 9600.00056061,782.7 : 1 1,3440.00078491,273.0 : 1 xAABB ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 2}^{2}\end{matrix}}$ 1,0800.00063071,584.5 : 1 1,5120.00088301,131.5 : 1 xAAB* ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 1}\end{matrix}}$ 27,3600.015978561.6 : 1 30,2400.017660455.6 : 1 xAB** ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}{51-4r \choose 2}\end{matrix}}$ 82,0800.047935419.9 : 1 70,5600.041207623.3 : 1 yyyAB ${\begin{matrix}{1 \choose 1}{3 \choose 3}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 2400.00014027,133.6 : 1 3360.00019625,095.1 : 1 yyAAB ${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 2,1600.0012615791.7 : 1 3,0240.0017660565.2 : 1 yyAB* ${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}{51-4r \choose 1}\end{matrix}}$ 13,6800.0079892124.2 : 1 15,1200.0088302112.2 : 1 yAAAB ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 1,4400.00084101,188.1 : 1 2,0160.0011774848.4 : 1 yAABB ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 2}^{2}\end{matrix}}$ 1,6200.00094611,056.0 : 1 2,2680.0013245754.0 : 1 yAAB* ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 1}\end{matrix}}$ 41,0400.023967740.7 : 1 45,3600.026490636.7 : 1 yAB** ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}{51-4r \choose 2}\end{matrix}}$ 123,1200.071903112.9 : 1 105,8400.061811515.2 : 1 Total 367,3200.21451803.7 : 1 348,8660.20374073.9 : 1 No low AAAAB ${\begin{matrix}{r-2 \choose 1}{4 \choose 4}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 1200.000070115,019.2 : 1 1680.000061316,306.7 : 1 AAAA* ${\begin{matrix}{r-2 \choose 1}{4 \choose 4}{51-4r \choose 1}\end{matrix}}$ 1140.000066615,853.7 : 1 1050.000057217,471.5 : 1 AAABB ${\begin{matrix}{r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 2}\end{matrix}}$ 7200.00042052,377.2 : 1 1,0080.00058871,697.7 : 1 AAAB* ${\begin{matrix}{r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 1}{51-4r \choose 1}\end{matrix}}$ 10,0800.0058868168.9 : 1 9,1200.0053262186.8 : 1 AABB* ${\begin{matrix}{r-2 \choose 2}{4 \choose 2}^{2}{51-4r \choose 1}\end{matrix}}$ 11,3400.0066227150.0 : 1 10,2600.0059919165.9 : 1 AAA** ${\begin{matrix}{r-2 \choose 1}{4 \choose 3}{51-4r \choose 2}\end{matrix}}$ 2,9400.0017170581.4 : 1 4,1040.0023968416.2 : 1 AAB** ${\begin{matrix}{r-2 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 2}\end{matrix}}$ 105,8400.061811515.2 : 1 123,1200.071903112.9 : 1 xxyyy ${\begin{matrix}{1 \choose 1}{2 \choose 2}{1 \choose 1}{3 \choose 3}\end{matrix}}$ 10.00000061,712,303.0 : 1 10.00000061,712,303.0 : 1 xxyy* ${\begin{matrix}{1 \choose 1}{2 \choose 2}{1 \choose 1}{3 \choose 2}{51-4r \choose 1}\end{matrix}}$ 570.000033330,039.4 : 1 450.000026338,050.2 : 1 xxy** ${\begin{matrix}{1 \choose 1}{2 \choose 2}{1 \choose 1}{3 \choose 1}{51-4r \choose 2}\end{matrix}}$ 5130.00029963,336.8 : 1 3150.00018405,434.9 : 1 xxAAA ${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 3}\end{matrix}}$ 240.000014071,345.0 : 1 280.000016461,152.7 : 1 xxAA* ${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 2}{51-4r \choose 1}\end{matrix}}$ 6840.00039952,502.4 : 1 6300.00036792,716.9 : 1 xxA** ${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 2}\end{matrix}}$ 4,1040.0023968416.2 : 1 2,9400.0017170581.4 : 1 xyyy* ${\begin{matrix}{1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 3}{51-4r \choose 1}\end{matrix}}$ 380.000022245,059.6 : 1 300.000017557,075.8 : 1 xyy** ${\begin{matrix}{1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 2}{51-4r \choose 2}\end{matrix}}$ 1,0260.00059921,667.9 : 1 6300.00036792,716.9 : 1 xAAAA ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 4}\end{matrix}}$ 120.0000070142,691.0 : 1 140.0000082122,306.4 : 1 xAAA* ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 3}{51-4r \choose 1}\end{matrix}}$ 9120.00053261,876.5 : 1 8400.00049062,037.5 : 1 xAA** ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 2}{51-4r \choose 2}\end{matrix}}$ 12,3120.0071903138.1 : 1 8,8200.0051510193.1 : 1 yyyAA ${\begin{matrix}{1 \choose 1}{3 \choose 3}{r-2 \choose 1}{4 \choose 2}\end{matrix}}$ 360.000021047,563.0 : 1 420.000024540,768.1 : 1 yyyA* ${\begin{matrix}{1 \choose 1}{3 \choose 3}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 1}\end{matrix}}$ 4560.00026633,754.1 : 1 4200.00024534,075.9 : 1 yyy** ${\begin{matrix}{1 \choose 1}{3 \choose 3}{51-4r \choose 2}\end{matrix}}$ 1710.000099910,012.5 : 1 1050.000061316,306.7 : 1 yyAAA ${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 3}\end{matrix}}$ 720.000042023,781.0 : 1 840.000049120,383.6 : 1 yyAA* ${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2}{51-4r \choose 1}\end{matrix}}$ 2,0520.0011984833.5 : 1 1,8900.0011038905.0 : 1 yyA** ${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 2}\end{matrix}}$ 12,3120.0071903138.1 : 1 8,8200.0051510193.1 : 1 yAAAA ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 4}\end{matrix}}$ 180.000010595,127.0 : 1 210.000012381,537.3 : 1 yAAA* ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 3}{51-4r \choose 1}\end{matrix}}$ 1,3680.00079891,250.7 : 1 1,2600.00073591,358.0 : 1 yAA** ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{51-4r \choose 2}\end{matrix}}$ 18,4680.010785591.7 : 1 13,2300.0077264128.4 : 1  ??*** ${\begin{matrix}{4r-3 \choose 2}{51-4r \choose 3}\end{matrix}}$ 393,4140.22975713.4 : 1 240,2400.14030226.1 : 1  ?**** ${\begin{matrix}{4r-3 \choose 1}{51-4r \choose 4}\end{matrix}}$ 112,4040.065644914.2 : 1 45,0450.026306737.0 : 1 ***** ${\begin{matrix}{51-4r \choose 5}\end{matrix}}$ 11,6280.0067909146.3 : 1 3,0030.0017538569.2 : 1 Total 719,6400.42027581.4 : 1 459,9340.26860532.7 : 1

## Derivations for making low hands with a low three of a kind and a low card

Starting hands with a low three of a kind and a low card (xxxy) can make a low hand when at least three other low ranks beside x and y appear on the board. With no high cards in the hand, there are $52-4r-0=52-4r$ high cards and $4r-4+0=4r-4$ low cards left in the deck. The following tables show the derivations when holding a low three of a kind and a low card.
    Derivations for low hand shape xxxy (a low three of a kind and a low card) on the flop Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Make low ABC ${\begin{matrix}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 1,2800.074005612.5 : 1 2,2400.12950976.7 : 1 Miss low xAB ${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 2400.013876071.1 : 1 3360.019426550.5 : 1 yAB ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 7200.041628123.0 : 1 1,0080.058279416.2 : 1 xyA ${\begin{matrix}{1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 720.0041628239.2 : 1 840.0048566204.9 : 1 Total 1,0320.059667015.8 : 1 1,4280.082562411.1 : 1 No low AAA ${\begin{matrix}{r-2 \choose 1}{4 \choose 3}\end{matrix}}$ 240.0013876719.7 : 1 280.0016189616.7 : 1 AAB ${\begin{matrix}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 7200.041628123.0 : 1 1,0080.058279416.2 : 1 xyy ${\begin{matrix}{1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 2}\end{matrix}}$ 30.00017355,764.3 : 1 30.00017355,764.3 : 1 xAA ${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-2 \choose 1}{4 \choose 2}\end{matrix}}$ 360.0020814479.4 : 1 420.0024283410.8 : 1 yyy ${\begin{matrix}{1 \choose 1}{3 \choose 3}\end{matrix}}$ 10.000057817,295.0 : 1 10.000057817,295.0 : 1 yyA ${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 1080.0062442159.1 : 1 1260.0072849136.3 : 1 yAA ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}\end{matrix}}$ 2160.012488479.1 : 1 2520.014569867.6 : 1  ??* ${\begin{matrix}{4r-4 \choose 2}{52-4r \choose 1}\end{matrix}}$ 7,5600.43709531.3 : 1 7,9360.45883441.2 : 1  ?** ${\begin{matrix}{4r-4 \choose 1}{52-4r \choose 2}\end{matrix}}$ 5,3200.30758562.3 : 1 3,8400.22201673.5 : 1 *** ${\begin{matrix}{52-4r \choose 3}\end{matrix}}$ 1,1400.065911214.2 : 1 5600.032377429.9 : 1 Total 14,9840.86632750.2 : 1 13,6280.78792780.3 : 1
    Derivations for low hand shape xxxy (a low three of a kind and a low card) on the turn Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Make low ABCD ${\begin{matrix}{r-2 \choose 4}{4 \choose 1}^{4}\end{matrix}}$ 8,9600.046047920.7 : 1 17,9200.09209589.9 : 1 AABC ${\begin{matrix}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 10,0800.051803918.3 : 1 16,1280.082886211.1 : 1 ABC* ${\begin{matrix}{r-2 \choose 3}{4 \choose 1}^{3}{52-4r \choose 1}\end{matrix}}$ 25,6000.13156546.6 : 1 35,8400.18419164.4 : 1 xABC ${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 1,2800.0065783151.0 : 1 2,2400.011512085.9 : 1 yABC ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 3,8400.019734849.7 : 1 6,7200.034535928.0 : 1 Total 40,3200.20721553.8 : 1 63,8400.32809132.0 : 1 Miss low xyyA ${\begin{matrix}{1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 720.00037002,701.5 : 1 840.00043172,315.4 : 1 xyAA ${\begin{matrix}{1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}\end{matrix}}$ 1080.00055501,800.7 : 1 1260.00064751,543.3 : 1 xAAB ${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 7200.0037003269.3 : 1 1,0080.0051804192.0 : 1 xyAB ${\begin{matrix}{1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 7200.0037003269.3 : 1 1,0080.0051804192.0 : 1 xyA* ${\begin{matrix}{1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 1,4400.0074006134.1 : 1 1,3440.0069072143.8 : 1 xAB* ${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-2 \choose 2}{4 \choose 1}^{2}{52-4r \choose 1}\end{matrix}}$ 4,8000.024668539.5 : 1 5,3760.027628735.2 : 1 yyAB ${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 7200.0037003269.3 : 1 1,0080.0051804192.0 : 1 yAAB ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 2,1600.011100889.1 : 1 3,0240.015541263.3 : 1 yAB* ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^{2}{52-4r \choose 1}\end{matrix}}$ 14,4000.074005612.5 : 1 16,1280.082886211.1 : 1 Total 25,1400.12920146.7 : 1 29,1060.14958375.7 : 1 No low AAAA ${\begin{matrix}{r-2 \choose 1}{4 \choose 4}\end{matrix}}$ 60.000030832,429.0 : 1 70.000036027,796.1 : 1 AAAB ${\begin{matrix}{r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 4800.0024669404.4 : 1 6720.0034536288.6 : 1 AABB ${\begin{matrix}{r-2 \choose 2}{4 \choose 2}^{2}\end{matrix}}$ 5400.0027752359.3 : 1 7560.0038853256.4 : 1 AAA* ${\begin{matrix}{r-2 \choose 1}{4 \choose 3}{52-4r \choose 1}\end{matrix}}$ 4800.0024669404.4 : 1 4480.0023024433.3 : 1 AAB* ${\begin{matrix}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 14,4000.074005612.5 : 1 16,1280.082886211.1 : 1 xyyy ${\begin{matrix}{1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 3}\end{matrix}}$ 10.0000051194,579.0 : 1 10.0000051194,579.0 : 1 xyy* ${\begin{matrix}{1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 2}{52-4r \choose 1}\end{matrix}}$ 600.00030843,242.0 : 1 480.00024674,052.8 : 1 xAAA ${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-2 \choose 1}{4 \choose 3}\end{matrix}}$ 240.00012338,106.5 : 1 280.00014396,948.3 : 1 xAA* ${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-2 \choose 1}{4 \choose 2}{52-4r \choose 1}\end{matrix}}$ 7200.0037003269.3 : 1 6720.0034536288.6 : 1 yyyA ${\begin{matrix}{1 \choose 1}{3 \choose 3}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 240.00012338,106.5 : 1 280.00014396,948.3 : 1 yyy* ${\begin{matrix}{1 \choose 1}{3 \choose 3}{52-4r \choose 1}\end{matrix}}$ 200.00010289,728.0 : 1 160.000082212,160.3 : 1 yyAA ${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2}\end{matrix}}$ 1080.00055501,800.7 : 1 1260.00064751,543.3 : 1 yyA* ${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 1,4400.0074006134.1 : 1 1,3440.0069072143.8 : 1 yAAA ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 3}\end{matrix}}$ 720.00037002,701.5 : 1 840.00043172,315.4 : 1 yAA* ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{52-4r \choose 1}\end{matrix}}$ 2,1600.011100889.1 : 1 2,0160.010360895.5 : 1  ??** ${\begin{matrix}{4r-4 \choose 2}{52-4r \choose 2}\end{matrix}}$ 71,8200.36910271.7 : 1 59,5200.30588962.3 : 1  ?*** ${\begin{matrix}{4r-4 \choose 1}{52-4r \choose 3}\end{matrix}}$ 31,9200.16404565.1 : 1 17,9200.09209589.9 : 1 **** ${\begin{matrix}{52-4r \choose 4}\end{matrix}}$ 4,8450.024899839.2 : 1 1,8200.0093535105.9 : 1 Total 129,1200.66358310.5 : 1 101,6340.52232500.9 : 1
    Derivations for low hand shape xxxy (a low three of a kind and a low card) on the river Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Make low ABCDE ${\begin{matrix}{r-2 \choose 5}{4 \choose 1}^{5}\end{matrix}}$ 6,1440.0035881277.7 : 1 21,5040.012558578.6 : 1 AABCD ${\begin{matrix}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 23,0400.013455673.3 : 1 53,7600.031396330.9 : 1 ABCD* ${\begin{matrix}{r-2 \choose 4}{4 \choose 1}^{4}{52-4r \choose 1}\end{matrix}}$ 76,8000.044851821.3 : 1 143,3600.083723510.9 : 1 AABBC ${\begin{matrix}{r-2 \choose 2}{4 \choose 2}^{2}{r-4 \choose 1}{4 \choose 1}\end{matrix}}$ 8,6400.0050458197.2 : 1 15,1200.0088302112.2 : 1 AABC* ${\begin{matrix}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^{2}{52-4r \choose 1}\end{matrix}}$ 115,2000.067277813.9 : 1 161,2800.09418899.6 : 1 AAABC ${\begin{matrix}{r-2 \choose 1}{4 \choose 3}{r-3 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 3,8400.0022426444.9 : 1 6,7200.0039245253.8 : 1 ABC** ${\begin{matrix}{r-2 \choose 3}{4 \choose 1}^{3}{52-4r \choose 2}\end{matrix}}$ 243,2000.14203096.0 : 1 268,8000.15698155.4 : 1 xABCD ${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-2 \choose 4}{4 \choose 1}^{4}\end{matrix}}$ 3,8400.0022426444.9 : 1 8,9600.0052327190.1 : 1 xABC* ${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-2 \choose 3}{4 \choose 1}^{3}{52-4r \choose 1}\end{matrix}}$ 25,6000.014950665.9 : 1 35,8400.020930946.8 : 1 xAABC ${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 5,7600.0033639296.3 : 1 10,0800.0058868168.9 : 1 xyABC ${\begin{matrix}{1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 3,8400.0022426444.9 : 1 6,7200.0039245253.8 : 1 yyABC ${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 3,8400.0022426444.9 : 1 6,7200.0039245253.8 : 1 yABCD ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-2 \choose 4}{4 \choose 1}^{4}\end{matrix}}$ 11,5200.0067278147.6 : 1 26,8800.015698162.7 : 1 yABC* ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-2 \choose 3}{4 \choose 1}^{3}{52-4r \choose 1}\end{matrix}}$ 76,8000.044851821.3 : 1 107,5200.062792614.9 : 1 yAABC ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 17,2800.010091798.1 : 1 30,2400.017660455.6 : 1 Total 625,3440.36520621.7 : 1 903,5040.52765400.9 : 1 Miss low xyyyA ${\begin{matrix}{1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 3}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 240.000014071,345.0 : 1 280.000016461,152.7 : 1 xyyAA ${\begin{matrix}{1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 3}\end{matrix}}$ 1080.000063115,853.7 : 1 1260.000073613,588.7 : 1 xyyAB ${\begin{matrix}{1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 7200.00042052,377.2 : 1 1,0080.00058871,697.7 : 1 xyyA* ${\begin{matrix}{1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2}{52-4r \choose 1}\end{matrix}}$ 1,4400.00084101,188.1 : 1 1,3440.00078491,273.0 : 1 xyAAA ${\begin{matrix}{1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 3}\end{matrix}}$ 720.000042023,781.0 : 1 840.000049120,383.6 : 1 xyAAB ${\begin{matrix}{1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 2,1600.0012615791.7 : 1 3,0240.0017660565.2 : 1 xyAA* ${\begin{matrix}{1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{52-4r \choose 1}\end{matrix}}$ 2,1600.0012615791.7 : 1 2,0160.0011774848.4 : 1 xyAB* ${\begin{matrix}{1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^{2}{52-4r \choose 1}\end{matrix}}$ 14,4000.0084097117.9 : 1 16,1280.0094189105.2 : 1 xyA** ${\begin{matrix}{1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 2}\end{matrix}}$ 13,6800.0079892124.2 : 1 10,0800.0058868168.9 : 1 xAAAB ${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 4800.00028033,566.3 : 1 6720.00039252,547.1 : 1 xAABB ${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 2}{4 \choose 2}^{2}\end{matrix}}$ 5400.00031543,169.9 : 1 7560.00044152,264.0 : 1 xAAB* ${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 14,4000.0084097117.9 : 1 16,1280.0094189105.2 : 1 xAB** ${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}{52-4r \choose 2}\end{matrix}}$ 45,6000.026630836.6 : 1 40,3200.023547241.5 : 1 yyyAB ${\begin{matrix}{1 \choose 1}{3 \choose 3}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 2400.00014027,133.6 : 1 3360.00019625,095.1 : 1 yyAAB ${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 2,1600.0012615791.7 : 1 3,0240.0017660565.2 : 1 yyAB* ${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}{52-4r \choose 1}\end{matrix}}$ 14,4000.0084097117.9 : 1 16,1280.0094189105.2 : 1 yAAAB ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 1,4400.00084101,188.1 : 1 2,0160.0011774848.4 : 1 yAABB ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 2}^{2}\end{matrix}}$ 1,6200.00094611,056.0 : 1 2,2680.0013245754.0 : 1 yAAB* ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 43,2000.025229238.6 : 1 48,3840.028256734.4 : 1 yAB** ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}{52-4r \choose 2}\end{matrix}}$ 136,8000.079892411.5 : 1 120,9600.070641713.2 : 1 Total 295,6440.17265864.8 : 1 284,8300.16634315.0 : 1 No low AAAAB ${\begin{matrix}{r-2 \choose 1}{4 \choose 4}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 1200.000070115,019.2 : 1 1680.000061316,306.7 : 1 AAAA* ${\begin{matrix}{r-2 \choose 1}{4 \choose 4}{52-4r \choose 1}\end{matrix}}$ 1200.000070114,268.2 : 1 1120.000065415,287.4 : 1 AAABB ${\begin{matrix}{r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 2}\end{matrix}}$ 7200.00042052,377.2 : 1 1,0080.00058871,697.7 : 1 AAAB* ${\begin{matrix}{r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 9,6000.0056065177.4 : 1 10,7520.0062793158.3 : 1 AABB* ${\begin{matrix}{r-2 \choose 2}{4 \choose 2}^{2}{52-4r \choose 1}\end{matrix}}$ 10,8000.0063073157.5 : 1 12,0960.0070642140.6 : 1 AAA** ${\begin{matrix}{r-2 \choose 1}{4 \choose 3}{52-4r \choose 2}\end{matrix}}$ 4,5600.0026631374.5 : 1 3,3600.0019623508.6 : 1 AAB** ${\begin{matrix}{r-2 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 2}\end{matrix}}$ 136,8000.079892411.5 : 1 120,9600.070641713.2 : 1 xyyy* ${\begin{matrix}{1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 3}{52-4r \choose 1}\end{matrix}}$ 200.000011785,614.2 : 1 160.0000093107,018.0 : 1 xyy** ${\begin{matrix}{1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 2}{52-4r \choose 2}\end{matrix}}$ 5700.00033293,003.0 : 1 3600.00021024,755.4 : 1 xAAAA ${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-2 \choose 1}{4 \choose 4}\end{matrix}}$ 60.0000035285,383.0 : 1 70.0000041244,613.9 : 1 xAAA* ${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-2 \choose 1}{4 \choose 3}{52-4r \choose 1}\end{matrix}}$ 4800.00028033,566.3 : 1 4480.00026163,821.1 : 1 xAA** ${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-2 \choose 1}{4 \choose 2}{52-4r \choose 2}\end{matrix}}$ 6,8400.0039946249.3 : 1 5,0400.0029434338.7 : 1 yyyAA ${\begin{matrix}{1 \choose 1}{3 \choose 3}{r-2 \choose 1}{4 \choose 2}\end{matrix}}$ 360.000021047,563.0 : 1 420.000024540,768.1 : 1 yyyA* ${\begin{matrix}{1 \choose 1}{3 \choose 3}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 4800.00028033,566.3 : 1 4480.00026163,821.1 : 1 yyy** ${\begin{matrix}{1 \choose 1}{3 \choose 3}{52-4r \choose 2}\end{matrix}}$ 1900.00011109,011.1 : 1 1200.000070114,268.2 : 1 yyAAA ${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 3}\end{matrix}}$ 720.000042023,781.0 : 1 840.000049120,383.6 : 1 yyAA* ${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2}{52-4r \choose 1}\end{matrix}}$ 2,1600.0012615791.7 : 1 2,0160.0011774848.4 : 1 yyA** ${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 2}\end{matrix}}$ 13,6800.0079892124.2 : 1 10,0800.0058868168.9 : 1 yAAAA ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 4}\end{matrix}}$ 180.000010595,127.0 : 1 210.000012381,537.3 : 1 yAAA* ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 3}{52-4r \choose 1}\end{matrix}}$ 1,4400.00084101,188.1 : 1 1,3440.00078491,273.0 : 1 yAA** ${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{52-4r \choose 2}\end{matrix}}$ 20,5200.011983982.4 : 1 15,1200.0088302112.2 : 1  ??*** ${\begin{matrix}{4r-4 \choose 2}{52-4r \choose 3}\end{matrix}}$ 430,9200.25166093.0 : 1 277,7600.16221425.2 : 1  ?**** ${\begin{matrix}{4r-4 \choose 1}{52-4r \choose 4}\end{matrix}}$ 135,6600.079226611.6 : 1 58,2400.034012728.4 : 1 ***** ${\begin{matrix}{52-4r \choose 5}\end{matrix}}$ 15,5040.0090545109.4 : 1 4,3680.0025509391.0 : 1 Total 791,3160.46213521.2 : 1 523,9700.30600292.3 : 1

## Derivations for making low hands with two low pair

Starting hands with two low pair (xxyy) can make a low hand when at least three other low ranks beside x and y appear on the board. With no high cards in the hand, there are $52-4r-0=52-4r$ high cards and $4r-4+0=4r-4$ low cards left in the deck. The following tables show the derivations when holding two low pair.
    Derivations for low hand shape xxyy (two low pair) on the flop Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Make low ABC ${\begin{matrix}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 1,2800.074005612.5 : 1 2,2400.12950976.7 : 1 Miss low xAB ${\begin{matrix}{2 \choose 1}{2 \choose 1}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 9600.055504217.0 : 1 1,3440.077705811.9 : 1 xyA ${\begin{matrix}{2 \choose 2}{2 \choose 1}^{2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 960.0055504179.2 : 1 1120.0064755153.4 : 1 Total 1,0560.061054615.4 : 1 1,4560.084181310.9 : 1 No low AAA ${\begin{matrix}{r-2 \choose 1}{4 \choose 3}\end{matrix}}$ 240.0013876719.7 : 1 280.0016189616.7 : 1 AAB ${\begin{matrix}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 7200.041628123.0 : 1 1,0080.058279416.2 : 1 xxy ${\begin{matrix}{2 \choose 1}{2 \choose 2}{1 \choose 1}{2 \choose 1}\end{matrix}}$ 40.00023134,323.0 : 1 40.00023134,323.0 : 1 xxA ${\begin{matrix}{2 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 480.0027752359.3 : 1 560.0032377307.9 : 1 xAA ${\begin{matrix}{2 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 2}\end{matrix}}$ 1440.0083256119.1 : 1 1680.0097132102.0 : 1  ??* ${\begin{matrix}{4r-4 \choose 2}{52-4r \choose 1}\end{matrix}}$ 7,5600.43709531.3 : 1 7,9360.45883441.2 : 1  ?** ${\begin{matrix}{4r-4 \choose 1}{52-4r \choose 2}\end{matrix}}$ 5,3200.30758562.3 : 1 3,8400.22201673.5 : 1 *** ${\begin{matrix}{52-4r \choose 3}\end{matrix}}$ 1,1400.065911214.2 : 1 5600.032377429.9 : 1 Total 14,9600.86493990.2 : 1 13,6000.78630900.3 : 1
    Derivations for low hand shape xxyy (two low pair) on the turn Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Make low ABCD ${\begin{matrix}{r-2 \choose 4}{4 \choose 1}^{4}\end{matrix}}$ 3,8400.019734849.7 : 1 8,9600.046047920.7 : 1 AABC ${\begin{matrix}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 5,7600.029602232.8 : 1 10,0800.051803918.3 : 1 ABC* ${\begin{matrix}{r-2 \choose 3}{4 \choose 1}^{3}{52-4r \choose 1}\end{matrix}}$ 25,6000.13156546.6 : 1 35,8400.18419164.4 : 1 xABC ${\begin{matrix}{2 \choose 1}{2 \choose 1}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 5,1200.026313137.0 : 1 8,9600.046047920.7 : 1 Total 40,3200.20721553.8 : 1 63,8400.32809132.0 : 1 Miss low xxyA ${\begin{matrix}{2 \choose 1}{2 \choose 2}{1 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 960.00049342,025.9 : 1 1120.00057561,736.3 : 1 xxAB ${\begin{matrix}{2 \choose 1}{2 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 4800.0024669404.4 : 1 6720.0034536288.6 : 1 xyAA ${\begin{matrix}{2 \choose 2}{2 \choose 1}^{2}{r-2 \choose 1}{4 \choose 2}\end{matrix}}$ 1440.00074011,350.3 : 1 1680.00086341,157.2 : 1 xAAB ${\begin{matrix}{2 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 2,8800.014801166.6 : 1 4,0320.020721647.3 : 1 xyAB ${\begin{matrix}{2 \choose 2}{2 \choose 1}^{2}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 9600.0049337201.7 : 1 1,3440.0069072143.8 : 1 xyA* ${\begin{matrix}{2 \choose 2}{2 \choose 1}^{2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 1,9200.0098674100.3 : 1 1,7920.0092096107.6 : 1 xAB* ${\begin{matrix}{2 \choose 1}{2 \choose 1}{r-2 \choose 2}{4 \choose 1}^{2}{52-4r \choose 1}\end{matrix}}$ 19,2000.09867419.1 : 1 21,5040.11051508.0 : 1 Total 25,6800.13197666.6 : 1 29,6240.15224595.6 : 1 No low AAAA ${\begin{matrix}{r-2 \choose 1}{4 \choose 4}\end{matrix}}$ 60.000030832,429.0 : 1 70.000036027,796.1 : 1 AAAB ${\begin{matrix}{r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 4800.0024669404.4 : 1 6720.0034536288.6 : 1 AABB ${\begin{matrix}{r-2 \choose 2}{4 \choose 2}^{2}\end{matrix}}$ 5400.0027752359.3 : 1 7560.0038853256.4 : 1 AAA* ${\begin{matrix}{r-2 \choose 1}{4 \choose 3}{52-4r \choose 1}\end{matrix}}$ 4800.0024669404.4 : 1 4480.0023024433.3 : 1 AAB* ${\begin{matrix}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 14,4000.074005612.5 : 1 16,1280.082886211.1 : 1 xxyy ${\begin{matrix}{2 \choose 2}{2 \choose 2}^{2}\end{matrix}}$ 10.0000051194,579.0 : 1 10.0000051194,579.0 : 1 xxy* ${\begin{matrix}{2 \choose 1}{2 \choose 2}{1 \choose 1}{2 \choose 1}{52-4r \choose 1}\end{matrix}}$ 800.00041112,431.3 : 1 640.00032893,039.3 : 1 xxAA ${\begin{matrix}{2 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 2}\end{matrix}}$ 720.00037002,701.5 : 1 840.00043172,315.4 : 1 xxA* ${\begin{matrix}{2 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 9600.0049337201.7 : 1 8960.0046048216.2 : 1 xAAA ${\begin{matrix}{2 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 3}\end{matrix}}$ 960.00049342,025.9 : 1 1120.00057561,736.3 : 1 xAA* ${\begin{matrix}{2 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 2}{52-4r \choose 1}\end{matrix}}$ 2,8800.014801166.6 : 1 2,6880.013814471.4 : 1  ??** ${\begin{matrix}{4r-4 \choose 2}{52-4r \choose 2}\end{matrix}}$ 71,8200.36910271.7 : 1 59,5200.30588962.3 : 1  ?*** ${\begin{matrix}{4r-4 \choose 1}{52-4r \choose 3}\end{matrix}}$ 31,9200.16404565.1 : 1 17,9200.09209589.9 : 1 **** ${\begin{matrix}{52-4r \choose 4}\end{matrix}}$ 4,8450.024899839.2 : 1 1,8200.0093535105.9 : 1 Total 128,5800.66080790.5 : 1 101,1160.51966290.9 : 1
    Derivations for low hand shape xxyy (two low pair) on the river Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Make low ABCDE ${\begin{matrix}{r-2 \choose 5}{4 \choose 1}^{5}\end{matrix}}$ 6,1440.0035881277.7 : 1 21,5040.012558578.6 : 1 AABCD ${\begin{matrix}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 23,0400.013455673.3 : 1 53,7600.031396330.9 : 1 ABCD* ${\begin{matrix}{r-2 \choose 4}{4 \choose 1}^{4}{52-4r \choose 1}\end{matrix}}$ 76,8000.044851821.3 : 1 143,3600.083723510.9 : 1 AABBC ${\begin{matrix}{r-2 \choose 2}{4 \choose 2}^{2}{r-4 \choose 1}{4 \choose 1}\end{matrix}}$ 8,6400.0050458197.2 : 1 15,1200.0088302112.2 : 1 AABC* ${\begin{matrix}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^{2}{52-4r \choose 1}\end{matrix}}$ 115,2000.067277813.9 : 1 161,2800.09418899.6 : 1 AAABC ${\begin{matrix}{r-2 \choose 1}{4 \choose 3}{r-3 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 3,8400.0022426444.9 : 1 6,7200.0039245253.8 : 1 ABC** ${\begin{matrix}{r-2 \choose 3}{4 \choose 1}^{3}{52-4r \choose 2}\end{matrix}}$ 243,2000.14203096.0 : 1 268,8000.15698155.4 : 1 xABCD ${\begin{matrix}{2 \choose 1}{2 \choose 1}{r-2 \choose 4}{4 \choose 1}^{4}\end{matrix}}$ 15,3600.0089704110.5 : 1 35,8400.020930946.8 : 1 xABC* ${\begin{matrix}{2 \choose 1}{2 \choose 1}{r-2 \choose 3}{4 \choose 1}^{3}{52-4r \choose 1}\end{matrix}}$ 102,4000.059802515.7 : 1 143,3600.083723510.9 : 1 xAABC ${\begin{matrix}{2 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 23,0400.013455673.3 : 1 40,3200.023547241.5 : 1 xxABC ${\begin{matrix}{2 \choose 1}{2 \choose 2}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 2,5600.0014951667.9 : 1 4,4800.0026164381.2 : 1 xyABC ${\begin{matrix}{2 \choose 2}{2 \choose 1}^{2}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 5,1200.0029901333.4 : 1 8,9600.0052327190.1 : 1 Total 625,3440.36520621.7 : 1 903,5040.52765400.9 : 1 Miss low xxyyA ${\begin{matrix}{2 \choose 2}{2 \choose 2}^{2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 240.000014071,345.0 : 1 280.000016461,152.7 : 1 xxyAB ${\begin{matrix}{2 \choose 1}{2 \choose 2}{1 \choose 1}{2 \choose 1}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 9600.00056061,782.7 : 1 1,3440.00078491,273.0 : 1 xxyAA ${\begin{matrix}{2 \choose 1}{2 \choose 2}{1 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 2}\end{matrix}}$ 1440.000084111,890.0 : 1 1680.000098110,191.3 : 1 xxyA* ${\begin{matrix}{2 \choose 1}{2 \choose 2}{1 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 1,9200.0011213890.8 : 1 1,7920.0010465954.5 : 1 xxAAB ${\begin{matrix}{2 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 1,4400.00084101,188.1 : 1 2,0160.0011774848.4 : 1 xxAB* ${\begin{matrix}{2 \choose 1}{2 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}{52-4r \choose 1}\end{matrix}}$ 9,6000.0056065177.4 : 1 10,7520.0062793158.3 : 1 xyAAA ${\begin{matrix}{2 \choose 2}{2 \choose 1}^{2}{r-2 \choose 1}{4 \choose 3}\end{matrix}}$ 960.000056117,835.5 : 1 1120.000065415,287.4 : 1 xyAAB ${\begin{matrix}{2 \choose 2}{2 \choose 1}^{2}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 2,8800.0016819593.6 : 1 4,0320.0023547423.7 : 1 xyAA* ${\begin{matrix}{2 \choose 2}{2 \choose 1}^{2}{r-2 \choose 1}{4 \choose 2}{52-4r \choose 1}\end{matrix}}$ 2,8800.0016819593.6 : 1 2,6880.0015698636.0 : 1 xyAB* ${\begin{matrix}{2 \choose 2}{2 \choose 1}^{2}{r-2 \choose 2}{4 \choose 1}^{2}{52-4r \choose 1}\end{matrix}}$ 19,2000.011213088.2 : 1 21,5040.012558578.6 : 1 xyA** ${\begin{matrix}{2 \choose 2}{2 \choose 1}^{2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 2}\end{matrix}}$ 18,2400.010652392.9 : 1 13,4400.0078491126.4 : 1 xAAAB ${\begin{matrix}{2 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$ 1,9200.0011213890.8 : 1 2,6880.0015698636.0 : 1 xAABB ${\begin{matrix}{2 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 2}^{2}\end{matrix}}$ 2,1600.0012615791.7 : 1 3,0240.0017660565.2 : 1 xAAB* ${\begin{matrix}{2 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 57,6000.033638928.7 : 1 64,5120.037675625.5 : 1 xAB** ${\begin{matrix}{2 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}{52-4r \choose 2}\end{matrix}}$ 182,4000.10652318.4 : 1 161,2800.09418899.6 : 1 Total 301,4640.17605754.7 : 1 289,3800.16900044.9 : 1 No low AAAAB ${\begin{matrix}{r-2 \choose 1}{4 \choose 4}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 1200.000070114,268.2 : 1 1680.000098110,191.3 : 1 AAAA* ${\begin{matrix}{r-2 \choose 1}{4 \choose 4}{52-4r \choose 1}\end{matrix}}$ 1200.000070114,268.2 : 1 1120.000065415,287.4 : 1 AAABB ${\begin{matrix}{r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 2}\end{matrix}}$ 7200.00042052,377.2 : 1 1,0080.00058871,697.7 : 1 AAAB* ${\begin{matrix}{r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 9,6000.0056065177.4 : 1 10,7520.0062793158.3 : 1 AABB* ${\begin{matrix}{r-2 \choose 2}{4 \choose 2}^{2}{52-4r \choose 1}\end{matrix}}$ 10,8000.0063073157.5 : 1 12,0960.0070642140.6 : 1 AAA** ${\begin{matrix}{r-2 \choose 1}{4 \choose 3}{52-4r \choose 2}\end{matrix}}$ 4,5600.0026631374.5 : 1 3,3600.0019623508.6 : 1 AAB** ${\begin{matrix}{r-2 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 2}\end{matrix}}$ 136,8000.079892411.5 : 1 120,9600.070641713.2 : 1 xxyy* ${\begin{matrix}{2 \choose 2}{2 \choose 2}^{2}{52-4r \choose 1}\end{matrix}}$ 200.000011785,614.2 : 1 160.0000093107,018.0 : 1 xxy** ${\begin{matrix}{2 \choose 1}{2 \choose 2}{1 \choose 1}{2 \choose 1}{52-4r \choose 2}\end{matrix}}$ 7600.00044382,252.0 : 1 4800.00028033,566.3 : 1 xxAAA ${\begin{matrix}{2 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 3}\end{matrix}}$ 480.000028035,672.0 : 1 560.000032730,575.9 : 1 xxAA* ${\begin{matrix}{2 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 2}{52-4r \choose 1}\end{matrix}}$ 1,4400.00084101,188.1 : 1 1,3440.00078491,273.0 : 1 xxA** ${\begin{matrix}{2 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 2}\end{matrix}}$ 9,1200.0053262186.8 : 1 6,7200.0039245253.8 : 1 xAAAA ${\begin{matrix}{2 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 4}\end{matrix}}$ 240.000014071,345.0 : 1 280.000016461,152.7 : 1 xAAA* ${\begin{matrix}{2 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 3}{52-4r \choose 1}\end{matrix}}$ 1,9200.0011213890.8 : 1 1,7920.0010465954.5 : 1 xAA** ${\begin{matrix}{2 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 2}{52-4r \choose 2}\end{matrix}}$ 27,3600.015978561.6 : 1 20,1600.011773683.9 : 1  ??*** ${\begin{matrix}{4r-4 \choose 2}{52-4r \choose 3}\end{matrix}}$ 430,9200.25166093.0 : 1 277,7600.16221425.2 : 1  ?**** ${\begin{matrix}{4r-4 \choose 1}{52-4r \choose 4}\end{matrix}}$ 135,6600.079226611.6 : 1 58,2400.034012728.4 : 1 ***** ${\begin{matrix}{52-4r \choose 5}\end{matrix}}$ 15,5040.0090545109.4 : 1 4,3680.0025509391.0 : 1 Total 785,4960.45873631.2 : 1 519,4200.30334572.3 : 1

## Derivations for making low hands with three low ranks and a high card

Starting hands with three low ranks and a high card (xyz*) can make a low hand when either at least three other low ranks beside x, y and z appear on the board; or when two other low ranks beside x, y and z and at least one of the low ranks x, y or z appears. With one high card in the hand, there are $52-4r-1=51-4r$ high cards and $4r-4+1=4r-3$ low cards left in the deck. The following tables show the derivations when holding three low ranks and a high card.
    Derivations for low hand shape xyz* (three low ranks and a high card) on the flop Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Make low ABC ${\begin{matrix}{r-3 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 6400.037002826.0 : 1 1,2800.074005612.5 : 1 xAB ${\begin{matrix}{3 \choose 1}{3 \choose 1}{r-3 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 1,4400.083256211.0 : 1 2,1600.12488447.0 : 1 Total 2,0800.12025907.3 : 1 3,4400.19888994.0 : 1 Miss low xyA ${\begin{matrix}{3 \choose 2}{3 \choose 1}^{2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 5400.031221131.0 : 1 6480.037465325.7 : 1 xyz ${\begin{matrix}{3 \choose 3}{3 \choose 1}^{3}\end{matrix}}$ 270.0015611639.6 : 1 270.0015611639.6 : 1 Total 5670.032782129.5 : 1 6750.039026424.6 : 1 No low AAA ${\begin{matrix}{r-3 \choose 1}{4 \choose 3}\end{matrix}}$ 200.0011563863.8 : 1 240.0013876719.7 : 1 AAB ${\begin{matrix}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1}\end{matrix}}$ 4800.027752135.0 : 1 7200.041628123.0 : 1 xxx ${\begin{matrix}{3 \choose 1}{3 \choose 3}\end{matrix}}$ 30.00017355,764.3 : 1 30.00017355,764.3 : 1 xxy ${\begin{matrix}{3 \choose 1}{3 \choose 2}{2 \choose 1}{3 \choose 1}\end{matrix}}$ 540.0031221319.3 : 1 540.0031221319.3 : 1 xxA ${\begin{matrix}{3 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 1800.010407095.1 : 1 2160.012488479.1 : 1 xAA ${\begin{matrix}{3 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}\end{matrix}}$ 2700.015610563.1 : 1 3240.018732752.4 : 1  ??* ${\begin{matrix}{4r-3 \choose 2}{51-4r \choose 1}\end{matrix}}$ 7,7140.44599911.2 : 1 7,9200.45790931.2 : 1  ?** ${\begin{matrix}{4r-3 \choose 1}{51-4r \choose 2}\end{matrix}}$ 4,9590.28671372.5 : 1 3,4650.20033534.0 : 1 *** ${\begin{matrix}{51-4r \choose 3}\end{matrix}}$ 9690.056024516.8 : 1 4550.026306737.0 : 1 Total 14,6490.84695880.2 : 1 13,1810.76208370.3 : 1
    Derivations for low hand shape xyz* (three low ranks and a high card) on the turn Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Make low ABCD ${\begin{matrix}{r-3 \choose 4}{4 \choose 1}^{4}\end{matrix}}$ 1,2800.0065783151.0 : 1 3,8400.019734849.7 : 1 AABC ${\begin{matrix}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 2,8800.014801166.6 : 1 5,7600.029602232.8 : 1 ABC* ${\begin{matrix}{r-3 \choose 3}{4 \choose 1}^{3}{51-4r \choose 1}\end{matrix}}$ 12,1600.062493615.0 : 1 19,2000.09867419.1 : 1 xABC ${\begin{matrix}{3 \choose 1}{3 \choose 1}{r-3 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 5,7600.029602232.8 : 1 11,5200.059204415.9 : 1 xxAB ${\begin{matrix}{3 \choose 1}{3 \choose 2}{r-3 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 1,4400.0074006134.1 : 1 2,1600.011100889.1 : 1 xAAB ${\begin{matrix}{3 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1}\end{matrix}}$ 4,3200.022201744.0 : 1 6,4800.033302529.0 : 1 xyAB ${\begin{matrix}{3 \choose 2}{3 \choose 1}^{2}{r-3 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 4,3200.022201744.0 : 1 6,4800.033302529.0 : 1 xAB* ${\begin{matrix}{3 \choose 1}{3 \choose 1}{r-3 \choose 2}{4 \choose 1}^{2}{51-4r \choose 1}\end{matrix}}$ 27,3600.14061056.1 : 1 32,4000.16651255.0 : 1 Total 59,5200.30588962.3 : 1 87,8400.45143391.2 : 1 Miss low xxyA ${\begin{matrix}{3 \choose 1}{3 \choose 2}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 1,0800.0055504179.2 : 1 1,2960.0066605149.1 : 1 xyAA ${\begin{matrix}{3 \choose 2}{3 \choose 1}^{2}{r-3 \choose 1}{4 \choose 2}\end{matrix}}$ 8100.0041628239.2 : 1 9720.0049954199.2 : 1 xyA* ${\begin{matrix}{3 \choose 2}{3 \choose 1}^{2}{r-3 \choose 1}{4 \choose 1}{51-4r \choose 1}\end{matrix}}$ 10,2600.052729018.0 : 1 9,7200.049953719.0 : 1 xxyz ${\begin{matrix}{3 \choose 1}{3 \choose 2}{2 \choose 2}{3 \choose 1}^{2}\end{matrix}}$ 810.00041632,401.2 : 1 810.00041632,401.2 : 1 xyzA ${\begin{matrix}{3 \choose 3}{3 \choose 1}^{3}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 5400.0027752359.3 : 1 6480.0033302299.3 : 1 xyz* ${\begin{matrix}{3 \choose 3}{3 \choose 1}^{3}{51-4r \choose 1}\end{matrix}}$ 5130.0026364378.3 : 1 4050.0020814479.4 : 1 Total 13,2840.068270113.6 : 1 13,1220.067437613.8 : 1 No low AAAA ${\begin{matrix}{r-3 \choose 1}{4 \choose 4}\end{matrix}}$ 50.000025738,915.0 : 1 60.000030832,429.0 : 1 AAAB ${\begin{matrix}{r-3 \choose 1}{4 \choose 3}{r-4 \choose 1}{4 \choose 1}\end{matrix}}$ 3200.0016446607.1 : 1 4800.0024669404.4 : 1 AABB ${\begin{matrix}{r-3 \choose 2}{4 \choose 2}^{2}\end{matrix}}$ 3600.0018501539.5 : 1 5400.0027752359.3 : 1 AAA* ${\begin{matrix}{r-3 \choose 1}{4 \choose 3}{51-4r \choose 1}\end{matrix}}$ 3800.0019529511.1 : 1 3600.0018501539.5 : 1 AAB* ${\begin{matrix}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1}{51-4r \choose 1}\end{matrix}}$ 9,1200.046870220.3 : 1 10,8000.055504217.0 : 1 xxxy ${\begin{matrix}{3 \choose 1}{3 \choose 3}{2 \choose 1}{3 \choose 1}\end{matrix}}$ 180.000092510,809.0 : 1 180.000092510,809.0 : 1 xxxA ${\begin{matrix}{3 \choose 1}{3 \choose 3}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 600.00030843,242.0 : 1 720.00037002,701.5 : 1 xxx* ${\begin{matrix}{3 \choose 1}{3 \choose 3}{51-4r \choose 1}\end{matrix}}$ 570.00029293,412.7 : 1 450.00023134,323.0 : 1 xxyy ${\begin{matrix}{3 \choose 2}{3 \choose 2}^{2}\end{matrix}}$ 270.00013887,205.7 : 1 270.00013887,205.7 : 1 xxy* ${\begin{matrix}{3 \choose 1}{3 \choose 2}{2 \choose 1}{3 \choose 1}{51-4r \choose 1}\end{matrix}}$ 1,0260.0052729188.6 : 1 8100.0041628239.2 : 1 xxAA ${\begin{matrix}{3 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 2}\end{matrix}}$ 2700.0013876719.7 : 1 3240.0016651599.6 : 1 xxA* ${\begin{matrix}{3 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 1}{51-4r \choose 1}\end{matrix}}$ 3,4200.017576355.9 : 1 3,2400.016651259.1 : 1 xAAA ${\begin{matrix}{3 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 3}\end{matrix}}$ 1800.00092511,080.0 : 1 2160.0011101899.8 : 1 xAA* ${\begin{matrix}{3 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}{51-4r \choose 1}\end{matrix}}$ 5,1300.026364536.9 : 1 4,8600.024976939.0 : 1  ??** ${\begin{matrix}{4r-3 \choose 2}{51-4r \choose 2}\end{matrix}}$ 69,4260.35679931.8 : 1 55,4400.28492142.5 : 1  ?*** ${\begin{matrix}{4r-3 \choose 1}{51-4r \choose 3}\end{matrix}}$ 28,1010.14441875.9 : 1 15,0150.077166212.0 : 1 **** ${\begin{matrix}{51-4r \choose 4}\end{matrix}}$ 3,8760.019919849.2 : 1 1,3650.0070151141.5 : 1 Total 121,7760.62584030.6 : 1 93,6180.48112861.1 : 1
    Derivations for low hand shape xyz* (three low ranks and a high card) on the river Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Make low ABCDE ${\begin{matrix}{r-3 \choose 5}{4 \choose 1}^{5}\end{matrix}}$ 1,0240.00059801,671.2 : 1 6,1440.0035881277.7 : 1 AABCD ${\begin{matrix}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 7,6800.0044852222.0 : 1 23,0400.013455673.3 : 1 ABCD* ${\begin{matrix}{r-3 \choose 4}{4 \choose 1}^{4}{51-4r \choose 1}\end{matrix}}$ 24,3200.014203169.4 : 1 57,6000.033638928.7 : 1 AABBC ${\begin{matrix}{r-3 \choose 2}{4 \choose 2}^{2}{r-4 \choose 1}{4 \choose 1}\end{matrix}}$ 4,3200.0025229395.4 : 1 8,6400.0050458197.2 : 1 AABC* ${\begin{matrix}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 2}{4 \choose 1}^{2}{51-4r \choose 1}\end{matrix}}$ 54,7200.031956930.3 : 1 86,4000.050458318.8 : 1 AAABC ${\begin{matrix}{r-3 \choose 1}{4 \choose 3}{r-4 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 1,9200.0011213890.8 : 1 3,8400.0022426444.9 : 1 ABC** ${\begin{matrix}{r-3 \choose 3}{4 \choose 1}^{3}{51-4r \choose 2}\end{matrix}}$ 109,4400.063913914.6 : 1 134,4000.078490711.7 : 1 xABCD ${\begin{matrix}{3 \choose 1}{3 \choose 1}{r-3 \choose 4}{4 \choose 1}^{4}\end{matrix}}$ 11,5200.0067278147.6 : 1 34,5600.020183348.5 : 1 xABC* ${\begin{matrix}{3 \choose 1}{3 \choose 1}{r-3 \choose 3}{4 \choose 1}^{3}{51-4r \choose 1}\end{matrix}}$ 109,4400.063913914.6 : 1 172,8000.10091678.9 : 1 xAABC ${\begin{matrix}{3 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 25,9200.015137565.1 : 1 51,8400.030275032.0 : 1 xxABC ${\begin{matrix}{3 \choose 1}{3 \choose 2}{r-3 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 5,7600.0033639296.3 : 1 11,5200.0067278147.6 : 1 xyABC ${\begin{matrix}{3 \choose 2}{3 \choose 1}^{2}{r-3 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 17,2800.010091798.1 : 1 34,5600.020183348.5 : 1 xxxAB ${\begin{matrix}{3 \choose 1}{3 \choose 3}{r-3 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 4800.00028033,566.3 : 1 7200.00042052,377.2 : 1 xxyAB ${\begin{matrix}{3 \choose 1}{3 \choose 2}{2 \choose 1}{3 \choose 1}{r-3 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 8,6400.0050458197.2 : 1 12,9600.0075687131.1 : 1 xxAAB ${\begin{matrix}{3 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1}\end{matrix}}$ 4,3200.0025229395.4 : 1 6,4800.0037844263.2 : 1 xxAB* ${\begin{matrix}{3 \choose 1}{3 \choose 2}{r-3 \choose 2}{4 \choose 1}^{2}{51-4r \choose 1}\end{matrix}}$ 27,3600.015978561.6 : 1 32,4000.018921951.8 : 1 xyAAB ${\begin{matrix}{3 \choose 2}{3 \choose 1}^{2}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1}\end{matrix}}$ 12,9600.0075687131.1 : 1 19,4400.011353187.1 : 1 xyAB* ${\begin{matrix}{3 \choose 2}{3 \choose 1}^{2}{r-3 \choose 2}{4 \choose 1}^{2}{51-4r \choose 1}\end{matrix}}$ 82,0800.047935419.9 : 1 97,2000.056765616.6 : 1 xAAAB ${\begin{matrix}{3 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 2,8800.0016819593.6 : 1 4,3200.0025229395.4 : 1 xAABB ${\begin{matrix}{3 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 2}^{2}\end{matrix}}$ 3,2400.0018922527.5 : 1 4,8600.0028383351.3 : 1 xAAB* ${\begin{matrix}{3 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}{51-4r \choose 1}\end{matrix}}$ 82,0800.047935419.9 : 1 97,2000.056765616.6 : 1 xAB** ${\begin{matrix}{3 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}{51-4r \choose 2}\end{matrix}}$ 246,2400.14380626.0 : 1 226,8000.13245316.5 : 1 xyzAB ${\begin{matrix}{3 \choose 3}{3 \choose 1}^{3}{r-3 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 4,3200.0025229395.4 : 1 6,4800.0037844263.2 : 1 Total 847,9440.49520651.0 : 1 1,134,2040.66238470.5 : 1 Miss low xxxyA ${\begin{matrix}{3 \choose 1}{3 \choose 3}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 3600.00021024,755.4 : 1 4320.00025233,962.7 : 1 xxyyA ${\begin{matrix}{3 \choose 2}{3 \choose 2}^{2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 5400.00031543,169.9 : 1 6480.00037842,641.4 : 1 xxyAA ${\begin{matrix}{3 \choose 1}{3 \choose 2}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}\end{matrix}}$ 1,6200.00094611,056.0 : 1 1,9440.0011353879.8 : 1 xxyA* ${\begin{matrix}{3 \choose 1}{3 \choose 2}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1}{51-4r \choose 1}\end{matrix}}$ 20,5200.011983982.4 : 1 19,4400.011353187.1 : 1 xyAAA ${\begin{matrix}{3 \choose 2}{3 \choose 1}^{2}{r-3 \choose 1}{4 \choose 3}\end{matrix}}$ 5400.00031543,169.9 : 1 6480.00037842,641.4 : 1 xyAA* ${\begin{matrix}{3 \choose 2}{3 \choose 1}^{2}{r-3 \choose 1}{4 \choose 2}{51-4r \choose 1}\end{matrix}}$ 15,3900.0089879110.3 : 1 14,5800.0085148116.4 : 1 xyA** ${\begin{matrix}{3 \choose 2}{3 \choose 1}^{2}{r-3 \choose 1}{4 \choose 1}{51-4r \choose 2}\end{matrix}}$ 92,3400.053927317.5 : 1 68,0400.039735924.2 : 1 xxxyz ${\begin{matrix}{3 \choose 1}{3 \choose 3}{2 \choose 2}{3 \choose 1}^{2}\end{matrix}}$ 270.000015863,417.7 : 1 270.000015863,417.7 : 1 xxyyz ${\begin{matrix}{3 \choose 2}{3 \choose 2}^{2}{1 \choose 1}{3 \choose 1}\end{matrix}}$ 810.000047321,138.6 : 1 810.000047321,138.6 : 1 xxyzA ${\begin{matrix}{3 \choose 1}{3 \choose 2}{2 \choose 2}{3 \choose 1}^{2}\end{matrix}}$ 1,6200.00094611,056.0 : 1 1,9440.0011353879.8 : 1 xxyz* ${\begin{matrix}{3 \choose 1}{3 \choose 2}{2 \choose 2}{3 \choose 1}^{2}{51-4r \choose 1}\end{matrix}}$ 1,5390.00089881,111.6 : 1 1,2150.00070961,408.3 : 1 xyzAA ${\begin{matrix}{3 \choose 3}{3 \choose 1}^{3}{r-3 \choose 1}{4 \choose 2}\end{matrix}}$ 8100.00047302,113.0 : 1 9720.00056771,760.6 : 1 xyzA* ${\begin{matrix}{3 \choose 3}{3 \choose 1}^{3}{r-3 \choose 1}{4 \choose 1}{51-4r \choose 1}\end{matrix}}$ 10,2600.0059919165.9 : 1 9,7200.0056766175.2 : 1 xyz** ${\begin{matrix}{3 \choose 3}{3 \choose 1}^{3}{51-4r \choose 2}\end{matrix}}$ 4,6170.0026964369.9 : 1 2,8350.0016557603.0 : 1 Total 150,2640.087755410.4 : 1 122,5260.071556213.0 : 1 No low AAAAB ${\begin{matrix}{r-3 \choose 1}{4 \choose 4}{r-4 \choose 1}{4 \choose 1}\end{matrix}}$ 800.000046721,402.8 : 1 1200.000070114,268.2 : 1 AAAA* ${\begin{matrix}{r-3 \choose 1}{4 \choose 4}{51-4r \choose 1}\end{matrix}}$ 950.000055518,023.3 : 1 900.000052619,024.6 : 1 AAABB ${\begin{matrix}{r-3 \choose 1}{4 \choose 3}{r-4 \choose 1}{4 \choose 2}\end{matrix}}$ 4800.00028033,566.3 : 1 7200.00042052,377.2 : 1 AAAB* ${\begin{matrix}{r-3 \choose 1}{4 \choose 3}{r-4 \choose 1}{4 \choose 1}{51-4r \choose 1}\end{matrix}}$ 6,0800.0035508280.6 : 1 7,2000.0042049236.8 : 1 AABB* ${\begin{matrix}{r-3 \choose 2}{4 \choose 2}^{2}{51-4r \choose 1}\end{matrix}}$ 6,8400.0039946249.3 : 1 8,1000.0047305210.4 : 1 AAA** ${\begin{matrix}{r-3 \choose 1}{4 \choose 3}{51-4r \choose 2}\end{matrix}}$ 3,4200.0019973499.7 : 1 2,5200.0014717678.5 : 1 AAB** ${\begin{matrix}{r-3 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}{51-4r \choose 2}\end{matrix}}$ 82,0800.047935419.9 : 1 75,6000.044151021.6 : 1 xxxyy ${\begin{matrix}{3 \choose 1}{3 \choose 3}{2 \choose 1}{3 \choose 2}\end{matrix}}$ 180.000010595,127.0 : 1 180.000010595,127.0 : 1 xxxy* ${\begin{matrix}{3 \choose 1}{3 \choose 3}{2 \choose 1}{3 \choose 1}{51-4r \choose 1}\end{matrix}}$ 3420.00019975,005.7 : 1 2700.00015776,340.9 : 1 xxxAA ${\begin{matrix}{3 \choose 1}{3 \choose 3}{r-3 \choose 1}{4 \choose 2}\end{matrix}}$ 900.000052619,024.6 : 1 1080.000063115,853.7 : 1 xxxA* ${\begin{matrix}{3 \choose 1}{3 \choose 3}{r-3 \choose 1}{4 \choose 1}{51-4r \choose 1}\end{matrix}}$ 1,1400.00066581,501.0 : 1 1,0800.00063071,584.5 : 1 xxx** ${\begin{matrix}{3 \choose 1}{3 \choose 3}{51-4r \choose 2}\end{matrix}}$ 5130.00029963,336.8 : 1 3150.00018405,434.9 : 1 xxyy* ${\begin{matrix}{3 \choose 2}{3 \choose 2}^{2}{51-4r \choose 1}\end{matrix}}$ 5130.00029963,336.8 : 1 4050.00023654,226.9 : 1 xxy** ${\begin{matrix}{3 \choose 1}{3 \choose 2}{2 \choose 1}{3 \choose 1}{51-4r \choose 2}\end{matrix}}$ 9,2340.0053927184.4 : 1 5,6700.0033113301.0 : 1 xxAAA ${\begin{matrix}{3 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 3}\end{matrix}}$ 1800.00010519,511.8 : 1 2160.00012617,926.3 : 1 xxAA* ${\begin{matrix}{3 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 2}{51-4r \choose 1}\end{matrix}}$ 5,1300.0029960332.8 : 1 4,8600.0028383351.3 : 1 xxA** ${\begin{matrix}{3 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 1}{51-4r \choose 2}\end{matrix}}$ 30,7800.017975854.6 : 1 22,6800.013245374.5 : 1 xAAAA ${\begin{matrix}{3 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 4}\end{matrix}}$ 450.000026338,050.2 : 1 540.000031531,708.3 : 1 xAAA* ${\begin{matrix}{3 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 3}{51-4r \choose 1}\end{matrix}}$ 3,4200.0019973499.7 : 1 3,2400.0018922527.5 : 1 xAA** ${\begin{matrix}{3 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}{51-4r \choose 2}\end{matrix}}$ 46,1700.026963736.1 : 1 34,0200.019868049.3 : 1  ??*** ${\begin{matrix}{4r-3 \choose 2}{51-4r \choose 3}\end{matrix}}$ 393,4140.22975713.4 : 1 240,2400.14030226.1 : 1  ?**** ${\begin{matrix}{4r-3 \choose 1}{51-4r \choose 4}\end{matrix}}$ 112,4040.065644914.2 : 1 45,0450.026306737.0 : 1 ***** ${\begin{matrix}{51-4r \choose 5}\end{matrix}}$ 11,6280.0067909146.3 : 1 3,0030.0017538569.2 : 1 Total 714,0960.41703811.4 : 1 455,5740.26605912.8 : 1

## Derivations for making low hands with a low pair and two low ranks

Starting hands with a low pair and two low ranks (xxyz) can make a low hand when either at least three other low ranks beside x, y and z appear on the board; or when two other low ranks beside x, y and z and at least one of the low ranks x, y or z appears. With no high cards in the hand, there are $52-4r-0=52-4r$ high cards and $4r-4+0=4r-4$ low cards left in the deck. The following tables show the derivations when holding a low pair and two low ranks.
    Derivations for low hand shape xxyz (a low pair and two low ranks) on the flop Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Make low ABC ${\begin{matrix}{r-3 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 6400.037002826.0 : 1 1,2800.074005612.5 : 1 xAB ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-3 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 3200.018501453.1 : 1 4800.027752135.0 : 1 yAB ${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-3 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 9600.055504217.0 : 1 1,4400.083256211.0 : 1 Total 1,9200.11100838.0 : 1 3,2000.18501394.4 : 1 Miss low xyA ${\begin{matrix}{1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 2400.013876071.1 : 1 2880.016651295.1 : 1 yzA ${\begin{matrix}{2 \choose 2}{3 \choose 1}^{2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 1800.010407059.1 : 1 2160.012488479.1 : 1 xyz ${\begin{matrix}{1 \choose 1}{2 \choose 1}{2 \choose 2}{3 \choose 1}^{2}\end{matrix}}$ 180.0010407959.9 : 1 180.0010407959.9 : 1 Total 4380.025323838.5 : 1 5220.030180432.1 : 1 No low AAA ${\begin{matrix}{r-3 \choose 1}{4 \choose 3}\end{matrix}}$ 200.0011563863.8 : 1 240.0013876719.7 : 1 AAB ${\begin{matrix}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1}\end{matrix}}$ 4800.027752135.0 : 1 7200.041628123.0 : 1 xxy ${\begin{matrix}{1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 1}\end{matrix}}$ 60.00034692,881.7 : 1 60.00034692,881.7 : 1 xxA ${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 200.0011563863.8 : 1 240.0013876719.7 : 1 xyy ${\begin{matrix}{1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 2}\end{matrix}}$ 120.00069381,440.3 : 1 120.00069381,440.3 : 1 xAA ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-3 \choose 1}{4 \choose 2}\end{matrix}}$ 600.0034690287.3 : 1 720.0041628239.2 : 1 yyy ${\begin{matrix}{2 \choose 1}{3 \choose 3}\end{matrix}}$ 20.00011568,647.0 : 1 20.00011568,647.0 : 1 yyA ${\begin{matrix}{2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 1200.0069380143.1 : 1 1440.0083256119.1 : 1 yAA ${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}\end{matrix}}$ 1800.010407059.1 : 1 2160.012488479.1 : 1 yyz ${\begin{matrix}{2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}\end{matrix}}$ 180.0010407959.9 : 1 180.0010407959.9 : 1  ??* ${\begin{matrix}{4r-4 \choose 2}{52-4r \choose 1}\end{matrix}}$ 7,5600.43709531.3 : 1 7,9360.45883441.2 : 1  ?** ${\begin{matrix}{4r-4 \choose 1}{52-4r \choose 2}\end{matrix}}$ 5,3200.30758562.3 : 1 3,8400.22201673.5 : 1 *** ${\begin{matrix}{52-4r \choose 3}\end{matrix}}$ 1,1400.065911214.2 : 1 5600.032377429.9 : 1 Total 14,9380.86366790.2 : 1 13,5740.78480570.3 : 1
    Derivations for low hand shape xxyz (a low pair and two low ranks) on the turn Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Make low ABCD ${\begin{matrix}{r-3 \choose 4}{4 \choose 1}^{4}\end{matrix}}$ 1,2800.0065783151.0 : 1 3,8400.019734849.7 : 1 AABC ${\begin{matrix}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 2,8800.014801166.6 : 1 5,7600.029602232.8 : 1 ABC* ${\begin{matrix}{r-3 \choose 3}{4 \choose 1}^{3}{52-4r \choose 1}\end{matrix}}$ 12,8000.065782714.2 : 1 20,4800.10525238.5 : 1 xABC ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-3 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 1,2800.0065783151.0 : 1 2,5600.013156575.0 : 1 yABC ${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-3 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 6,7200.034535928.0 : 1 7,6800.039469624.3 : 1 xxAB ${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-3 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 1600.00082231,215.1 : 1 2400.0012334809.8 : 1 xAAB ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1}\end{matrix}}$ 9600.0049337201.7 : 1 1,4400.0074006134.1 : 1 xyAB ${\begin{matrix}{1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 1}{r-3 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 1,9200.0098674100.3 : 1 2,8800.014801166.6 : 1 xAB* ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-3 \choose 2}{4 \choose 1}^{2}{52-4r \choose 1}\end{matrix}}$ 6,4000.032891429.4 : 1 7,6800.039469624.3 : 1 yyAB ${\begin{matrix}{2 \choose 1}{3 \choose 2}{r-3 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 9600.0049337201.7 : 1 1,4400.0074006134.1 : 1 yAAB ${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1}\end{matrix}}$ 2,8800.014801166.6 : 1 4,3200.022201744.0 : 1 yAB* ${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-3 \choose 2}{4 \choose 1}^{2}{52-4r \choose 1}\end{matrix}}$ 19,2000.09867419.1 : 1 23,0400.11840897.4 : 1 yzAB ${\begin{matrix}{2 \choose 2}{3 \choose 1}^{2}{r-3 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 1,4400.0074006134.1 : 1 2,1600.011100889.1 : 1 Total 56,0000.28779942.5 : 1 83,5200.42923221.3 : 1 Miss low xxyA ${\begin{matrix}{1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 1200.00061671,620.5 : 1 1440.00074011,350.3 : 1 xyyA ${\begin{matrix}{1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 2400.0012334809.8 : 1 2880.0014801674.6 : 1 xyAA ${\begin{matrix}{1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}\end{matrix}}$ 3600.0018501539.5 : 1 4320.0022202449.4 : 1 xyA* ${\begin{matrix}{1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 4,8000.024668539.5 : 1 4,6080.023681841.2 : 1 yyzA ${\begin{matrix}{2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 3600.0018501539.5 : 1 4320.0022202449.4 : 1 yzAA ${\begin{matrix}{2 \choose 2}{3 \choose 1}^{2}{r-3 \choose 1}{4 \choose 2}\end{matrix}}$ 2700.0013876719.7 : 1 3240.0016651599.6 : 1 yzA* ${\begin{matrix}{2 \choose 2}{3 \choose 1}^{2}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 3,6000.018501453.1 : 1 3,4560.017761355.3 : 1 xxyz ${\begin{matrix}{1 \choose 1}{2 \choose 2}{2 \choose 2}{3 \choose 1}^{2}\end{matrix}}$ 90.000046321,619.0 : 1 90.000046321,619.0 : 1 xyyz ${\begin{matrix}{1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}\end{matrix}}$ 360.00018505,404.0 : 1 360.00018505,404.0 : 1 xyzA ${\begin{matrix}{1 \choose 1}{2 \choose 1}{2 \choose 2}{3 \choose 1}^{2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 3600.0018501539.5 : 1 4320.0022202449.4 : 1 xyz* ${\begin{matrix}{1 \choose 1}{2 \choose 1}{2 \choose 2}{3 \choose 1}^{2}{51-4r \choose 1}\end{matrix}}$ 3600.0018501539.5 : 1 2880.0014801674.6 : 1 Total 10,5150.054039517.5 : 1 10,4490.053700317.6 : 1 No low AAAA ${\begin{matrix}{r-3 \choose 1}{4 \choose 4}\end{matrix}}$ 50.000025738,915.0 : 1 60.000030832,429.0 : 1 AAAB ${\begin{matrix}{r-3 \choose 1}{4 \choose 3}{r-4 \choose 1}{4 \choose 1}\end{matrix}}$ 3200.0016446607.1 : 1 4800.0024669404.4 : 1 AABB ${\begin{matrix}{r-3 \choose 2}{4 \choose 2}^{2}\end{matrix}}$ 3600.0018501539.5 : 1 5400.0027752359.3 : 1 AAA* ${\begin{matrix}{r-3 \choose 1}{4 \choose 3}{52-4r \choose 1}\end{matrix}}$ 4000.0020557485.5 : 1 3840.0019735505.7 : 1 AAB* ${\begin{matrix}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 9,6000.049337019.3 : 1 11,5200.059204415.9 : 1 xxyy ${\begin{matrix}{1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 2}\end{matrix}}$ 60.000030832,429.0 : 1 60.000030832,429.0 : 1 xxy* ${\begin{matrix}{1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 1}{52-4r \choose 1}\end{matrix}}$ 1200.00061671,620.5 : 1 960.00049342,025.9 : 1 xxAA ${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-3 \choose 1}{4 \choose 2}\end{matrix}}$ 300.00015426,485.0 : 1 360.00018505,404.0 : 1 xxA* ${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 4000.0020557485.5 : 1 3840.0019735505.7 : 1 xyyy ${\begin{matrix}{1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 3}\end{matrix}}$ 40.000020648,644.0 : 1 40.000020648,644.0 : 1 xyy* ${\begin{matrix}{1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 2}{52-4r \choose 1}\end{matrix}}$ 2400.0012334809.8 : 1 1920.00098671,012.4 : 1 xAAA ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-3 \choose 1}{4 \choose 3}\end{matrix}}$ 400.00020564,863.5 : 1 480.00024674,052.8 : 1 xAA* ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-3 \choose 1}{4 \choose 2}{52-4r \choose 1}\end{matrix}}$ 1,2000.0061671161.2 : 1 1,1520.0059204167.9 : 1 yyyA ${\begin{matrix}{2 \choose 1}{3 \choose 3}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 400.00020564,863.5 : 1 480.00024674,052.8 : 1 yyy* ${\begin{matrix}{2 \choose 1}{3 \choose 3}{52-4r \choose 1}\end{matrix}}$ 400.00020564,863.5 : 1 320.00016456,079.6 : 1 yyAA ${\begin{matrix}{2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 2}\end{matrix}}$ 1800.00092511,080.0 : 1 2160.0011101899.8 : 1 yyA* ${\begin{matrix}{2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 2,4000.012334380.1 : 1 2,3040.011840983.5 : 1 yAAA ${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 3}\end{matrix}}$ 1200.00061671,620.5 : 1 1440.00074011,350.3 : 1 yAA* ${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}{52-4r \choose 1}\end{matrix}}$ 3,6000.018501453.1 : 1 3,4560.017761355.3 : 1 yyyz ${\begin{matrix}{2 \choose 1}{3 \choose 3}{1 \choose 1}{3 \choose 1}\end{matrix}}$ 60.000030832,429.0 : 1 60.000030832,429.0 : 1 yyzz ${\begin{matrix}{2 \choose 2}{3 \choose 2}^{2}\end{matrix}}$ 90.000046321,619.0 : 1 90.000046321,619.0 : 1 yyz* ${\begin{matrix}{2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{52-4r \choose 1}\end{matrix}}$ 3600.0018501539.5 : 1 2880.0014801674.6 : 1  ??** ${\begin{matrix}{4r-4 \choose 2}{52-4r \choose 2}\end{matrix}}$ 71,8200.36910271.7 : 1 59,5200.30588962.3 : 1  ?*** ${\begin{matrix}{4r-4 \choose 1}{52-4r \choose 3}\end{matrix}}$ 31,9200.16404565.1 : 1 17,9200.09209589.9 : 1 **** ${\begin{matrix}{52-4r \choose 4}\end{matrix}}$ 4,8450.024899839.2 : 1 1,8200.0093535105.9 : 1 Total 128,0650.65816120.5 : 1 100,6110.51706750.9 : 1
    Derivations for low hand shape xxyz (a low pair and two low ranks) on the river Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Make low ABCDE ${\begin{matrix}{r-3 \choose 5}{4 \choose 1}^{5}\end{matrix}}$ 1,0240.00059801,671.2 : 1 6,1440.0035881277.7 : 1 AABCD ${\begin{matrix}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 7,6800.004485230.9 : 1 23,0400.013455673.3 : 1 ABCD* ${\begin{matrix}{r-3 \choose 4}{4 \choose 1}^{4}{52-4r \choose 1}\end{matrix}}$ 25,6000.0149506222.0 : 1 61,4400.035881526.9 : 1 AABBC ${\begin{matrix}{r-3 \choose 2}{4 \choose 2}^{2}{r-4 \choose 1}{4 \choose 1}\end{matrix}}$ 4,3200.0025229395.4 : 1 8,6400.0050458197.2 : 1 AABC* ${\begin{matrix}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 2}{4 \choose 1}^{2}{52-4r \choose 1}\end{matrix}}$ 57,6000.033638928.7 : 1 92,1600.053822217.6 : 1 AAABC ${\begin{matrix}{r-3 \choose 1}{4 \choose 3}{r-4 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 1,9200.0011213890.8 : 1 3,8400.0022426444.9 : 1 ABC** ${\begin{matrix}{r-3 \choose 3}{4 \choose 1}^{3}{52-4r \choose 2}\end{matrix}}$ 121,6000.071015413.1 : 1 153,6000.089703710.1 : 1 xABCD ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-3 \choose 4}{4 \choose 1}^{4}\end{matrix}}$ 2,5600.0014951667.9 : 1 7,6800.0044852222.0 : 1 xABC* ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-3 \choose 3}{4 \choose 1}^{3}{52-4r \choose 1}\end{matrix}}$ 25,6000.014950665.9 : 1 40,9600.023921040.8 : 1 xAABC ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 5,7600.0033639296.3 : 1 11,5200.0067278147.6 : 1 xxABC ${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-3 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 6400.00037382,674.5 : 1 1,2800.00074751,336.7 : 1 xyABC ${\begin{matrix}{1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 1}{r-3 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 7,6800.0044852222.0 : 1 15,3600.0089704110.5 : 1 yyABC ${\begin{matrix}{2 \choose 1}{3 \choose 2}{r-3 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 3,8400.0022426444.9 : 1 7,6800.0044852222.0 : 1 yABCD ${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-3 \choose 4}{4 \choose 1}^{4}\end{matrix}}$ 7,6800.0044852222.0 : 1 23,0400.013455673.3 : 1 yABC* ${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-3 \choose 3}{4 \choose 1}^{3}{52-4r \choose 1}\end{matrix}}$ 76,8000.044851821.3 : 1 122,8800.071763012.9 : 1 yAABC ${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 17,2800.010091798.1 : 1 34,5600.020183348.5 : 1 xxyAB ${\begin{matrix}{1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 1}{r-3 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 9600.00056061,782.7 : 1 1,4400.00084101,188.1 : 1 xxAAB ${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1}\end{matrix}}$ 4800.00028033,566.3 : 1 7200.00042052,377.2 : 1 xxAB* ${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-3 \choose 2}{4 \choose 1}^{2}{52-4r \choose 1}\end{matrix}}$ 3,2000.0018688534.1 : 1 3,8400.0022426444.9 : 1 xyyAB ${\begin{matrix}{1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1}\end{matrix}}$ 1,9200.0011213890.8 : 1 2,8800.0016819593.6 : 1 xyAAB ${\begin{matrix}{1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1}\end{matrix}}$ 5,7600.0033639296.3 : 1 8,6400.0050458197.2 : 1 xyAB* ${\begin{matrix}{1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 1}{r-3 \choose 2}{4 \choose 1}^{2}{52-4r \choose 1}\end{matrix}}$ 38,4000.022425943.6 : 1 46,0800.026911136.2 : 1 xAAAB ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 6400.00037382,674.5 : 1 9600.00056061,782.7 : 1 xAABB ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 2}^{2}\end{matrix}}$ 7200.00042052,377.2 : 1 1,0800.00063071,584.5 : 1 xAAB* ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 19,2000.011213088.2 : 1 23,0400.013455673.3 : 1 xAB** ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}{52-4r \choose 2}\end{matrix}}$ 60,8000.035507727.2 : 1 57,6000.033638928.7 : 1 yyyAB ${\begin{matrix}{2 \choose 1}{3 \choose 3}{r-3 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 3200.00018695,350.0 : 1 4800.00028033,566.3 : 1 yyAAB ${\begin{matrix}{2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1}\end{matrix}}$ 2,8800.0016819593.6 : 1 4,3200.0025229395.4 : 1 yyAB* ${\begin{matrix}{2 \choose 1}{3 \choose 2}{r-3 \choose 2}{4 \choose 1}^{2}{52-4r \choose 1}\end{matrix}}$ 19,2000.011213088.2 : 1 23,0400.013455673.3 : 1 yAAAB ${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 1,9200.0011213890.8 : 1 2,8800.0016819593.6 : 1 yAABB ${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 2}^{2}\end{matrix}}$ 1,6200.00094611,056.0 : 1 3,2400.0018922527.5 : 1 yAAB* ${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 57,6000.033638928.7 : 1 69,1200.040366723.8 : 1 yAB** ${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}{52-4r \choose 2}\end{matrix}}$ 182,4000.10652318.4 : 1 172,8000.10091678.9 : 1 yyzAB ${\begin{matrix}{2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{r-3 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 2,8800.0016819593.6 : 1 4,3200.0025229395.4 : 1 yzAAB ${\begin{matrix}{2 \choose 2}{3 \choose 1}^{2}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1}\end{matrix}}$ 4,3200.0025229395.4 : 1 6,4800.0037844263.2 : 1 yzABC ${\begin{matrix}{2 \choose 2}{3 \choose 1}^{2}{r-3 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 5,7600.0033639296.3 : 1 11,5200.0067278147.6 : 1 yzAB* ${\begin{matrix}{2 \choose 2}{3 \choose 1}^{2}{r-3 \choose 2}{4 \choose 1}^{2}{52-4r \choose 1}\end{matrix}}$ 28,8000.016819458.5 : 1 34,5600.020183348.5 : 1 xyzAB ${\begin{matrix}{1 \choose 1}{2 \choose 2}{2 \choose 2}{3 \choose 1}^{2}{r-3 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 2,8800.0016819593.6 : 1 4,3200.0025229395.4 : 1 Total 810,7840.47350471.1 : 1 1,097,1840.64076470.6 : 1 Miss low xxyyA ${\begin{matrix}{1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 1200.000070114,268.2 : 1 1440.000084111,890.0 : 1 xxyAA ${\begin{matrix}{1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}\end{matrix}}$ 1800.00010519,511.8 : 1 2160.00012617,926.3 : 1 xxyA* ${\begin{matrix}{1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 2,4000.0014016712.5 : 1 2,3040.0013456742.2 : 1 xyyyA ${\begin{matrix}{1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 3}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 800.000046721,402.8 : 1 960.000056117,835.5 : 1 xyyAA ${\begin{matrix}{1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 3}\end{matrix}}$ 3600.00021024,755.4 : 1 4320.00025233,962.7 : 1 xyyA* ${\begin{matrix}{1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 2}{52-4r \choose 1}\end{matrix}}$ 4,8000.0028032355.7 : 1 4,6080.0026911370.6 : 1 xyAAA ${\begin{matrix}{1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 3}\end{matrix}}$ 2400.00014027,133.6 : 1 2880.00016825,944.5 : 1 xyAA* ${\begin{matrix}{1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}{52-4r \choose 1}\end{matrix}}$ 7,2000.0042049236.8 : 1 6,9120.0040367246.7 : 1 xyA** ${\begin{matrix}{1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 2}\end{matrix}}$ 45,6000.026630836.6 : 1 34,5600.020183348.5 : 1 yyyzA ${\begin{matrix}{2 \choose 1}{3 \choose 3}{1 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 1200.000070114,268.2 : 1 1440.000084111,890.0 : 1 yyzzA ${\begin{matrix}{2 \choose 2}{3 \choose 2}^{2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 1800.00010519,511.8 : 1 2160.00012617,926.3 : 1 yyzAA ${\begin{matrix}{2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}\end{matrix}}$ 5400.00031543,169.9 : 1 6480.00037842,641.4 : 1 yyzA* ${\begin{matrix}{2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 7,2000.0042049236.8 : 1 6,9120.0040367246.7 : 1 yzAAA ${\begin{matrix}{2 \choose 2}{3 \choose 1}^{2}{r-3 \choose 1}{4 \choose 3}\end{matrix}}$ 1800.00010519,511.8 : 1 2160.00012617,926.3 : 1 yzAA* ${\begin{matrix}{2 \choose 2}{3 \choose 1}^{2}{r-3 \choose 1}{4 \choose 2}{52-4r \choose 1}\end{matrix}}$ 5,4000.0031536316.1 : 1 5,1840.0030275329.3 : 1 yzA** ${\begin{matrix}{2 \choose 2}{3 \choose 1}^{2}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 2}\end{matrix}}$ 34,2000.019973149.1 : 1 25,9200.015137565.1 : 1 xxyyz ${\begin{matrix}{1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}\end{matrix}}$ 180.000010595,127.0 : 1 180.000010595,127.0 : 1 xxyzA ${\begin{matrix}{1 \choose 1}{2 \choose 2}{2 \choose 2}{3 \choose 1}^{2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 1800.00010519,511.8 : 1 2160.00012617,926.3 : 1 xxyz* ${\begin{matrix}{1 \choose 1}{2 \choose 2}{2 \choose 2}{3 \choose 1}^{2}{52-4r \choose 1}\end{matrix}}$ 1800.00010519,511.8 : 1 1440.000084111,890.0 : 1 xyyyz ${\begin{matrix}{1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 3}{1 \choose 1}{3 \choose 1}\end{matrix}}$ 120.000007014,2691 : 1 120.000007014,2691 : 1 xyyzz ${\begin{matrix}{1 \choose 1}{2 \choose 1}{2 \choose 2}{3 \choose 2}^{2}\end{matrix}}$ 180.000010595,127.0 : 1 180.000010595,127.0 : 1 xyyzA ${\begin{matrix}{1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$ 7200.00042052,377.2 : 1 8640.00050461,980.8 : 1 xyyz* ${\begin{matrix}{1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{52-4r \choose 1}\end{matrix}}$ 7200.00042052,377.2 : 1 5760.00033642,971.8 : 1 xyzAA ${\begin{matrix}{1 \choose 1}{2 \choose 1}{2 \choose 2}{3 \choose 1}^{2}{r-3 \choose 1}{4 \choose 2}\end{matrix}}$ 5400.00031543,169.9 : 1 6480.00037842,641.4 : 1 xyzA* ${\begin{matrix}{1 \choose 1}{2 \choose 1}{2 \choose 2}{3 \choose 1}^{2}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 7,2000.0042049236.8 : 1 6,9120.0040367246.7 : 1 xyz** ${\begin{matrix}{1 \choose 1}{2 \choose 1}{2 \choose 2}{3 \choose 1}^{2}{52-4r \choose 2}\end{matrix}}$ 3,4200.0019973499.7 : 1 2,1600.0012615791.7 : 1 Total 121,8080.071136913.1 : 1 100,3680.058615816.1 : 1 No low AAAAB ${\begin{matrix}{r-3 \choose 1}{4 \choose 4}{r-4 \choose 1}{4 \choose 1}\end{matrix}}$ 800.000046721,402.8 : 1 1200.000070115,019.2 : 1 AAAA* ${\begin{matrix}{r-3 \choose 1}{4 \choose 4}{52-4r \choose 1}\end{matrix}}$ 1000.000058417,122.0 : 1 960.000056117,835.5 : 1 AAABB ${\begin{matrix}{r-3 \choose 1}{4 \choose 3}{r-4 \choose 1}{4 \choose 2}\end{matrix}}$ 4800.00028033,566.3 : 1 7200.00042052,377.2 : 1 AAAB* ${\begin{matrix}{r-3 \choose 1}{4 \choose 3}{r-4 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 6,4000.0037377266.5 : 1 7,6800.004485230.9 : 1 AABB* ${\begin{matrix}{r-3 \choose 2}{4 \choose 2}^{2}{52-4r \choose 1}\end{matrix}}$ 7,2000.0042049236.8 : 1 8,6400.0050458197.2 : 1 AAA** ${\begin{matrix}{r-3 \choose 1}{4 \choose 3}{52-4r \choose 2}\end{matrix}}$ 3,8000.0022192449.6 : 1 2,8800.0016819593.6 : 1 AAB** ${\begin{matrix}{r-3 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 2}\end{matrix}}$ 91,2000.053261617.8 : 1 86,4000.050458318.8 : 1 xxyyy ${\begin{matrix}{1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 3}\end{matrix}}$ 20.0000012856151.0 : 1 20.0000012856151.0 : 1 xxyy* ${\begin{matrix}{1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 2}{52-4r \choose 1}\end{matrix}}$ 1200.000070114,268.2 : 1 960.000056117,835.5 : 1 xxy** ${\begin{matrix}{1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 1}{52-4r \choose 2}\end{matrix}}$ 1,1400.00066581,501.0 : 1 7200.00042052,377.2 : 1 xxAAA ${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-3 \choose 1}{4 \choose 3}\end{matrix}}$ 200.000011785,614.2 : 1 240.000014071,345.0 : 1 xxAA* ${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-3 \choose 1}{4 \choose 2}{52-4r \choose 1}\end{matrix}}$ 6000.00035042,852.8 : 1 5760.00033642,971.8 : 1 xxA** ${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 2}\end{matrix}}$ 3,8000.0022192449.6 : 1 2,8800.0016819593.6 : 1 xyyy* ${\begin{matrix}{1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 3}{52-4r \choose 1}\end{matrix}}$ 800.000046721,402.8 : 1 640.000037426,753.8 : 1 xyy** ${\begin{matrix}{1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 2}{52-4r \choose 2}\end{matrix}}$ 2,2800.0013315750.0 : 1 1,4400.00084101,188.1 : 1 xAAAA ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-3 \choose 1}{4 \choose 4}\end{matrix}}$ 100.0000058171,229.4 : 1 120.0000070142,691.0 : 1 xAAA* ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-3 \choose 1}{4 \choose 3}{52-4r \choose 1}\end{matrix}}$ 8000.00046722,139.4 : 1 7680.00044852,228.6 : 1 xAA** ${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-3 \choose 1}{4 \choose 2}{52-4r \choose 2}\end{matrix}}$ 11,4000.0066577149.2 : 1 8,6400.0050458197.2 : 1 yyyAA ${\begin{matrix}{2 \choose 1}{3 \choose 3}{r-3 \choose 1}{4 \choose 2}\end{matrix}}$ 600.000035028,537.4 : 1 720.000042023,781.0 : 1 yyyA* ${\begin{matrix}{2 \choose 1}{3 \choose 3}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 8000.00046722,139.4 : 1 7680.00044852,228.6 : 1 yyy** ${\begin{matrix}{2 \choose 1}{3 \choose 3}{52-4r \choose 2}\end{matrix}}$ 3800.00022194,505.1 : 1 2400.00014027,133.6 : 1 yyAAA ${\begin{matrix}{2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 3}\end{matrix}}$ 1200.000070114,268.2 : 1 1440.000084111,890.0 : 1 yyAA* ${\begin{matrix}{2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 2}{52-4r \choose 1}\end{matrix}}$ 3,6000.0021024474.6 : 1 3,4560.0020183494.5 : 1 yyA** ${\begin{matrix}{2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 2}\end{matrix}}$ 22,8000.013315474.1 : 1 17,2800.010091798.1 : 1 yAAAA ${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 4}\end{matrix}}$ 300.000017557,075.8 : 1 360.000021047,563.0 : 1 yAAA* ${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 3}{52-4r \choose 1}\end{matrix}}$ 2,4000.0014016712.5 : 1 2,3040.0013456742.2 : 1 yAA** ${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}{52-4r \choose 2}\end{matrix}}$ 34,2000.019973149.1 : 1 25,9200.015137565.1 : 1 yyyzz ${\begin{matrix}{2 \choose 1}{3 \choose 3}{1 \choose 1}{3 \choose 2}\end{matrix}}$ 60.0000035285,383.0 : 1 60.0000035285,383.0 : 1 yyyz* ${\begin{matrix}{2 \choose 1}{3 \choose 3}{1 \choose 1}{3 \choose 1}{52-4r \choose 1}\end{matrix}}$ 1200.000070114,268.2 : 1 960.000056117,835.5 : 1 yyzz* ${\begin{matrix}{2 \choose 2}{3 \choose 2}^{2}{52-4r \choose 1}\end{matrix}}$ 1800.00010519,511.8 : 1 1440.000084111,890.0 : 1 yyz** ${\begin{matrix}{2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{52-4r \choose 2}\end{matrix}}$ 3,4200.0019973499.7 : 1 2,1600.0012615791.7 : 1  ??*** ${\begin{matrix}{4r-4 \choose 2}{52-4r \choose 3}\end{matrix}}$ 430,9200.25166093.0 : 1 277,7600.16221425.2 : 1  ?**** ${\begin{matrix}{4r-4 \choose 1}{52-4r \choose 4}\end{matrix}}$ 135,6600.079226611.6 : 1 58,2400.034012728.4 : 1 ***** ${\begin{matrix}{52-4r \choose 5}\end{matrix}}$ 15,5040.0090545109.4 : 1 4,3680.0025509391.0 : 1 Total 779,7120.45535841.2 : 1 514,7520.30061952.3 : 1

## Derivations for making low hands with four low ranks

Starting hands with four low ranks (xyzr) can make a low hand when at least three low ranks appear on the board with at least one low rank beside x, y, z or r. With no high cards in the hand, there are $52-4r-0=52-4r$ high cards and $4r-4+0=4r-4$ low cards left in the deck. The following tables show the derivations when holding four low ranks.
    Derivations for low hand shape xyzr (four low ranks) on the flop Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Make low ABC ${\begin{matrix}{r-4 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 2560.014801166.6 : 1 6400.037002826.0 : 1 xAB ${\begin{matrix}{4 \choose 1}{3 \choose 1}{r-4 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 1,1520.066605014.0 : 1 1,9200.11100838.0 : 1 xyA ${\begin{matrix}{4 \choose 2}{3 \choose 1}^{2}{r-4 \choose 1}{4 \choose 1}\end{matrix}}$ 8640.049953719.0 : 1 1,0800.062442215.0 : 1 Total 2,2720.13135996.6 : 1 3,6400.21045333.8 : 1 Miss low xyz ${\begin{matrix}{4 \choose 3}{3 \choose 1}^{3}\end{matrix}}$ 1080.0062442159.1 : 1 1080.0062442159.1 : 1 No low AAA ${\begin{matrix}{r-4 \choose 1}{4 \choose 3}\end{matrix}}$ 160.00092511,080.0 : 1 200.0011563863.8 : 1 AAB ${\begin{matrix}{r-4 \choose 1}{4 \choose 2}{r-5 \choose 1}{4 \choose 1}\end{matrix}}$ 2880.016651259.1 : 1 4800.027752135.0 : 1 xxx ${\begin{matrix}{4 \choose 1}{3 \choose 3}\end{matrix}}$ 40.00023134,323.0 : 1 40.00023134,323.0 : 1 xxy ${\begin{matrix}{4 \choose 1}{3 \choose 2}{3 \choose 1}{3 \choose 1}\end{matrix}}$ 1080.0062442159.1 : 1 1080.0062442159.1 : 1 xxA ${\begin{matrix}{4 \choose 1}{3 \choose 2}{r-4 \choose 1}{4 \choose 1}\end{matrix}}$ 1920.011100889.1 : 1 2400.013876071.1 : 1 xAA ${\begin{matrix}{4 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 2}\end{matrix}}$ 2880.016651259.1 : 1 3600.020814147.0 : 1  ??* ${\begin{matrix}{4r-4 \choose 2}{52-4r \choose 1}\end{matrix}}$ 7,5600.43709531.3 : 1 7,9360.45883441.2 : 1  ?** ${\begin{matrix}{4r-4 \choose 1}{52-4r \choose 2}\end{matrix}}$ 5,3200.30758562.3 : 1 3,8400.22201673.5 : 1 *** ${\begin{matrix}{52-4r \choose 3}\end{matrix}}$ 1,1400.065911214.2 : 1 5600.032377429.9 : 1 Total 14,9160.86239590.2 : 1 13,5480.78330250.3 : 1
    Derivations for low hand shape xyzr (four low ranks) on the turn Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Make low ABCD ${\begin{matrix}{r-4 \choose 4}{4 \choose 1}^{4}\end{matrix}}$ 2560.0013157759.1 : 1 1,2800.0065783151.0 : 1 AABC ${\begin{matrix}{r-4 \choose 1}{4 \choose 2}{r-5 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 1,1520.0059204167.9 : 1 2,8800.014801166.6 : 1 ABC* ${\begin{matrix}{r-4 \choose 3}{4 \choose 1}^{3}{52-4r \choose 1}\end{matrix}}$ 5,1200.026313137.0 : 1 10,2400.052626218.0 : 1 xABC ${\begin{matrix}{4 \choose 1}{3 \choose 1}{r-4 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 3,0720.015787962.3 : 1 7,6800.039469624.3 : 1 xxAB ${\begin{matrix}{4 \choose 1}{3 \choose 2}{r-4 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 1,1520.0059204167.9 : 1 1,9200.0098674100.3 : 1 xAAB ${\begin{matrix}{4 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 2}{r-5 \choose 1}{4 \choose 1}\end{matrix}}$ 3,4560.017761355.3 : 1 5,7600.029602232.8 : 1 xyAB ${\begin{matrix}{4 \choose 2}{3 \choose 1}^{2}{r-4 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 5,1840.026642036.5 : 1 8,6400.044403321.5 : 1 xAB* ${\begin{matrix}{4 \choose 1}{3 \choose 1}{r-4 \choose 2}{4 \choose 1}^{2}{52-4r \choose 1}\end{matrix}}$ 23,0400.11840897.4 : 1 30,7200.15787855.3 : 1 xxyA ${\begin{matrix}{4 \choose 1}{3 \choose 2}{3 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 1}\end{matrix}}$ 1,7280.0088807111.6 : 1 2,1600.011100889.1 : 1 xyAA ${\begin{matrix}{4 \choose 2}{3 \choose 1}^{2}{r-4 \choose 1}{4 \choose 2}\end{matrix}}$ 1,2960.0066605149.1 : 1 1,6200.0083256119.1 : 1 xyA* ${\begin{matrix}{4 \choose 2}{3 \choose 1}^{2}{r-4 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 17,2800.088806710.3 : 1 17,2800.088806710.3 : 1 xyzA ${\begin{matrix}{4 \choose 3}{3 \choose 1}^{3}{r-4 \choose 1}{4 \choose 1}\end{matrix}}$ 1,7280.0088807111.6 : 1 2,1600.011100889.1 : 1 Total 64,4640.33129822.0 : 1 92,3400.47456061.1 : 1 Miss low xxyz ${\begin{matrix}{4 \choose 1}{3 \choose 2}{3 \choose 2}{3 \choose 1}^{2}\end{matrix}}$ 3240.0016651599.6 : 1 3240.0016651599.6 : 1 xyzr ${\begin{matrix}{4 \choose 4}{3 \choose 1}^{4}\end{matrix}}$ 810.00041632,401.2 : 1 810.00041632,401.2 : 1 xyz* ${\begin{matrix}{4 \choose 3}{3 \choose 1}^{3}{52-4r \choose 1}\end{matrix}}$ 2,1600.011100889.1 : 1 1,7280.0088807111.6 : 1 Total 2,5650.13182274.9 : 1 2,1330.10962190.2 : 1 No low AAAA ${\begin{matrix}{r-4 \choose 1}{4 \choose 4}\end{matrix}}$ 40.000020648,644.0 : 1 50.000025738,915.0 : 1 AAAB ${\begin{matrix}{r-4 \choose 1}{4 \choose 3}{r-5 \choose 1}{4 \choose 1}\end{matrix}}$ 1920.00098671,012.4 : 1 3200.0016446607.1 : 1 AABB ${\begin{matrix}{r-4 \choose 2}{4 \choose 2}^{2}\end{matrix}}$ 2160.0011101899.8 : 1 3600.0018501539.5 : 1 AAA* ${\begin{matrix}{r-4 \choose 1}{4 \choose 3}{52-4r \choose 1}\end{matrix}}$ 3200.0016446607.1 : 1 3200.0016446607.1 : 1 AAB* ${\begin{matrix}{r-4 \choose 1}{4 \choose 2}{r-5 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 5,7600.029602232.8 : 1 7,6800.039469624.3 : 1 xxxy ${\begin{matrix}{4 \choose 1}{3 \choose 3}{3 \choose 1}{3 \choose 1}\end{matrix}}$ 360.00018505,404.0 : 1 360.00018505,404.0 : 1 xxxA ${\begin{matrix}{4 \choose 1}{3 \choose 3}{r-4 \choose 1}{4 \choose 1}\end{matrix}}$ 640.00032893,039.3 : 1 800.00041112,431.3 : 1 xxx* ${\begin{matrix}{4 \choose 1}{3 \choose 3}{52-4r \choose 1}\end{matrix}}$ 800.00041112,431.3 : 1 640.00032893,039.3 : 1 xxyy ${\begin{matrix}{4 \choose 2}{3 \choose 2}^{2}\end{matrix}}$ 540.00027753,602.3 : 1 540.00027753,602.3 : 1 xxy* ${\begin{matrix}{4 \choose 1}{3 \choose 2}{3 \choose 1}{3 \choose 1}{52-4r \choose 1}\end{matrix}}$ 2,1600.011100889.1 : 1 1,7280.0088807111.6 : 1 xxAA ${\begin{matrix}{4 \choose 1}{3 \choose 2}{r-4 \choose 1}{4 \choose 2}\end{matrix}}$ 2700.0013876719.7 : 1 2880.0014801674.6 : 1 xxA* ${\begin{matrix}{4 \choose 1}{3 \choose 2}{r-4 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 3,8400.019734849.7 : 1 3,8400.019734849.7 : 1 xAAA ${\begin{matrix}{4 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 3}\end{matrix}}$ 1920.00098671,012.4 : 1 2400.0012334809.8 : 1 xAA* ${\begin{matrix}{4 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 2}{52-4r \choose 1}\end{matrix}}$ 5,7600.029602232.8 : 1 5,7600.029602232.8 : 1  ??** ${\begin{matrix}{4r-4 \choose 2}{52-4r \choose 2}\end{matrix}}$ 71,8200.36910271.7 : 1 59,5200.30588962.3 : 1  ?*** ${\begin{matrix}{4r-4 \choose 1}{52-4r \choose 3}\end{matrix}}$ 31,9200.16404565.1 : 1 17,9200.09209589.9 : 1 **** ${\begin{matrix}{52-4r \choose 4}\end{matrix}}$ 4,8450.024899839.2 : 1 1,8200.0093535105.9 : 1 Total 127,5510.65551960.5 : 1 100,1070.51447730.9 : 1
    Derivations for low hand shape xyzr (four low ranks) on the river Hand to makeBoardDerivation 8-high qualifier (r = 8) 9-high qualifier (r = 9) CombosProbabilityOddsCombosProbabilityOdds Make low ABCDE ${\begin{matrix}{r-4 \choose 5}{4 \choose 1}^{5}\end{matrix}}$ $^{(see#1below)}$ 00.0000000∞ : 1 1,0240.00059801,671.2 : 1 AABCD ${\begin{matrix}{r-4 \choose 1}{4 \choose 2}{r-5 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 1,5360.00089701,113.8 : 1 7,6800.0044852222.0 : 1 ABCD* ${\begin{matrix}{r-4 \choose 4}{4 \choose 1}^{4}{52-4r \choose 1}\end{matrix}}$ 5,1200.0029901333.4 : 1 20,4800.011960582.6 : 1 AABBC ${\begin{matrix}{r-4 \choose 2}{4 \choose 2}^{2}{r-5 \choose 1}{4 \choose 1}\end{matrix}}$ 1,7280.0010092989.9 : 1 4,3200.0025229395.4 : 1 AABC* ${\begin{matrix}{r-4 \choose 1}{4 \choose 2}{r-5 \choose 2}{4 \choose 1}^{2}{52-4r \choose 1}\end{matrix}}$ 23,0400.013455673.3 : 1 46,0800.026911136.2 : 1 AAABC ${\begin{matrix}{r-4 \choose 1}{4 \choose 3}{r-5 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 7680.00044852,228.6 : 1 1,9200.0011213890.8 : 1 ABC** ${\begin{matrix}{r-4 \choose 3}{4 \choose 1}^{3}{52-4r \choose 2}\end{matrix}}$ 48,6400.028406234.2 : 1 76,8000.044851821.3 : 1 xABCD ${\begin{matrix}{4 \choose 1}{3 \choose 1}{r-4 \choose 4}{4 \choose 1}^{4}\end{matrix}}$ 3,0720.0017941556.4 : 1 23,0400.013455673.3 : 1 xABC* ${\begin{matrix}{4 \choose 1}{3 \choose 1}{r-4 \choose 3}{4 \choose 1}^{3}{52-4r \choose 1}\end{matrix}}$ 61,4400.035881526.9 : 1 122,8800.071763012.9 : 1 xAABC ${\begin{matrix}{4 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 2}{r-5 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 13,8240.0080733122.9 : 1 34,5600.020183348.5 : 1 xxABC ${\begin{matrix}{4 \choose 1}{3 \choose 2}{r-4 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 3,0720.0017941556.4 : 1 7,6800.0044852222.0 : 1 xyABC ${\begin{matrix}{4 \choose 2}{3 \choose 1}^{2}{r-4 \choose 3}{4 \choose 1}^{3}\end{matrix}}$ 13,8240.0080733122.9 : 1 34,5600.020183348.5 : 1 xxxAB ${\begin{matrix}{4 \choose 1}{3 \choose 3}{r-4 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 3840.00022434,458.1 : 1 6400.00037382,674.5 : 1 xxyAB ${\begin{matrix}{4 \choose 1}{3 \choose 2}{3 \choose 1}{3 \choose 1}{r-4 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 10,3680.0060550164.2 : 1 17,2800.010091798.1 : 1 xxAAB ${\begin{matrix}{4 \choose 1}{3 \choose 2}{r-4 \choose 1}{4 \choose 2}{r-5 \choose 1}{4 \choose 1}\end{matrix}}$ 3,4560.0020183494.5 : 1 5,7600.0033639296.3 : 1 xxAB* ${\begin{matrix}{4 \choose 1}{3 \choose 2}{r-4 \choose 2}{4 \choose 1}^{2}{52-4r \choose 1}\end{matrix}}$ 23,0400.013455673.3 : 1 30,7200.017940754.7 : 1 xyAAB ${\begin{matrix}{4 \choose 2}{3 \choose 1}^{2}{r-4 \choose 1}{4 \choose 2}{r-5 \choose 1}{4 \choose 1}\end{matrix}}$ 15,5520.0090825109.1 : 1 25,9200.015137565.1 : 1 xyAB* ${\begin{matrix}{4 \choose 2}{3 \choose 1}^{2}{r-4 \choose 2}{4 \choose 1}^{2}{52-4r \choose 1}\end{matrix}}$ 103,6800.060550015.5 : 1 138,2400.080733311.4 : 1 xAAAB ${\begin{matrix}{4 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-4 \choose 1}{4 \choose 1}\end{matrix}}$ 2,3040.0013456742.2 : 1 3,8400.0022426444.9 : 1 xAABB ${\begin{matrix}{4 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 2}^{2}\end{matrix}}$ 2,5920.0015137659.6 : 1 4,3200.0025229395.4 : 1 xAAB* ${\begin{matrix}{4 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 69,1200.040366723.8 : 1 92,1600.053822217.6 : 1 xAB** ${\begin{matrix}{4 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}{52-4r \choose 2}\end{matrix}}$ 218,8800.12782786.8 : 1 230,4000.13455556.4 : 1 xyzAB ${\begin{matrix}{4 \choose 3}{3 \choose 1}^{3}{r-4 \choose 2}{4 \choose 1}^{2}\end{matrix}}$ 10,3680.0060550164.2 : 1 17,2800.010091798.1 : 1 xxxyA ${\begin{matrix}{4 \choose 1}{3 \choose 3}{3 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 1}\end{matrix}}$ 5760.00033642,971.8 : 1 7200.00042052,377.2 : 1 xxyyA ${\begin{matrix}{4 \choose 2}{3 \choose 2}^{2}{r-4 \choose 1}{4 \choose 1}\end{matrix}}$ 8640.00050461,980.8 : 1 1,0800.00063071,584.5 : 1 xxyAA ${\begin{matrix}{4 \choose 1}{3 \choose 2}{3 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 2}\end{matrix}}$ 1,6200.00094611,056.0 : 1 2,5920.0015137659.6 : 1 xxyA* ${\begin{matrix}{4 \choose 1}{3 \choose 2}{3 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 34,5600.020183348.5 : 1 34,5600.020183348.5 : 1 xyAAA ${\begin{matrix}{4 \choose 2}{3 \choose 1}^{2}{r-4 \choose 1}{4 \choose 3}\end{matrix}}$ 8640.00050461,980.8 : 1 1,0800.00063071,584.5 : 1 xyAA* ${\begin{matrix}{4 \choose 2}{3 \choose 1}^{2}{r-4 \choose 1}{4 \choose 2}{52-4r \choose 1}\end{matrix}}$ 25,9200.015137565.1 : 1 25,9200.015137565.1 : 1 xyA** ${\begin{matrix}{4 \choose 2}{3 \choose 1}^{2}{r-4 \choose 1}{4 \choose 1}{52-4r \choose 2}\end{matrix}}$ 164,1600.09587089.4 : 1 129,6000.075687512.2 : 1 xxyzA ${\begin{matrix}{4 \choose 1}{3 \choose 2}{3 \choose 2}{3 \choose 1}^{2}\end{matrix}}$ 5,1840.0030275329.3 : 1 6,4800.0037844263.2 : 1 xyzAA ${\begin{matrix}{4 \choose 3}{3 \choose 1}^{3}{r-4 \choose 1}{4 \choose 2}\end{matrix}}$ 2,5920.0015137659.6 : 1 3,2400.0018922527.5 : 1 xyzA* ${\begin{matrix}{4 \choose 3}{3 \choose 1}^{3}{r-4 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 34,5600.020183348.5 : 1 34,5600.020183348.5 : 1 xyzrA ${\begin{matrix}{4 \choose 4}{3 \choose 1}^{4}{r-4 \choose 1}{4 \choose 1}\end{matrix}}$ 1,2960.00075691,320.2 : 1 1,6200.00094611,056.0 : 1 Total 908,9760.53084970.9 : 1 1,182,0040.69030030.4 : 1 Miss low xxxyz ${\begin{matrix}{4 \choose 1}{3 \choose 3}{3 \choose 2}{3 \choose 1}^{2}\end{matrix}}$ 1080.000063115,853.7 : 1 1080.000063115,853.7 : 1 xxyyz ${\begin{matrix}{4 \choose 2}{3 \choose 2}^{2}{2 \choose 1}{3 \choose 1}\end{matrix}}$ 3240.00018925,283.9 : 1 3240.00018925,283.9 : 1 xxyz* ${\begin{matrix}{4 \choose 1}{3 \choose 2}{3 \choose 2}{3 \choose 1}^{2}{52-4r \choose 1}\end{matrix}}$ 6,4800.0037844263.2 : 1 5,1840.0030275329.3 : 1 xyz** ${\begin{matrix}{4 \choose 3}{3 \choose 1}^{3}{52-4r \choose 2}\end{matrix}}$ 20,5200.011983982.4 : 1 12,9600.0075687131.1 : 1 xxyzr ${\begin{matrix}{4 \choose 1}{3 \choose 2}{3 \choose 3}{3 \choose 1}^{3}\end{matrix}}$ 3240.00018925,283.9 : 1 3240.00018925,283.9 : 1 xyzr* ${\begin{matrix}{4 \choose 4}{3 \choose 1}^{4}{52-4r \choose 2}\end{matrix}}$ 1,6200.00094611,056.0 : 1 1,2960.00075691,320.2 : 1 Total 29,3760.017155857.3 : 1 20,1960.011794683.8 : 1 No low AAAAB ${\begin{matrix}{r-4 \choose 1}{4 \choose 4}{r-5 \choose 1}{4 \choose 1}\end{matrix}}$ 480.000028035,672.0 : 1 800.000046721,402.8 : 1 AAAA* ${\begin{matrix}{r-4 \choose 1}{4 \choose 4}{52-4r \choose 1}\end{matrix}}$ 800.000046721,402.8 : 1 800.000046721,402.8 : 1 AAABB ${\begin{matrix}{r-4 \choose 1}{4 \choose 3}{r-5 \choose 1}{4 \choose 2}\end{matrix}}$ 2880.00016825,944.5 : 1 4800.00028033,566.3 : 1 AAAB* ${\begin{matrix}{r-4 \choose 1}{4 \choose 3}{r-5 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 3,8400.0022426444.9 : 1 5,1200.0029901333.4 : 1 AABB* ${\begin{matrix}{r-4 \choose 2}{4 \choose 2}^{2}{52-4r \choose 1}\end{matrix}}$ 4,3200.0025229395.4 : 1 5,7600.0033639296.3 : 1 AAA** ${\begin{matrix}{r-4 \choose 1}{4 \choose 3}{52-4r \choose 2}\end{matrix}}$ 3,0400.0017754562.3 : 1 2,4000.0014016712.5 : 1 AAB** ${\begin{matrix}{r-4 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1}{52-4r \choose 2}\end{matrix}}$ 54,7200.031956930.3 : 1 57,6000.033638928.7 : 1 xxxyy ${\begin{matrix}{4 \choose 1}{3 \choose 3}{3 \choose 1}{3 \choose 2}\end{matrix}}$ 360.000021047,563.0 : 1 360.000021047,563.0 : 1 xxxy* ${\begin{matrix}{4 \choose 1}{3 \choose 3}{3 \choose 1}{3 \choose 1}{52-4r \choose 1}\end{matrix}}$ 7200.00042052,377.2 : 1 5760.00033642,971.8 : 1 xxxAA ${\begin{matrix}{4 \choose 1}{3 \choose 3}{r-4 \choose 1}{4 \choose 2}\end{matrix}}$ 960.000056117,835.5 : 1 1200.000070114,268.2 : 1 xxxA* ${\begin{matrix}{4 \choose 1}{3 \choose 3}{r-4 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$ 1,2800.00074751,336.7 : 1 1,2800.00074751,336.7 : 1 xxx** ${\begin{matrix}{4 \choose 1}{3 \choose 3}{52-4r \choose 2}\end{matrix}}$ 7600.00044382,252.0 : 1 4800.00028033,566.3 : 1 xxyy* ${\begin{matrix}{4 \choose 2}{3 \choose 2}^{2}{52-4r \choose 1}\end{matrix}}$ 1,0800.00063071,584.5 : 1 8640.00050461,980.8 : 1 xxy** ${\begin{matrix}{4 \choose 1}{3 \choose 2}{3 \choose 1}{3 \choose 1}{52-4r \choose 2}\end{matrix}}$ 20,5200.011983982.4 : 1 12,9600.0075687131.1 : 1 xxAAA ${\begin{matrix}{4 \choose 1}{3 \choose 2}{r-4 \choose 1}{4 \choose 3}\end{matrix}}$ 1920.00011218,917.3 : 1 2400.00014027,133.6 : 1 xxAA* ${\begin{matrix}{4 \choose 1}{3 \choose 2}{r-4 \choose 1}{4 \choose 2}{52-4r \choose 1}\end{matrix}}$ 5,7600.0033639296.3 : 1 5,7600.0033639296.3 : 1 xxA** ${\begin{matrix}{4 \choose 1}{3 \choose 2}{r-4 \choose 1}{4 \choose 1}{52-4r \choose 2}\end{matrix}}$ 36,4800.021304645.9 : 1 28,8000.016819458.5 : 1 xAAAA ${\begin{matrix}{4 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 4}\end{matrix}}$ 480.000028035,672.0 : 1 600.000035028,537.4 : 1 xAAA* ${\begin{matrix}{4 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 3}{52-4r \choose 1}\end{matrix}}$ 3,8400.0022426444.9 : 1 3,8400.0022426444.9 : 1 xAA** ${\begin{matrix}{4 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 2}{52-4r \choose 2}\end{matrix}}$ 54,7200.031956930.3 : 1 43,2000.025229238.6 : 1  ??*** ${\begin{matrix}{4r-4 \choose 2}{52-4r \choose 3}\end{matrix}}$ 430,9200.25166093.0 : 1 277,7600.16221425.2 : 1  ?**** ${\begin{matrix}{4r-4 \choose 1}{52-4r \choose 4}\end{matrix}}$ 135,6600.079226611.6 : 1 58,2400.034012728.4 : 1 ***** ${\begin{matrix}{52-4r \choose 5}\end{matrix}}$ 15,5040.0090545109.4 : 1 4,3680.0025509391.0 : 1 Total 773,9520.45199451.2 : 1 510,1040.29790502.4 : 1
* The board ABCDE is not possible with an 8-high qualifier since there are only 8 − 4 = 4 low ranks not represented in the hand xyzr. In this case, by definition the binomial coefficient ${\begin{matrix}{8-5 \choose 4}=0\end{matrix}}.$