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

< Poker‎ | Omaha

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 ${\displaystyle h}$ is the number of high cards in the hand (${\displaystyle h}$ = 0, 1, 2, 3, 4) and ${\displaystyle r}$ is the low qualifier (${\displaystyle r}$ = 8, 9), then ${\displaystyle 4r-4+h}$ is the number of low cards remaining in the deck and ${\displaystyle 52-4r-h}$ 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:

Flop Turn River
Board Derivation Board Derivation Board Derivation
??* ${\displaystyle {\begin{matrix}{4r-4+h \choose 2}{52-4r-h \choose 1}\end{matrix}}}$ ??** ${\displaystyle {\begin{matrix}{4r-4+h \choose 2}{52-4r-h \choose 2}\end{matrix}}}$ ??*** ${\displaystyle {\begin{matrix}{4r-4+h \choose 2}{52-4r-h \choose 3}\end{matrix}}}$
?** ${\displaystyle {\begin{matrix}{4r-4+h \choose 1}{52-4r-h \choose 2}\end{matrix}}}$ ?*** ${\displaystyle {\begin{matrix}{4r-4+h \choose 1}{52-4r-h \choose 3}\end{matrix}}}$ ?**** ${\displaystyle {\begin{matrix}{4r-4+h \choose 1}{52-4r-h \choose 4}\end{matrix}}}$
*** ${\displaystyle {\begin{matrix}{52-4r-h \choose 3}\end{matrix}}}$ **** ${\displaystyle {\begin{matrix}{52-4r-h \choose 4}\end{matrix}}}$ ***** ${\displaystyle {\begin{matrix}{52-4r-h \choose 5}\end{matrix}}}$

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 ${\displaystyle {\begin{matrix}{48 \choose 3}=17,296\end{matrix}}}$ boards on the flop, ${\displaystyle {\begin{matrix}{48 \choose 4}=194,580\end{matrix}}}$ boards on the turn, and ${\displaystyle {\begin{matrix}{48 \choose 5}=1,712,304\end{matrix}}}$ 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 ${\displaystyle 52-4r-4=48-4r}$ high cards and ${\displaystyle 4r-4+4=4r}$ 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 make Board Derivation 8-high qualifier (r = 8) 9-high qualifier (r = 9)
Combos Probability Odds Combos Probability Odds
Miss low ABC ${\displaystyle {\begin{matrix}{r \choose 3}{4 \choose 1}^{3}\end{matrix}}}$ 3,584 0.2072155 3.8 : 1 5,376 0.3108233 2.2 : 1
No low AAA ${\displaystyle {\begin{matrix}{r \choose 1}{4 \choose 3}\end{matrix}}}$ 32 0.0018501 539.5 : 1 36 0.0020814 479.4 : 1
AAB ${\displaystyle {\begin{matrix}{r \choose 1}{4 \choose 2}{r-1 \choose 1}{4 \choose 1}\end{matrix}}}$ 1,344 0.0777058 11.9 : 1 1,728 0.0999075 9.0 : 1
??* ${\displaystyle {\begin{matrix}{4r \choose 2}{48-4r \choose 1}\end{matrix}}}$ 7,168 0.4144311 1.4 : 1 6,912 0.3996300 1.5 : 1
?** ${\displaystyle {\begin{matrix}{4r \choose 1}{48-4r \choose 2}\end{matrix}}}$ 7,936 0.4588344 1.2 : 1 7,560 0.4370953 1.3 : 1
*** ${\displaystyle {\begin{matrix}{48-4r \choose 3}\end{matrix}}}$ 560 0.0323774 29.9 : 1 220 0.0127197 77.6 : 1
Total 13,712 0.7927845 0.3 : 1 11,920 0.6891767 0.5 : 1
Derivations for low hand shape **** (four high cards) on the turn
Hand to make Board Derivation 8-high qualifier (r = 8) 9-high qualifier (r = 9)
Combos Probability Odds Combos Probability Odds
Miss low ABCD ${\displaystyle {\begin{matrix}{r \choose 4}{4 \choose 1}^{4}\end{matrix}}}$ 17,920 0.0920958 9.9 : 1 32,256 0.1657724 5.0 : 1
AABC ${\displaystyle {\begin{matrix}{r \choose 1}{4 \choose 2}{r-1 \choose 2}{4 \choose 1}^{2}\end{matrix}}}$ 16,128 0.0828862 11.1 : 1 24,192 0.1243293 7.0 : 1
ABC* ${\displaystyle {\begin{matrix}{r \choose 3}{4 \choose 1}^{3}{48-4r \choose 1}\end{matrix}}}$ 57,344 0.2947065 2.4 : 1 64,512 0.3315449 2.0 : 1
Total 91,392 0.7927845 1.1 : 1 120,960 0.6216466 0.6 : 1
No low AAAA ${\displaystyle {\begin{matrix}{r \choose 1}{4 \choose 4}\end{matrix}}}$ 8 0.0000411 24,321.5 : 1 9 0.0000463 21,619.0 : 1
AAAB ${\displaystyle {\begin{matrix}{r \choose 1}{4 \choose 3}{r-1 \choose 1}{4 \choose 1}\end{matrix}}}$ 896 0.0046048 216.2 : 1 1,152 0.0059204 167.9 : 1
AABB ${\displaystyle {\begin{matrix}{r \choose 2}{4 \choose 2}^{2}\end{matrix}}}$ 1,008 0.0051804 192.0 : 1 1,296 0.0066605 149.1 : 1
AAA* ${\displaystyle {\begin{matrix}{r \choose 1}{4 \choose 3}{48-4r \choose 1}\end{matrix}}}$ 512 0.0026313 379.0 : 1 432 0.0022202 449.4 : 1
AAB* ${\displaystyle {\begin{matrix}{r \choose 1}{4 \choose 2}{r-1 \choose 1}{4 \choose 1}{48-4r \choose 1}\end{matrix}}}$ 21,504 0.1105150 8.0 : 1 20,736 0.1065680 8.4 : 1
??** ${\displaystyle {\begin{matrix}{4r \choose 2}{48-4r \choose 2}\end{matrix}}}$ 59,520 0.3058896 2.3 : 1 41,580 0.2136910 3.7 : 1
?*** ${\displaystyle {\begin{matrix}{4r \choose 1}{48-4r \choose 3}\end{matrix}}}$ 17,920 0.0920958 9.9 : 1 7,920 0.0407031 23.6 : 1
**** ${\displaystyle {\begin{matrix}{48-4r \choose 4}\end{matrix}}}$ 1,820 0.0093535 105.9 : 1 495 0.0025439 392.1 : 1
Total 103,188 0.5303114 0.9 : 1 73,620 0.3783534 1.6 : 1
Derivations for low hand shape **** (four high cards) on the river
Hand to make Board Derivation 8-high qualifier (r = 8) 9-high qualifier (r = 9)
Combos Probability Odds Combos Probability Odds
Miss low ABCDE ${\displaystyle {\begin{matrix}{r \choose 5}{4 \choose 1}^{5}\end{matrix}}}$ 57,344 0.0334894 28.9 : 1 129,024 0.0753511 12.3 : 1
AABCD ${\displaystyle {\begin{matrix}{r \choose 1}{4 \choose 2}{r-1 \choose 3}{4 \choose 1}^{3}\end{matrix}}}$ 107,520 0.0627926 14.9 : 1 193,536 0.1130267 7.8 : 1
ABCD* ${\displaystyle {\begin{matrix}{r \choose 4}{4 \choose 1}^{4}{48-4r \choose 1}\end{matrix}}}$ 286,720 0.1674469 5.0 : 1 387,072 0.2260533 3.4 : 1
AABBC ${\displaystyle {\begin{matrix}{r \choose 2}{4 \choose 2}^{2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}}$ 24,192 0.0141283 69.8 : 1 36,288 0.0211925 46.2 : 1
AAABC ${\displaystyle {\begin{matrix}{r \choose 1}{4 \choose 3}{r-1 \choose 2}{4 \choose 1}^{2}\end{matrix}}}$ 10,752 0.0062793 158.3 : 1 16,128 0.0094189 105.2 : 1
AABC* ${\displaystyle {\begin{matrix}{r \choose 1}{4 \choose 2}{r-1 \choose 2}{4 \choose 1}^{2}{48-4r \choose 1}\end{matrix}}}$ 258,048 0.1507022 5.6 : 1 290,304 0.1695400 4.9 : 1
ABC** ${\displaystyle {\begin{matrix}{r \choose 3}{4 \choose 1}^{3}{48-4r \choose 2}\end{matrix}}}$ 430,080 0.2511704 3.0 : 1 354,816 0.2072155 3.8 : 1
Total 1,174,656 0.6860090 0.5 : 1 1,407,168 0.8217980 0.2 : 1
No low AAAAB ${\displaystyle {\begin{matrix}{r \choose 1}{4 \choose 4}{r-1 \choose 1}{4 \choose 1}\end{matrix}}}$ 224 0.0001308 7,643.2 : 1 288 0.0001682 5,944.5 : 1
AAAA* ${\displaystyle {\begin{matrix}{r \choose 1}{4 \choose 4}{48-4r \choose 1}\end{matrix}}}$ 128 0.0000748 13,376.4 : 1 108 0.0000631 15,853.7 : 1
AAABB ${\displaystyle {\begin{matrix}{r \choose 1}{4 \choose 3}{r-1 \choose 1}{4 \choose 2}\end{matrix}}}$ 1,344 0.0007849 1,273.0 : 1 1,728 0.0010092 989.9 : 1
AAAB* ${\displaystyle {\begin{matrix}{r \choose 1}{4 \choose 3}{r-1 \choose 1}{4 \choose 1}{48-4r \choose 1}\end{matrix}}}$ 14,336 0.0083723 118.4 : 1 13,824 0.0080733 122.9 : 1
AABB* ${\displaystyle {\begin{matrix}{r \choose 2}{4 \choose 2}^{2}{48-4r \choose 1}\end{matrix}}}$ 16,128 0.0094189 105.2 : 1 15,552 0.0090825 109.1 : 1
AAA** ${\displaystyle {\begin{matrix}{r \choose 1}{4 \choose 3}{48-4r \choose 2}\end{matrix}}}$ 3,840 0.0022426 444.9 : 1 2,376 0.0013876 719.7 : 1
AAB** ${\displaystyle {\begin{matrix}{r \choose 1}{4 \choose 2}{r-1 \choose 1}{4 \choose 1}{48-4r \choose 2}\end{matrix}}}$ 161,280 0.0941889 9.6 : 1 114,048 0.0666050 14.0 : 1
??*** ${\displaystyle {\begin{matrix}{4r \choose 2}{48-4r \choose 3}\end{matrix}}}$ 277,760 0.1622142 5.2 : 1 138,600 0.0809436 11.4 : 1
?**** ${\displaystyle {\begin{matrix}{4r \choose 1}{48-4r \choose 4}\end{matrix}}}$ 58,240 0.0340127 28.4 : 1 17,820 0.0104070 95.1 : 1
***** ${\displaystyle {\begin{matrix}{48-4r \choose 5}\end{matrix}}}$ 4,368 0.0025509 391.0 : 1 792 0.0004625 2,161.0 : 1
Total 537,648 0.3139910 2.2 : 1 305,136 0.1782020 4.6 : 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 ${\displaystyle 52-4r-3=49-4r}$ high cards and ${\displaystyle 4r-4+3=4r-1}$ 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 make Board Derivation 8-high qualifier (r = 8) 9-high qualifier (r = 9)
Combos Probability Odds Combos Probability Odds
Miss low ABC ${\displaystyle {\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}\end{matrix}}}$ 2,240 0.1295097 6.7 : 1 3,584 0.2072155 3.8 : 1
xAB ${\displaystyle {\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}\end{matrix}}}$ 1,008 0.0582794 16.2 : 1 1,344 0.0777058 11.9 : 1
Total 3,248 0.1877891 4.3 : 1 4,928 0.2849214 2.5 : 1
No low AAA ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 3}\end{matrix}}}$ 28 0.0016189 616.7 : 1 32 0.0018501 539.5 : 1
AAB ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}}$ 1,008 0.0582794 16.2 : 1 1,344 0.0777058 11.9 : 1
xxx ${\displaystyle {\begin{matrix}{1 \choose 1}{3 \choose 3}\end{matrix}}}$ 1 0.0000578 17,295.0 : 1 1 0.0000578 17,295.0 : 1
xxA ${\displaystyle {\begin{matrix}{1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 1}\end{matrix}}}$ 84 0.0048566 204.9 : 1 96 0.0055504 179.2 : 1
xAA ${\displaystyle {\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}\end{matrix}}}$ 126 0.0072849 136.3 : 1 144 0.0083256 119.1 : 1
??* ${\displaystyle {\begin{matrix}{4r-1 \choose 2}{49-4r \choose 1}\end{matrix}}}$ 7,905 0.4570421 1.2 : 1 7,735 0.4472132 1.2 : 1
?** ${\displaystyle {\begin{matrix}{4r-1 \choose 1}{49-4r \choose 2}\end{matrix}}}$ 4,216 0.2437558 3.1 : 1 2,730 0.1578400 5.3 : 1
*** ${\displaystyle {\begin{matrix}{49-4r \choose 3}\end{matrix}}}$ 680 0.0393154 24.4 : 1 286 0.0165356 59.5 : 1
Total 14,048 0.8122109 0.2 : 1 12,368 0.7150786 0.4 : 1
Derivations for low hand shape x*** (one low card and three high cards) on the turn
Hand to make Board Derivation 8-high qualifier (r = 8) 9-high qualifier (r = 9)
Combos Probability Odds Combos Probability Odds
Miss low ABCD ${\displaystyle {\begin{matrix}{r-1 \choose 4}{4 \choose 1}^{4}\end{matrix}}}$ 8,960 0.0460479 20.7 : 1 17,920 0.0920958 9.9 : 1
AABC ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}}$ 10,080 0.0518039 18.3 : 1 16,128 0.0828862 11.1 : 1
ABC* ${\displaystyle {\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}{49-4r \choose 1}\end{matrix}}}$ 38,080 0.1957036 4.1 : 1 46,592 0.2394491 3.2 : 1
xxAB ${\displaystyle {\begin{matrix}{1 \choose 1}{3 \choose 2}{r-1 \choose 2}{4 \choose 1}^{2}\end{matrix}}}$ 1,008 0.0051804 192.0 : 1 1,344 0.0069072 143.8 : 1
xABC ${\displaystyle {\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 3}{4 \choose 1}^{3}\end{matrix}}}$ 6,720 0.0345359 28.0 : 1 10,752 0.0552575 17.1 : 1
xAAB ${\displaystyle {\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}}$ 3,024 0.0155412 63.3 : 1 4,032 0.0207216 47.3 : 1
xAB* ${\displaystyle {\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}{49-4r \choose 1}\end{matrix}}}$ 17,136 0.0880666 10.4 : 1 17,472 0.0897934 10.1 : 1
Total 85,008 0.4368794 1.3 : 1 114,240 0.5871107 0.7 : 1
No low AAAA ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 4}\end{matrix}}}$ 7 0.0000360 27,796.1 : 1 8 0.0000411 24,321.5 : 1
AAAB ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}\end{matrix}}}$ 672 0.0034536 288.6 : 1 896 0.0046048 216.2 : 1
AABB ${\displaystyle {\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}\end{matrix}}}$ 756 0.0038853 256.4 : 1 1,008 0.0051804 192.0 : 1
AAA* ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 3}{49-4r \choose 1}\end{matrix}}}$ 476 0.0024463 407.8 : 1 416 0.0021379 466.7 : 1
AAB* ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{49-4r \choose 1}\end{matrix}}}$ 17,136 0.0880666 10.4 : 1 17,472 0.0897934 10.1 : 1
xxx* ${\displaystyle {\begin{matrix}{1 \choose 1}{3 \choose 3}{49-4r \choose 1}\end{matrix}}}$ 17 0.0000874 11,444.9 : 1 13 0.0000668 14,966.7 : 1
xxxA ${\displaystyle {\begin{matrix}{1 \choose 1}{3 \choose 3}{r-1 \choose 1}{4 \choose 1}\end{matrix}}}$ 28 0.0001439 6,948.3 : 1 32 0.0001645 6,079.6 : 1
xxAA ${\displaystyle {\begin{matrix}{1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 2}\end{matrix}}}$ 126 0.0006475 1,543.3 : 1 144 0.0007401 1,350.3 : 1
xxA* ${\displaystyle {\begin{matrix}{1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 1}{49-4r \choose 1}\end{matrix}}}$ 1,428 0.0073389 135.3 : 1 1,248 0.0064138 154.9 : 1
xAAA ${\displaystyle {\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 3}\end{matrix}}}$ 84 0.0004317 2,315.4 : 1 96 0.0004934 2,025.9 : 1
xAA* ${\displaystyle {\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{49-4r \choose 1}\end{matrix}}}$ 2,142 0.0110083 89.8 : 1 1,872 0.0096207 102.9 : 1
??** ${\displaystyle {\begin{matrix}{4r-1 \choose 2}{49-4r \choose 2}\end{matrix}}}$ 63,240 0.3250077 2.1 : 1 46,410 0.2385137 3.2 : 1
?*** ${\displaystyle {\begin{matrix}{4r-1 \choose 1}{49-4r \choose 3}\end{matrix}}}$ 21,080 0.1083359 8.2 : 1 10,010 0.0514441 18.4 : 1
**** ${\displaystyle {\begin{matrix}{49-4r \choose 4}\end{matrix}}}$ 2,380 0.0122315 80.8 : 1 715 0.0036746 271.1 : 1
Total 109,572 0.5631206 0.8 : 1 80,340 0.4128893 1.4 : 1
Derivations for low hand shape x*** (one low card and three high cards) on the river
Hand to make Board Derivation 8-high qualifier (r = 8) 9-high qualifier (r = 9)
Combos Probability Odds Combos Probability Odds
Miss low ABCDE ${\displaystyle {\begin{matrix}{r-1 \choose 5}{4 \choose 1}^{5}\end{matrix}}}$ 21,504 0.0125585 78.6 : 1 57,344 0.0334894 28.9 : 1
AABCD ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}}$ 53,760 0.0313963 30.9 : 1 107,520 0.0627926 14.9 : 1
ABCD* ${\displaystyle {\begin{matrix}{r-1 \choose 4}{4 \choose 1}^{4}{49-4r \choose 1}\end{matrix}}}$ 152,320 0.0889562 10.2 : 1 232,960 0.1360506 6.4 : 1
AABBC ${\displaystyle {\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}}$ 15,120 0.0088302 112.2 : 1 24,192 0.0141283 69.8 : 1
AAABC ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}}$ 6,720 0.0039245 253.8 : 1 10,752 0.0062793 158.3 : 1
AABC* ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}{49-4r \choose 1}\end{matrix}}}$ 171,360 0.1000757 9.0 : 1 209,664 0.1224455 7.2 : 1
ABC** ${\displaystyle {\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}{49-4r \choose 2}\end{matrix}}}$ 304,640 0.1779123 4.6 : 1 279,552 0.1632607 5.1 : 1
xABCD ${\displaystyle {\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 4}{4 \choose 1}^{4}\end{matrix}}}$ 26,880 0.0156981 62.7 : 1 53,760 0.0313963 30.9 : 1
xABC* ${\displaystyle {\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 3}{4 \choose 1}^{3}{49-4r \choose 1}\end{matrix}}}$ 114,240 0.0667171 14.0 : 1 139,776 0.0816304 11.3 : 1
xAABC ${\displaystyle {\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}}$ 30,240 0.0176604 55.6 : 1 48,384 0.0282567 34.4 : 1
xAAB* ${\displaystyle {\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{49-4r \choose 1}\end{matrix}}}$ 51,408 0.0300227 32.3 : 1 52,416 0.0306114 31.7 : 1
xAAAB ${\displaystyle {\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}\end{matrix}}}$ 2,016 0.0011774 848.4 : 1 2,688 0.0015698 636.0 : 1
xAABB ${\displaystyle {\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 2}^{2}\end{matrix}}}$ 2,268 0.0013245 754.0 : 1 3,024 0.0017660 565.2 : 1
xAB** ${\displaystyle {\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}{49-4r \choose 2}\end{matrix}}}$ 137,088 0.0800605 11.5 : 1 104,832 0.0612228 15.3 : 1
xxABC ${\displaystyle {\begin{matrix}{1 \choose 1}{3 \choose 2}{r-1 \choose 3}{4 \choose 1}^{3}\end{matrix}}}$ 6,720 0.0039245 253.8 : 1 10,752 0.0062793 158.3 : 1
xxAB* ${\displaystyle {\begin{matrix}{1 \choose 1}{3 \choose 2}{r-1 \choose 2}{4 \choose 1}^{2}{49-4r \choose 1}\end{matrix}}}$ 17,136 0.0100076 98.9 : 1 17,472 0.0102038 97.0 : 1
xxAAB ${\displaystyle {\begin{matrix}{1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}}$ 3,024 0.0017660 565.2 : 1 4,032 0.0023547 423.7 : 1
xxxAB ${\displaystyle {\begin{matrix}{1 \choose 1}{3 \choose 3}{r-1 \choose 2}{4 \choose 1}^{2}\end{matrix}}}$ 336 0.0001962 5,095.1 : 1 448 0.0002616 3,821.1 : 1
Total 1,116,780 0.6522090 0.5 : 1 1,359,568 0.7939992 0.3 : 1
No low AAAAB ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 4}{r-2 \choose 1}{4 \choose 1}\end{matrix}}}$ 168 0.0000981 10,191.3 : 1 224 0.0001308 7,643.2 : 1
AAAA* ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 4}{49-4r \choose 1}\end{matrix}}}$ 119 0.0000695 14,388.1 : 1 104 0.0000607 16,463.5 : 1
AAABB ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 2}\end{matrix}}}$ 1,008 0.0005887 1,697.7 : 1 1,344 0.0007849 1,273.0 : 1
AAAB* ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}{49-4r \choose 1}\end{matrix}}}$ 11,424 0.0066717 148.9 : 1 11,648 0.0068025 146.0 : 1
AABB* ${\displaystyle {\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}{49-4r \choose 1}\end{matrix}}}$ 12,852 0.0075057 132.2 : 1 13,104 0.0076528 129.7 : 1
AAA** ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 3}{49-4r \choose 2}\end{matrix}}}$ 3,808 0.0022239 448.7 : 1 2,496 0.0014577 685.0 : 1
AAB** ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{49-4r \choose 2}\end{matrix}}}$ 137,088 0.0800605 11.5 : 1 104,832 0.0612228 15.3 : 1
xxxAA ${\displaystyle {\begin{matrix}{1 \choose 1}{3 \choose 3}{r-1 \choose 1}{4 \choose 2}\end{matrix}}}$ 42 0.0000245 40,768.1 : 1 48 0.0000280 35,672.0 : 1
xxxA* ${\displaystyle {\begin{matrix}{1 \choose 1}{3 \choose 3}{r-1 \choose 1}{4 \choose 1}{49-4r \choose 1}\end{matrix}}}$ 476 0.0002780 3,596.3 : 1 416 0.0002429 4,115.1 : 1
xxx** ${\displaystyle {\begin{matrix}{1 \choose 1}{3 \choose 3}{49-4r \choose 2}\end{matrix}}}$ 136 0.0000794 12,589.5 : 1 78 0.0000456 21,951.6 : 1
xxAAA ${\displaystyle {\begin{matrix}{1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 3}\end{matrix}}}$ 84 0.0000491 20,383.6 : 1 96 0.0000561 17,835.5 : 1
xxAA* ${\displaystyle {\begin{matrix}{1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 2}{49-4r \choose 1}\end{matrix}}}$ 2,142 0.0012509 798.4 : 1 1,872 0.0010933 913.7 : 1
xxA** ${\displaystyle {\begin{matrix}{1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 1}{49-4r \choose 2}\end{matrix}}}$ 11,424 0.0066717 148.9 : 1 7,488 0.0043731 227.7 : 1
xAAAA ${\displaystyle {\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 4}\end{matrix}}}$ 21 0.0000123 81,537.3 : 1 24 0.0000140 71,345.0 : 1
xAAA* ${\displaystyle {\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 3}{49-4r \choose 1}\end{matrix}}}$ 1,428 0.0008340 1,198.1 : 1 1,248 0.0007288 1,371.0 : 1
xAA** ${\displaystyle {\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{49-4r \choose 2}\end{matrix}}}$ 17,136 0.0100076 98.9 : 1 11,232 0.0065596 151.4 : 1
??*** ${\displaystyle {\begin{matrix}{4r-1 \choose 2}{49-4r \choose 3}\end{matrix}}}$ 316,200 0.1846635 4.4 : 1 170,170 0.0993807 9.1 : 1
?**** ${\displaystyle {\begin{matrix}{4r-1 \choose 1}{49-4r \choose 4}\end{matrix}}}$ 73,780 0.0430881 22.2 : 1 25,025 0.0146148 67.4 : 1
***** ${\displaystyle {\begin{matrix}{49-4r \choose 5}\end{matrix}}}$ 6,188 0.0036138 275.7 : 1 1,287 0.0007516 1,329.5 : 1
Total 595,524 0.3477910 1.9 : 1 352,736 0.2060008 3.9 : 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 ${\displaystyle 52-4r-2=50-4r}$ high cards and ${\displaystyle 4r-4+2=4r-2}$ 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 make Board Derivation 8-high qualifier (r = 8) 9-high qualifier (r = 9)
Combos Probability Odds Combos Probability Odds
Miss low ABC ${\displaystyle {\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}\end{matrix}}}$ 2,240 0.1295097 6.7 : 1 3,584 0.2072155 3.8 : 1
xAB ${\displaystyle {\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}\end{matrix}}}$ 672 0.0388529 24.7 : 1 896 0.0518039 18.3 : 1
Total 2,912 0.1683626 4.9 : 1 4,480 0.2590194 2.9 : 1
No low AAA ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 3}\end{matrix}}}$ 28 0.0016189 616.7 : 1 32 0.0018501 539.5 : 1
AAB ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}}$ 1,008 0.0582794 16.2 : 1 1,344 0.0777058 11.9 : 1
xxA ${\displaystyle {\begin{matrix}{1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 1}\end{matrix}}}$ 28 0.0016189 616.7 : 1 32 0.0018501 539.5 : 1
xAA ${\displaystyle {\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}\end{matrix}}}$ 84 0.0048566 204.9 : 1 96 0.0055504 179.2 : 1
??* ${\displaystyle {\begin{matrix}{4r-2 \choose 2}{50-4r \choose 1}\end{matrix}}}$ 7,830 0.4527058 1.2 : 1 7,854 0.4540934 1.2 : 1
?** ${\displaystyle {\begin{matrix}{4r-2 \choose 1}{50-4r \choose 2}\end{matrix}}}$ 4,590 0.2653793 2.8 : 1 3,094 0.1788853 4.6 : 1
*** ${\displaystyle {\begin{matrix}{50-4r \choose 3}\end{matrix}}}$ 816 0.0471785 20.2 : 1 364 0.0210453 46.5 : 1
Total 14,384 0.8316374 0.2 : 1 12,816 0.7409806 0.3 : 1
Derivations for low hand shape xx** (a low pair and two high cards) on the turn
Hand to make Board Derivation 8-high qualifier (r = 8) 9-high qualifier (r = 9)
Combos Probability Odds Combos Probability Odds
Miss low ABCD ${\displaystyle {\begin{matrix}{r-1 \choose 4}{4 \choose 1}^{4}\end{matrix}}}$ 8,960 0.0460479 20.7 : 1 17,920 0.0920958 9.9 : 1
AABC ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}}$ 10,080 0.0518039 18.3 : 1 16,128 0.0828862 11.1 : 1
ABC* ${\displaystyle {\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}{50-4r \choose 1}\end{matrix}}}$ 40,320 0.2072155 3.8 : 1 50,176 0.2578682 2.9 : 1
xxAB ${\displaystyle {\begin{matrix}{1 \choose 1}{2 \choose 2}{r-1 \choose 2}{4 \choose 1}^{2}\end{matrix}}}$ 336 0.0017268 578.1 : 1 448 0.0023024 433.3 : 1
xABC ${\displaystyle {\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 3}{4 \choose 1}^{3}\end{matrix}}}$ 4,480 0.0230239 42.4 : 1 7,168 0.0368383 26.1 : 1
xAAB ${\displaystyle {\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}}$ 2,016 0.0103608 95.5 : 1 2,688 0.0138144 71.4 : 1
xAB* ${\displaystyle {\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}{50-4r \choose 1}\end{matrix}}}$ 12,096 0.0621647 15.1 : 1 12,544 0.0644671 14.5 : 1
Total 78,288 0.4023435 1.5 : 1 107,072 0.5502724 0.8 : 1
No low AAAA ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 4}\end{matrix}}}$ 7 0.0000360 27,796.1 : 1 8 0.0000411 24,321.5 : 1
AAAB ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}\end{matrix}}}$ 672 0.0034536 288.6 : 1 896 0.0046048 216.2 : 1
AABB ${\displaystyle {\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}\end{matrix}}}$ 756 0.0038853 256.4 : 1 1,008 0.0051804 192.0 : 1
AAA* ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 3}{50-4r \choose 1}\end{matrix}}}$ 504 0.0025902 385.1 : 1 448 0.0023024 433.3 : 1
AAB* ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1}\end{matrix}}}$ 18,144 0.0932470 9.7 : 1 18,816 0.0967006 9.3 : 1
xxAA ${\displaystyle {\begin{matrix}{1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 2}\end{matrix}}}$ 42 0.0002158 4,631.9 : 1 48 0.0002467 4,052.8 : 1
xxA* ${\displaystyle {\begin{matrix}{1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 1}{50-4r \choose 1}\end{matrix}}}$ 504 0.0025902 385.1 : 1 448 0.0023024 433.3 : 1
xAAA ${\displaystyle {\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 3}\end{matrix}}}$ 56 0.0002878 3,473.6 : 1 64 0.0003289 3,039.3 : 1
xAA* ${\displaystyle {\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}{50-4r \choose 1}\end{matrix}}}$ 1,512 0.0077706 127.7 : 1 1,344 0.0069072 143.8 : 1
??** ${\displaystyle {\begin{matrix}{4r-2 \choose 2}{50-4r \choose 2}\end{matrix}}}$ 66,555 0.3420444 1.9 : 1 51,051 0.2623651 2.8 : 1
?*** ${\displaystyle {\begin{matrix}{4r-2 \choose 1}{50-4r \choose 3}\end{matrix}}}$ 24,480 0.1258094 6.9 : 1 12,376 0.0636037 14.7 : 1
**** ${\displaystyle {\begin{matrix}{50-4r \choose 4}\end{matrix}}}$ 3,060 0.0157262 62.6 : 1 1,001 0.0051444 193.4 : 1
Total 116,292 0.5976565 0.7 : 1 87,508 0.4497276 1.2 : 1
Derivations for low hand shape xx** (a low pair and two high cards) on the river
Hand to make Board Derivation 8-high qualifier (r = 8) 9-high qualifier (r = 9)
Combos Probability Odds Combos Probability Odds
Miss low ABCDE ${\displaystyle {\begin{matrix}{r-1 \choose 5}{4 \choose 1}^{5}\end{matrix}}}$ 21,504 0.0125585 78.6 : 1 57,344 0.0334894 28.9 : 1
AABCD ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}}$ 53,760 0.0313963 30.9 : 1 107,520 0.0627926 14.9 : 1
ABCD* ${\displaystyle {\begin{matrix}{r-1 \choose 4}{4 \choose 1}^{4}{50-4r \choose 1}\end{matrix}}}$ 161,280 0.0941889 9.6 : 1 250,880 0.1465160 5.8 : 1
AABBC ${\displaystyle {\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}}$ 15,120 0.0088302 112.2 : 1 24,192 0.0141283 69.8 : 1
AAABC ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}}$ 6,720 0.0039245 253.8 : 1 10,752 0.0062793 158.3 : 1
AABC* ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}{50-4r \choose 1}\end{matrix}}}$ 181,440 0.1059625 8.4 : 1 225,792 0.1318644 6.6 : 1
ABC** ${\displaystyle {\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}{50-4r \choose 2}\end{matrix}}}$ 342,720 0.2001514 4.0 : 1 326,144 0.1904709 4.3 : 1
xABCD ${\displaystyle {\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 4}{4 \choose 1}^{4}\end{matrix}}}$ 17,920 0.0104654 94.6 : 1 35,840 0.0209309 46.8 : 1
xABC* ${\displaystyle {\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 3}{4 \choose 1}^{3}{50-4r \choose 1}\end{matrix}}}$ 80,640 0.0470944 20.2 : 1 100,352 0.0586064 16.1 : 1
xAABC ${\displaystyle {\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}}$ 20,160 0.0117736 83.9 : 1 32,256 0.0188378 52.1 : 1
xAAB* ${\displaystyle {\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1}\end{matrix}}}$ 36,288 0.0211925 46.2 : 1 37,632 0.0219774 44.5 : 1
xAAAB ${\displaystyle {\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}\end{matrix}}}$ 1,344 0.0007849 1,273.0 : 1 1,792 0.0010465 954.5 : 1
xAABB ${\displaystyle {\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 2}^{2}\end{matrix}}}$ 1,512 0.0008830 1,131.5 : 1 2,016 0.0011774 848.4 : 1
xAB** ${\displaystyle {\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}{50-4r \choose 2}\end{matrix}}}$ 102,816 0.0600454 15.7 : 1 81,536 0.0476177 20.0 : 1
xxABC ${\displaystyle {\begin{matrix}{1 \choose 1}{2 \choose 2}{r-1 \choose 3}{4 \choose 1}^{3}\end{matrix}}}$ 2,240 0.0013082 763.4 : 1 3,584 0.0020931 476.8 : 1
xxAB* ${\displaystyle {\begin{matrix}{1 \choose 1}{2 \choose 2}{r-1 \choose 2}{4 \choose 1}^{2}{50-4r \choose 1}\end{matrix}}}$ 6,048 0.0035321 282.1 : 1 6,272 0.0036629 272.0 : 1
xxAAB ${\displaystyle {\begin{matrix}{1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}}$ 1,008 0.0005887 1,697.7 : 1 1,344 0.0007849 1,273.0 : 1
Total 1,052,520 0.6146806 0.6 : 1 1,305,248 0.7622759 0.3 : 1
No low AAAAB ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 4}{r-2 \choose 1}{4 \choose 1}\end{matrix}}}$ 168 0.0000981 10,191.3 : 1 224 0.0001308 7,643.2 : 1
AAAA* ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 4}{50-4r \choose 1}\end{matrix}}}$ 126 0.0000736 13,588.7 : 1 112 0.0000654 15,287.4 : 1
AAABB ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 2}\end{matrix}}}$ 1,008 0.0005887 1,697.7 : 1 1,344 0.0007849 1,273.0 : 1
AAAB* ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1}\end{matrix}}}$ 12,096 0.0070642 140.6 : 1 12,544 0.0073258 135.5 : 1
AABB* ${\displaystyle {\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}{50-4r \choose 1}\end{matrix}}}$ 13,608 0.0079472 124.8 : 1 14,112 0.0082415 120.3 : 1
AAA** ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 3}{50-4r \choose 2}\end{matrix}}}$ 4,284 0.0025019 398.7 : 1 2,912 0.0017006 587.0 : 1
AAB** ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 2}\end{matrix}}}$ 154,224 0.0900681 10.1 : 1 122,304 0.0714266 13.0 : 1
xxAAA ${\displaystyle {\begin{matrix}{1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 3}\end{matrix}}}$ 28 0.0000164 61,152.7 : 1 32 0.0000187 53,508.5 : 1
xxAA* ${\displaystyle {\begin{matrix}{1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 2}{50-4r \choose 1}\end{matrix}}}$ 756 0.0004415 2,264.0 : 1 672 0.0003925 2,547.1 : 1
xxA** ${\displaystyle {\begin{matrix}{1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 1}{50-4r \choose 2}\end{matrix}}}$ 4,284 0.0025019 398.7 : 1 2,912 0.0017006 587.0 : 1
xAAAA ${\displaystyle {\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 4}\end{matrix}}}$ 14 0.0000082 122,306.4 : 1 16 0.0000093 107,018.0 : 1
xAAA* ${\displaystyle {\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 3}{50-4r \choose 1}\end{matrix}}}$ 1,008 0.0005887 1,697.7 : 1 896 0.0005233 1,910.1 : 1
xAA** ${\displaystyle {\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}{50-4r \choose 2}\end{matrix}}}$ 12,852 0.0075057 132.2 : 1 8,736 0.0051019 195.0 : 1
??*** ${\displaystyle {\begin{matrix}{4r-2 \choose 2}{50-4r \choose 3}\end{matrix}}}$ 354,960 0.2072996 3.8 : 1 204,204 0.1192569 7.4 : 1
?**** ${\displaystyle {\begin{matrix}{4r-2 \choose 1}{50-4r \choose 4}\end{matrix}}}$ 91,800 0.0536120 17.7 : 1 34,034 0.0198761 49.3 : 1
***** ${\displaystyle {\begin{matrix}{50-4r \choose 5}\end{matrix}}}$ 8,568 0.0050038 198.8 : 1 2,002 0.0011692 854.3 : 1
Total 659,784 0.3853194 1.6 : 1 407,056 0.2377241 3.2 : 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 ${\displaystyle 52-4r-1=51-4r}$ high cards and ${\displaystyle 4r-4+1=4r-3}$ 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 make Board Derivation 8-high qualifier (r = 8) 9-high qualifier (r = 9)
Combos Probability Odds Combos Probability Odds
Miss low ABC ${\displaystyle {\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}\end{matrix}}}$ 2,240 0.1295097 6.7 : 1 3,584 0.2072155 3.8 : 1
xAB ${\displaystyle {\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}\end{matrix}}}$ 336 0.0194265 50.5 : 1 448 0.0259019 37.6 : 1
Total 2,576 0.1489362 5.7 : 1 4,032 0.2331175 3.3 : 1
No low AAA ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 3}\end{matrix}}}$ 28 0.0016189 616.7 : 1 32 0.0018501 539.5 : 1
AAB ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}}$ 1,008 0.0582794 16.2 : 1 1,344 0.0777058 11.9 : 1
xAA ${\displaystyle {\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 2}\end{matrix}}}$ 42 0.0024283 410.8 : 1 48 0.0027752 359.3 : 1
??* ${\displaystyle {\begin{matrix}{4r-3 \choose 2}{51-4r \choose 1}\end{matrix}}}$ 7,714 0.4459991 1.2 : 1 7,920 0.4579093 1.2 : 1
?** ${\displaystyle {\begin{matrix}{4r-3 \choose 1}{51-4r \choose 2}\end{matrix}}}$ 4,959 0.2867137 2.5 : 1 3,465 0.2003353 4.0 : 1
*** ${\displaystyle {\begin{matrix}{51-4r \choose 3}\end{matrix}}}$ 969 0.0560245 16.8 : 1 455 0.0263067 37.0 : 1
Total 14,720 0.8510638 0.2 : 1 13,264 0.7668825 0.3 : 1
Derivations for low hand shape xxx* (a low three of a kind and a high card) on the turn
Hand to make Board Derivation 8-high qualifier (r = 8) 9-high qualifier (r = 9)
Combos Probability Odds Combos Probability Odds
Miss low ABCD ${\displaystyle {\begin{matrix}{r-1 \choose 4}{4 \choose 1}^{4}\end{matrix}}}$ 8,960 0.0460479 20.7 : 1 17,920 0.0920958 9.9 : 1
AABC ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}}$ 10,080 0.0518039 18.3 : 1 16,128 0.0828862 11.1 : 1
ABC* ${\displaystyle {\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}{51-4r \choose 1}\end{matrix}}}$ 42,560 0.2187275 3.6 : 1 53,760 0.2762874 2.6 : 1
xABC ${\displaystyle {\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 3}{4 \choose 1}^{3}\end{matrix}}}$ 2,240 0.0115120 85.9 : 1 3,584 0.0184192 53.3 : 1
xAAB ${\displaystyle {\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}}$ 1,008 0.0051804 192.0 : 1 1,344 0.0069072 143.8 : 1
xAB* ${\displaystyle {\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}{51-4r \choose 1}\end{matrix}}}$ 6,384 0.0328091 29.5 : 1 6,720 0.0345359 28.0 : 1
Total 71,232 0.3660808 1.7 : 1 99,456 0.5111317 1.0 : 1
No low AAAA ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 4}\end{matrix}}}$ 7 0.0000360 27,796.1 : 1 8 0.0000411 24,321.5 : 1
AAAB ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}\end{matrix}}}$ 672 0.0034536 288.6 : 1 896 0.0046048 216.2 : 1
AABB ${\displaystyle {\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}\end{matrix}}}$ 756 0.0038853 256.4 : 1 1,008 0.0051804 192.0 : 1
AAA* ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 3}{51-4r \choose 1}\end{matrix}}}$ 532 0.0027341 364.8 : 1 480 0.0024669 404.4 : 1
AAB* ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 1}\end{matrix}}}$ 19,152 0.0984274 9.2 : 1 20,160 0.1036078 8.7 : 1
xAAA ${\displaystyle {\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 3}\end{matrix}}}$ 28 0.0001439 6,948.3 : 1 32 0.0001645 6,079.6 : 1
xAA* ${\displaystyle {\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 2}{51-4r \choose 1}\end{matrix}}}$ 798 0.0041011 242.8 : 1 720 0.0037003 269.3 : 1
??** ${\displaystyle {\begin{matrix}{4r-3 \choose 2}{51-4r \choose 2}\end{matrix}}}$ 69,426 0.3567993 1.8 : 1 55,440 0.2849214 2.5 : 1
?*** ${\displaystyle {\begin{matrix}{4r-3 \choose 1}{51-4r \choose 3}\end{matrix}}}$ 28,101 0.1444187 5.9 : 1 15,015 0.0771662 12.0 : 1
**** ${\displaystyle {\begin{matrix}{51-4r \choose 4}\end{matrix}}}$ 3,876 0.0199198 49.2 : 1 1,365 0.0070151 141.5 : 1
Total 123,348 0.6339192 0.6 : 1 95,124 0.4888683 1.0 : 1
Derivations for low hand shape xxx* (a low three of a kind and a high card) on the river
Hand to make Board Derivation 8-high qualifier (r = 8) 9-high qualifier (r = 9)
Combos Probability Odds Combos Probability Odds
Miss low ABCDE ${\displaystyle {\begin{matrix}{r-1 \choose 5}{4 \choose 1}^{5}\end{matrix}}}$ 21,504 0.0125585 78.6 : 1 57,344 0.0334894 28.9 : 1
AABCD ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}}$ 53,760 0.0313963 30.9 : 1 107,520 0.0627926 14.9 : 1
ABCD* ${\displaystyle {\begin{matrix}{r-1 \choose 4}{4 \choose 1}^{4}{51-4r \choose 1}\end{matrix}}}$ 170,240 0.0994216 9.1 : 1 268,800 0.1569815 5.4 : 1
AABBC ${\displaystyle {\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}}$ 15,120 0.0088302 112.2 : 1 24,192 0.0141283 69.8 : 1
AAABC ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}}$ 6,720 0.0039245 253.8 : 1 10,752 0.0062793 158.3 : 1
AABC* ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}{51-4r \choose 1}\end{matrix}}}$ 191,520 0.1118493 7.9 : 1 241,920 0.1412833 6.1 : 1
ABC** ${\displaystyle {\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}{51-4r \choose 2}\end{matrix}}}$ 383,040 0.2236986 3.5 : 1 376,320 0.2197741 3.6 : 1
xABCD ${\displaystyle {\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 4}{4 \choose 1}^{4}\end{matrix}}}$ 8,960 0.0052327 190.1 : 1 17,920 0.0104654 94.6 : 1
xABC* ${\displaystyle {\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 3}{4 \choose 1}^{3}{51-4r \choose 1}\end{matrix}}}$ 42,560 0.0248554 39.2 : 1 53,760 0.0313963 30.9 : 1
xAABC ${\displaystyle {\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}}$ 10,080 0.0058868 168.9 : 1 16,128 0.0094189 105.2 : 1
xAAB* ${\displaystyle {\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 1}\end{matrix}}}$ 19,152 0.0111849 88.4 : 1 20,160 0.0117736 83.9 : 1
xAAAB ${\displaystyle {\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}\end{matrix}}}$ 672 0.0003925 2,547.1 : 1 896 0.0005233 1,910.1 : 1
xAABB ${\displaystyle {\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 2}{4 \choose 2}^{2}\end{matrix}}}$ 756 0.0004415 2,264.0 : 1 1,008 0.0005887 1,697.7 : 1
xAB** ${\displaystyle {\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}{51-4r \choose 2}\end{matrix}}}$ 57,456 0.0335548 28.8 : 1 47,040 0.0274718 35.4 : 1
Total 981,540 0.5732277 0.7 : 1 1,243,760 0.7263663 0.4 : 1
No low AAAAB ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 4}{r-2 \choose 1}{4 \choose 1}\end{matrix}}}$ 168 0.0000981 10,191.3 : 1 224 0.0001308 7,643.2 : 1
AAAA* ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 4}{51-4r \choose 1}\end{matrix}}}$ 133 0.0000777 12,873.5 : 1 120 0.0000701 14,268.2 : 1
AAABB ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 2}\end{matrix}}}$ 1,008 0.0005887 1,697.7 : 1 1,344 0.0007849 1,273.0 : 1
AAAB* ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 1}\end{matrix}}}$ 12,768 0.0074566 133.1 : 1 13,440 0.0078491 126.4 : 1
AABB* ${\displaystyle {\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}{51-4r \choose 1}\end{matrix}}}$ 14,364 0.0083887 118.2 : 1 15,120 0.0088302 112.2 : 1
AAA** ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 3}{51-4r \choose 2}\end{matrix}}}$ 4,788 0.0027962 356.6 : 1 3,360 0.0019623 508.6 : 1
AAB** ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 2}\end{matrix}}}$ 172,368 0.1006644 8.9 : 1 141,120 0.0824153 11.1 : 1
xAAAA ${\displaystyle {\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 4}\end{matrix}}}$ 7 0.0000041 244,613.9 : 1 8 0.0000047 214,037.0 : 1
xAAA* ${\displaystyle {\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 3}{51-4r \choose 1}\end{matrix}}}$ 532 0.0003107 3,217.6 : 1 480 0.0002803 3,566.3 : 1
xAA** ${\displaystyle {\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 2}{51-4r \choose 2}\end{matrix}}}$ 7,182 0.0041943 237.4 : 1 5,040 0.0029434 338.7 : 1
??*** ${\displaystyle {\begin{matrix}{4r-3 \choose 2}{51-4r \choose 3}\end{matrix}}}$ 393,414 0.2297571 3.4 : 1 240,240 0.1403022 6.1 : 1
?**** ${\displaystyle {\begin{matrix}{4r-3 \choose 1}{51-4r \choose 4}\end{matrix}}}$ 112,404 0.0656449 14.2 : 1 45,045 0.0263067 37.0 : 1
***** ${\displaystyle {\begin{matrix}{51-4r \choose 5}\end{matrix}}}$ 11,628 0.0067909 146.3 : 1 3,003 0.0017538 569.2 : 1
Total 730,764 0.4267723 1.3 : 1 468,544 0.2736337 2.7 : 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 ${\displaystyle 52-4r-0=52-4r}$ high cards and ${\displaystyle 4r-4+0=4r-4}$ 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 make Board Derivation 8-high qualifier (r = 8) 9-high qualifier (r = 9)
Combos Probability Odds Combos Probability Odds
Miss low ABC ${\displaystyle {\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}\end{matrix}}}$ 2,240 0.1295097 6.7 : 1 3,584 0.2072155 3.8 : 1
No low AAA ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 3}\end{matrix}}}$ 28 0.0016189 616.7 : 1 32 0.0018501 539.5 : 1
AAB ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}}$ 1,008 0.0582794 16.2 : 1 1,344 0.0777058 11.9 : 1
??* ${\displaystyle {\begin{matrix}{4r-4 \choose 2}{52-4r \choose 1}\end{matrix}}}$ 7,560 0.4370953 1.3 : 1 7,936 0.4588344 1.2 : 1
?** ${\displaystyle {\begin{matrix}{4r-4 \choose 1}{52-4r \choose 2}\end{matrix}}}$ 5,320 0.3075856 2.3 : 1 3,840 0.2220167 3.5 : 1
*** ${\displaystyle {\begin{matrix}{52-4r \choose 3}\end{matrix}}}$ 1,140 0.0659112 14.2 : 1 560 0.0323774 29.9 : 1
Total 15,056 0.8704903 0.1 : 1 13,712 0.7927845 0.3 : 1
Derivations for low hand shape xxxx (a low four of a kind) on the turn
Hand to make Board Derivation 8-high qualifier (r = 8) 9-high qualifier (r = 9)
Combos Probability Odds Combos Probability Odds
Miss low ABCD ${\displaystyle {\begin{matrix}{r-1 \choose 4}{4 \choose 1}^{4}\end{matrix}}}$ 8,960 0.0460479 20.7 : 1 17,920 0.0920958 9.9 : 1
AABC ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}}$ 10,080 0.0518039 18.3 : 1 16,128 0.0828862 11.1 : 1
ABC* ${\displaystyle {\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}{52-4r \choose 1}\end{matrix}}}$ 44,800 0.2302395 3.3 : 1 57,344 0.2947065 2.4 : 1
Total 63,840 0.3280913 2.0 : 1 91,392 0.4696886 1.1 : 1
No low AAAA ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 4}\end{matrix}}}$ 7 0.0000360 27,796.1 : 1 8 0.0000411 24,321.5 : 1
AAAB ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}\end{matrix}}}$ 672 0.0034536 288.6 : 1 896 0.0046048 216.2 : 1
AABB ${\displaystyle {\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}\end{matrix}}}$ 756 0.0038853 256.4 : 1 1,008 0.0051804 192.0 : 1
AAA* ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 3}{52-4r \choose 1}\end{matrix}}}$ 560 0.0028780 346.5 : 1 512 0.0026313 379.0 : 1
AAB* ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}}$ 20,160 0.1036078 8.7 : 1 21,504 0.1105150 8.0 : 1
??** ${\displaystyle {\begin{matrix}{4r-4 \choose 2}{52-4r \choose 2}\end{matrix}}}$ 71,820 0.3691027 1.7 : 1 59,520 0.3058896 2.3 : 1
?*** ${\displaystyle {\begin{matrix}{4r-4 \choose 1}{52-4r \choose 3}\end{matrix}}}$ 31,920 0.1640456 5.1 : 1 17,920 0.0920958 9.9 : 1
**** ${\displaystyle {\begin{matrix}{52-4r \choose 4}\end{matrix}}}$ 4,845 0.0248998 39.2 : 1 1,820 0.0093535 105.9 : 1
Total 130,740 0.6719087 0.5 : 1 103,188 0.5303114 0.9 : 1
Derivations for low hand shape xxxx (a low four of a kind) on the river
Hand to make Board Derivation 8-high qualifier (r = 8) 9-high qualifier (r = 9)
Combos Probability Odds Combos Probability Odds
Miss low ABCDE ${\displaystyle {\begin{matrix}{r-1 \choose 5}{4 \choose 1}^{5}\end{matrix}}}$ 21,504 0.0125585 78.6 : 1 57,344 0.0334894 28.9 : 1
AABCD ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}}$ 53,760 0.0313963 30.9 : 1 107,520 0.0627926 14.9 : 1
ABCD* ${\displaystyle {\begin{matrix}{r-1 \choose 4}{4 \choose 1}^{4}{52-4r \choose 1}\end{matrix}}}$ 179,200 0.1046543 8.6 : 1 286,720 0.1674469 5.0 : 1
AABBC ${\displaystyle {\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}}$ 15,120 0.0088302 112.2 : 1 24,192 0.0141283 69.8 : 1
AAABC ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}}$ 6,720 0.0039245 253.8 : 1 10,752 0.0062793 158.3 : 1
AABC* ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}{52-4r \choose 1}\end{matrix}}}$ 201,600 0.1177361 7.5 : 1 258,048 0.1507022 5.6 : 1
ABC** ${\displaystyle {\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}{52-4r \choose 2}\end{matrix}}}$ 425,600 0.2485540 3.0 : 1 430,080 0.2511704 3.0 : 1
Total 903,504 0.5276540 0.9 : 1 1,174,656 0.6860090 0.5 : 1
No low AAAAB ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 4}{r-2 \choose 1}{4 \choose 1}\end{matrix}}}$ 168 0.0000981 10,191.3 : 1 224 0.0001308 7,643.2 : 1
AAAA* ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 4}{52-4r \choose 1}\end{matrix}}}$ 140 0.0000818 12,229.7 : 1 128 0.0000748 13,376.4 : 1
AAABB ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 2}\end{matrix}}}$ 1,008 0.0005887 1,697.7 : 1 1,344 0.0007849 1,273.0 : 1
AAAB* ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}}$ 13,440 0.0078491 126.4 : 1 14,336 0.0083723 118.4 : 1
AABB* ${\displaystyle {\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}{52-4r \choose 1}\end{matrix}}}$ 15,120 0.0088302 112.2 : 1 16,128 0.0094189 105.2 : 1
AAA** ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 3}{52-4r \choose 2}\end{matrix}}}$ 5,320 0.0031069 320.9 : 1 3,840 0.0022426 444.9 : 1
AAB** ${\displaystyle {\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 2}\end{matrix}}}$ 191,520 0.1118493 7.9 : 1 161,280 0.0941889 9.6 : 1
??*** ${\displaystyle {\begin{matrix}{4r-4 \choose 2}{52-4r \choose 3}\end{matrix}}}$ 430,920 0.2516609 3.0 : 1 277,760 0.1622142 5.2 : 1
?**** ${\displaystyle {\begin{matrix}{4r-4 \choose 1}{52-4r \choose 4}\end{matrix}}}$ 135,660 0.0792266 11.6 : 1 58,240 0.0340127 28.4 : 1
***** ${\displaystyle {\begin{matrix}{52-4r \choose 5}\end{matrix}}}$ 15,504 0.0090545 109.4 : 1 4,368 0.0025509 391.0 : 1
Total 808,800 0.4723460 1.1 : 1 537,648 0.3139910 2.2 : 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 ${\displaystyle 52-4r-2=50-4r}$ high cards and ${\displaystyle 4r-4+2=4r-2}$ 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 make Board Derivation 8-high qualifier (r = 8) 9-high qualifier (r = 9)
Combos Probability Odds Combos Probability Odds
Make low ABC ${\displaystyle {\begin{matrix}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}}$ 1,280 0.0740056 12.5 : 1 2,240 0.1295097 6.7 : 1
Miss low xAB ${\displaystyle {\begin{matrix}{2 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}}$ 1,440 0.0832562 11.0 : 1 2,016 0.1165587 7.6 : 1
xyA ${\displaystyle {\begin{matrix}{2 \choose 2}{3 \choose 1}^{2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}}$ 216 0.0124884 79.1 : 1 252 0.0145698 67.6 : 1
Total 1,656 0.0957447 9.4 : 1 2,268 0.1311286 6.6 : 1
No low AAA ${\displaystyle {\begin{matrix}{r-2 \choose 1}{4 \choose 3}\end{matrix}}}$ 24 0.0013876 719.7 : 1 28 0.0016189 616.7 : 1
AAB ${\displaystyle {\begin{matrix}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}}$ 720 0.0416281 23.0 : 1 1,008 0.0582794 16.2 : 1
xxx ${\displaystyle {\begin{matrix}{2 \choose 1}{3 \choose 3}\end{matrix}}}$ 2 0.0001156 8,647.0 : 1 2 0.0001156 8,647.0 : 1
xxy ${\displaystyle {\begin{matrix}{2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}\end{matrix}}}$ 18 0.0010407 959.9 : 1 18 0.0010407 959.9 : 1
xxA ${\displaystyle {\begin{matrix}{2 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}}$ 144 0.0083256 119.1 : 1 168 0.0097132 102.0 : 1
xAA ${\displaystyle {\begin{matrix}{2 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}\end{matrix}}}$ 216 0.0124884 79.1 : 1 252 0.0145698 67.6 : 1
??* ${\displaystyle {\begin{matrix}{4r-2 \choose 2}{50-4r \choose 1}\end{matrix}}}$ 7,830 0.4527058 1.2 : 1 7,854 0.4540934 1.2 : 1
?** ${\displaystyle {\begin{matrix}{4r-2 \choose 1}{50-4r \choose 2}\end{matrix}}}$ 4,590 0.2653793 2.8 : 1 3,094 0.1788853 4.6 : 1
*** ${\displaystyle {\begin{matrix}{50-4r \choose 3}\end{matrix}}}$ 816 0.0471785 20.2 : 1 364 0.0210453 46.5 : 1
Total 14,360 0.8302498 0.2 : 1 12,788 0.7393617 0.4 : 1
Derivations for low hand shape xy** (two low ranks and two high cards) on the turn
Hand to make Board Derivation 8-high qualifier (r = 8) 9-high qualifier (r = 9)
Combos Probability Odds Combos Probability Odds
Make low ABCD ${\displaystyle {\begin{matrix}{r-2 \choose 4}{4 \choose 1}^{4}\end{matrix}}}$ 3,840 0.0197348 49.7 : 1 8,960 0.0460479 20.7 : 1
AABC ${\displaystyle {\begin{matrix}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^{2}\end{matrix}}}$ 5,760 0.0296022 32.8 : 1 10,080 0.0518039 18.3 : 1
ABC* ${\displaystyle {\begin{matrix}{r-2 \choose 3}{4 \choose 1}^{3}{50-4r \choose 1}\end{matrix}}}$ 23,040 0.1184089 7.4 : 1 31,360 0.1611676 5.2 : 1
xABC ${\displaystyle {\begin{matrix}{2 \choose 1}{3 \choose 1}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}}$ 7,680 0.0394696 24.3 : 1 13,440 0.0690718 13.5 : 1
Total 40,320 0.2072155 3.8 : 1 63,840 0.3280913 2.0 : 1
Miss low xxyA ${\displaystyle {\begin{matrix}{2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1}\end{matrix}}}$ 432 0.0022202 449.4 : 1 504 0.0025902 385.1 : 1
xxAB ${\displaystyle {\begin{matrix}{2 \choose 1}{3 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}}$ 1,440 0.0074006 134.1 : 1 2,016 0.0103608 95.5 : 1
xyAA