# 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 $h$ 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:

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

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 make Board Derivation 8-high qualifier (r = 8) 9-high qualifier (r = 9)
Combos Probability Odds Combos Probability Odds
Miss low ABC $\begin{matrix} {r \choose 3}{4 \choose 1}^3 \end{matrix}$ 3,584 0.2072155 3.8 : 1 5,376 0.3108233 2.2 : 1
No low AAA $\begin{matrix} {r \choose 1}{4 \choose 3} \end{matrix}$ 32 0.0018501 539.5 : 1 36 0.0020814 479.4 : 1
AAB $\begin{matrix} {r \choose 1}{4 \choose 2}{r-1 \choose 1}{4 \choose 1} \end{matrix}$ 1,344 0.0777058 11.9 : 1 1,728 0.0999075 9.0 : 1
??* $\begin{matrix} {4r \choose 2}{48-4r \choose 1} \end{matrix}$ 7,168 0.4144311 1.4 : 1 6,912 0.3996300 1.5 : 1
?** $\begin{matrix} {4r \choose 1}{48-4r \choose 2} \end{matrix}$ 7,936 0.4588344 1.2 : 1 7,560 0.4370953 1.3 : 1
*** $\begin{matrix} {48-4r \choose 3} \end{matrix}$ 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 $\begin{matrix} {r \choose 4}{4 \choose 1}^4 \end{matrix}$ 17,920 0.0920958 9.9 : 1 32,256 0.1657724 5.0 : 1
AABC $\begin{matrix} {r \choose 1}{4 \choose 2}{r-1 \choose 2}{4 \choose 1}^2 \end{matrix}$ 16,128 0.0828862 11.1 : 1 24,192 0.1243293 7.0 : 1
ABC* $\begin{matrix} {r \choose 3}{4 \choose 1}^3{48-4r \choose 1} \end{matrix}$ 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 $\begin{matrix} {r \choose 1}{4 \choose 4} \end{matrix}$ 8 0.0000411 24,321.5 : 1 9 0.0000463 21,619.0 : 1
AAAB $\begin{matrix} {r \choose 1}{4 \choose 3}{r-1 \choose 1}{4 \choose 1} \end{matrix}$ 896 0.0046048 216.2 : 1 1,152 0.0059204 167.9 : 1
AABB $\begin{matrix} {r \choose 2}{4 \choose 2}^2 \end{matrix}$ 1,008 0.0051804 192.0 : 1 1,296 0.0066605 149.1 : 1
AAA* $\begin{matrix} {r \choose 1}{4 \choose 3}{48-4r \choose 1} \end{matrix}$ 512 0.0026313 379.0 : 1 432 0.0022202 449.4 : 1
AAB* $\begin{matrix} {r \choose 1}{4 \choose 2}{r-1 \choose 1}{4 \choose 1}{48-4r \choose 1} \end{matrix}$ 21,504 0.1105150 8.0 : 1 20,736 0.1065680 8.4 : 1
??** $\begin{matrix} {4r \choose 2}{48-4r \choose 2} \end{matrix}$ 59,520 0.3058896 2.3 : 1 41,580 0.2136910 3.7 : 1
?*** $\begin{matrix} {4r \choose 1}{48-4r \choose 3} \end{matrix}$ 17,920 0.0920958 9.9 : 1 7,920 0.0407031 23.6 : 1
**** $\begin{matrix} {48-4r \choose 4} \end{matrix}$ 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 $\begin{matrix} {r \choose 5}{4 \choose 1}^5 \end{matrix}$ 57,344 0.0334894 28.9 : 1 129,024 0.0753511 12.3 : 1
AABCD $\begin{matrix} {r \choose 1}{4 \choose 2}{r-1 \choose 3}{4 \choose 1}^3 \end{matrix}$ 107,520 0.0627926 14.9 : 1 193,536 0.1130267 7.8 : 1
ABCD* $\begin{matrix} {r \choose 4}{4 \choose 1}^4{48-4r \choose 1} \end{matrix}$ 286,720 0.1674469 5.0 : 1 387,072 0.2260533 3.4 : 1
AABBC $\begin{matrix} {r \choose 2}{4 \choose 2}^2{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 24,192 0.0141283 69.8 : 1 36,288 0.0211925 46.2 : 1
AAABC $\begin{matrix} {r \choose 1}{4 \choose 3}{r-1 \choose 2}{4 \choose 1}^2 \end{matrix}$ 10,752 0.0062793 158.3 : 1 16,128 0.0094189 105.2 : 1
AABC* $\begin{matrix} {r \choose 1}{4 \choose 2}{r-1 \choose 2}{4 \choose 1}^2{48-4r \choose 1} \end{matrix}$ 258,048 0.1507022 5.6 : 1 290,304 0.1695400 4.9 : 1
ABC** $\begin{matrix} {r \choose 3}{4 \choose 1}^3{48-4r \choose 2} \end{matrix}$ 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 $\begin{matrix} {r \choose 1}{4 \choose 4}{r-1 \choose 1}{4 \choose 1} \end{matrix}$ 224 0.0001308 7,643.2 : 1 288 0.0001682 5,944.5 : 1
AAAA* $\begin{matrix} {r \choose 1}{4 \choose 4}{48-4r \choose 1} \end{matrix}$ 128 0.0000748 13,376.4 : 1 108 0.0000631 15,853.7 : 1
AAABB $\begin{matrix} {r \choose 1}{4 \choose 3}{r-1 \choose 1}{4 \choose 2} \end{matrix}$ 1,344 0.0007849 1,273.0 : 1 1,728 0.0010092 989.9 : 1
AAAB* $\begin{matrix} {r \choose 1}{4 \choose 3}{r-1 \choose 1}{4 \choose 1}{48-4r \choose 1} \end{matrix}$ 14,336 0.0083723 118.4 : 1 13,824 0.0080733 122.9 : 1
AABB* $\begin{matrix} {r \choose 2}{4 \choose 2}^2{48-4r \choose 1} \end{matrix}$ 16,128 0.0094189 105.2 : 1 15,552 0.0090825 109.1 : 1
AAA** $\begin{matrix} {r \choose 1}{4 \choose 3}{48-4r \choose 2} \end{matrix}$ 3,840 0.0022426 444.9 : 1 2,376 0.0013876 719.7 : 1
AAB** $\begin{matrix} {r \choose 1}{4 \choose 2}{r-1 \choose 1}{4 \choose 1}{48-4r \choose 2} \end{matrix}$ 161,280 0.0941889 9.6 : 1 114,048 0.0666050 14.0 : 1
??*** $\begin{matrix} {4r \choose 2}{48-4r \choose 3} \end{matrix}$ 277,760 0.1622142 5.2 : 1 138,600 0.0809436 11.4 : 1
?**** $\begin{matrix} {4r \choose 1}{48-4r \choose 4} \end{matrix}$ 58,240 0.0340127 28.4 : 1 17,820 0.0104070 95.1 : 1
***** $\begin{matrix} {48-4r \choose 5} \end{matrix}$ 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 $52 - 4r - 3 = 49 - 4r$ 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 make Board Derivation 8-high qualifier (r = 8) 9-high qualifier (r = 9)
Combos Probability Odds Combos Probability Odds
Miss low ABC $\begin{matrix} {r-1 \choose 3}{4 \choose 1}^3 \end{matrix}$ 2,240 0.1295097 6.7 : 1 3,584 0.2072155 3.8 : 1
xAB $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 1}^2 \end{matrix}$ 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 $\begin{matrix} {r-1 \choose 1}{4 \choose 3} \end{matrix}$ 28 0.0016189 616.7 : 1 32 0.0018501 539.5 : 1
AAB $\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 1,008 0.0582794 16.2 : 1 1,344 0.0777058 11.9 : 1
xxx $\begin{matrix} {1 \choose 1}{3 \choose 3} \end{matrix}$ 1 0.0000578 17,295.0 : 1 1 0.0000578 17,295.0 : 1
xxA $\begin{matrix} {1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 1} \end{matrix}$ 84 0.0048566 204.9 : 1 96 0.0055504 179.2 : 1
xAA $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2} \end{matrix}$ 126 0.0072849 136.3 : 1 144 0.0083256 119.1 : 1
??* $\begin{matrix} {4r-1 \choose 2}{49-4r \choose 1} \end{matrix}$ 7,905 0.4570421 1.2 : 1 7,735 0.4472132 1.2 : 1
?** $\begin{matrix} {4r-1 \choose 1}{49-4r \choose 2} \end{matrix}$ 4,216 0.2437558 3.1 : 1 2,730 0.1578400 5.3 : 1
*** $\begin{matrix} {49-4r \choose 3} \end{matrix}$ 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 $\begin{matrix} {r-1 \choose 4}{4 \choose 1}^4 \end{matrix}$ 8,960 0.0460479 20.7 : 1 17,920 0.0920958 9.9 : 1
AABC $\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$ 10,080 0.0518039 18.3 : 1 16,128 0.0828862 11.1 : 1
ABC* $\begin{matrix} {r-1 \choose 3}{4 \choose 1}^3{49-4r \choose 1} \end{matrix}$ 38,080 0.1957036 4.1 : 1 46,592 0.2394491 3.2 : 1
xxAB $\begin{matrix} {1 \choose 1}{3 \choose 2}{r-1 \choose 2}{4 \choose 1}^2 \end{matrix}$ 1,008 0.0051804 192.0 : 1 1,344 0.0069072 143.8 : 1
xABC $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 3}{4 \choose 1}^3 \end{matrix}$ 6,720 0.0345359 28.0 : 1 10,752 0.0552575 17.1 : 1
xAAB $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 3,024 0.0155412 63.3 : 1 4,032 0.0207216 47.3 : 1
xAB* $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 1}^2{49-4r \choose 1} \end{matrix}$ 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 $\begin{matrix} {r-1 \choose 1}{4 \choose 4} \end{matrix}$ 7 0.0000360 27,796.1 : 1 8 0.0000411 24,321.5 : 1
AAAB $\begin{matrix} {r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 672 0.0034536 288.6 : 1 896 0.0046048 216.2 : 1
AABB $\begin{matrix} {r-1 \choose 2}{4 \choose 2}^2 \end{matrix}$ 756 0.0038853 256.4 : 1 1,008 0.0051804 192.0 : 1
AAA* $\begin{matrix} {r-1 \choose 1}{4 \choose 3}{49-4r \choose 1} \end{matrix}$ 476 0.0024463 407.8 : 1 416 0.0021379 466.7 : 1
AAB* $\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{49-4r \choose 1} \end{matrix}$ 17,136 0.0880666 10.4 : 1 17,472 0.0897934 10.1 : 1
xxx* $\begin{matrix} {1 \choose 1}{3 \choose 3}{49-4r \choose 1} \end{matrix}$ 17 0.0000874 11,444.9 : 1 13 0.0000668 14,966.7 : 1
xxxA $\begin{matrix} {1 \choose 1}{3 \choose 3}{r-1 \choose 1}{4 \choose 1} \end{matrix}$ 28 0.0001439 6,948.3 : 1 32 0.0001645 6,079.6 : 1
xxAA $\begin{matrix} {1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 2} \end{matrix}$ 126 0.0006475 1,543.3 : 1 144 0.0007401 1,350.3 : 1
xxA* $\begin{matrix} {1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 1}{49-4r \choose 1} \end{matrix}$ 1,428 0.0073389 135.3 : 1 1,248 0.0064138 154.9 : 1
xAAA $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 3} \end{matrix}$ 84 0.0004317 2,315.4 : 1 96 0.0004934 2,025.9 : 1
xAA* $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{49-4r \choose 1} \end{matrix}$ 2,142 0.0110083 89.8 : 1 1,872 0.0096207 102.9 : 1
??** $\begin{matrix} {4r-1 \choose 2}{49-4r \choose 2} \end{matrix}$ 63,240 0.3250077 2.1 : 1 46,410 0.2385137 3.2 : 1
?*** $\begin{matrix} {4r-1 \choose 1}{49-4r \choose 3} \end{matrix}$ 21,080 0.1083359 8.2 : 1 10,010 0.0514441 18.4 : 1
**** $\begin{matrix} {49-4r \choose 4} \end{matrix}$ 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 $\begin{matrix} {r-1 \choose 5}{4 \choose 1}^5 \end{matrix}$ 21,504 0.0125585 78.6 : 1 57,344 0.0334894 28.9 : 1
AABCD $\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$ 53,760 0.0313963 30.9 : 1 107,520 0.0627926 14.9 : 1
ABCD* $\begin{matrix} {r-1 \choose 4}{4 \choose 1}^4{49-4r \choose 1} \end{matrix}$ 152,320 0.0889562 10.2 : 1 232,960 0.1360506 6.4 : 1
AABBC $\begin{matrix} {r-1 \choose 2}{4 \choose 2}^2{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 15,120 0.0088302 112.2 : 1 24,192 0.0141283 69.8 : 1
AAABC $\begin{matrix} {r-1 \choose 1}{4 \choose 3}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$ 6,720 0.0039245 253.8 : 1 10,752 0.0062793 158.3 : 1
AABC* $\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^2{49-4r \choose 1} \end{matrix}$ 171,360 0.1000757 9.0 : 1 209,664 0.1224455 7.2 : 1
ABC** $\begin{matrix} {r-1 \choose 3}{4 \choose 1}^3{49-4r \choose 2} \end{matrix}$ 304,640 0.1779123 4.6 : 1 279,552 0.1632607 5.1 : 1
xABCD $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 4}{4 \choose 1}^4 \end{matrix}$ 26,880 0.0156981 62.7 : 1 53,760 0.0313963 30.9 : 1
xABC* $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 3}{4 \choose 1}^3{49-4r \choose 1} \end{matrix}$ 114,240 0.0667171 14.0 : 1 139,776 0.0816304 11.3 : 1
xAABC $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$ 30,240 0.0176604 55.6 : 1 48,384 0.0282567 34.4 : 1
xAAB* $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{49-4r \choose 1} \end{matrix}$ 51,408 0.0300227 32.3 : 1 52,416 0.0306114 31.7 : 1
xAAAB $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 2,016 0.0011774 848.4 : 1 2,688 0.0015698 636.0 : 1
xAABB $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 2}^2 \end{matrix}$ 2,268 0.0013245 754.0 : 1 3,024 0.0017660 565.2 : 1
xAB** $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 1}^2{49-4r \choose 2} \end{matrix}$ 137,088 0.0800605 11.5 : 1 104,832 0.0612228 15.3 : 1
xxABC $\begin{matrix} {1 \choose 1}{3 \choose 2}{r-1 \choose 3}{4 \choose 1}^3 \end{matrix}$ 6,720 0.0039245 253.8 : 1 10,752 0.0062793 158.3 : 1
xxAB* $\begin{matrix} {1 \choose 1}{3 \choose 2}{r-1 \choose 2}{4 \choose 1}^2{49-4r \choose 1} \end{matrix}$ 17,136 0.0100076 98.9 : 1 17,472 0.0102038 97.0 : 1
xxAAB $\begin{matrix} {1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 3,024 0.0017660 565.2 : 1 4,032 0.0023547 423.7 : 1
xxxAB $\begin{matrix} {1 \choose 1}{3 \choose 3}{r-1 \choose 2}{4 \choose 1}^2 \end{matrix}$ 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 $\begin{matrix} {r-1 \choose 1}{4 \choose 4}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 168 0.0000981 10,191.3 : 1 224 0.0001308 7,643.2 : 1
AAAA* $\begin{matrix} {r-1 \choose 1}{4 \choose 4}{49-4r \choose 1} \end{matrix}$ 119 0.0000695 14,388.1 : 1 104 0.0000607 16,463.5 : 1
AAABB $\begin{matrix} {r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 2} \end{matrix}$ 1,008 0.0005887 1,697.7 : 1 1,344 0.0007849 1,273.0 : 1
AAAB* $\begin{matrix} {r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}{49-4r \choose 1} \end{matrix}$ 11,424 0.0066717 148.9 : 1 11,648 0.0068025 146.0 : 1
AABB* $\begin{matrix} {r-1 \choose 2}{4 \choose 2}^2{49-4r \choose 1} \end{matrix}$ 12,852 0.0075057 132.2 : 1 13,104 0.0076528 129.7 : 1
AAA** $\begin{matrix} {r-1 \choose 1}{4 \choose 3}{49-4r \choose 2} \end{matrix}$ 3,808 0.0022239 448.7 : 1 2,496 0.0014577 685.0 : 1
AAB** $\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{49-4r \choose 2} \end{matrix}$ 137,088 0.0800605 11.5 : 1 104,832 0.0612228 15.3 : 1
xxxAA $\begin{matrix} {1 \choose 1}{3 \choose 3}{r-1 \choose 1}{4 \choose 2} \end{matrix}$ 42 0.0000245 40,768.1 : 1 48 0.0000280 35,672.0 : 1
xxxA* $\begin{matrix} {1 \choose 1}{3 \choose 3}{r-1 \choose 1}{4 \choose 1}{49-4r \choose 1} \end{matrix}$ 476 0.0002780 3,596.3 : 1 416 0.0002429 4,115.1 : 1
xxx** $\begin{matrix} {1 \choose 1}{3 \choose 3}{49-4r \choose 2} \end{matrix}$ 136 0.0000794 12,589.5 : 1 78 0.0000456 21,951.6 : 1
xxAAA $\begin{matrix} {1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 3} \end{matrix}$ 84 0.0000491 20,383.6 : 1 96 0.0000561 17,835.5 : 1
xxAA* $\begin{matrix} {1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 2}{49-4r \choose 1} \end{matrix}$ 2,142 0.0012509 798.4 : 1 1,872 0.0010933 913.7 : 1
xxA** $\begin{matrix} {1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 1}{49-4r \choose 2} \end{matrix}$ 11,424 0.0066717 148.9 : 1 7,488 0.0043731 227.7 : 1
xAAAA $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 4} \end{matrix}$ 21 0.0000123 81,537.3 : 1 24 0.0000140 71,345.0 : 1
xAAA* $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 3}{49-4r \choose 1} \end{matrix}$ 1,428 0.0008340 1,198.1 : 1 1,248 0.0007288 1,371.0 : 1
xAA** $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{49-4r \choose 2} \end{matrix}$ 17,136 0.0100076 98.9 : 1 11,232 0.0065596 151.4 : 1
??*** $\begin{matrix} {4r-1 \choose 2}{49-4r \choose 3} \end{matrix}$ 316,200 0.1846635 4.4 : 1 170,170 0.0993807 9.1 : 1
?**** $\begin{matrix} {4r-1 \choose 1}{49-4r \choose 4} \end{matrix}$ 73,780 0.0430881 22.2 : 1 25,025 0.0146148 67.4 : 1
***** $\begin{matrix} {49-4r \choose 5} \end{matrix}$ 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 $52 - 4r - 2 = 50 - 4r$ 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 make Board Derivation 8-high qualifier (r = 8) 9-high qualifier (r = 9)
Combos Probability Odds Combos Probability Odds
Miss low ABC $\begin{matrix} {r-1 \choose 3}{4 \choose 1}^3 \end{matrix}$ 2,240 0.1295097 6.7 : 1 3,584 0.2072155 3.8 : 1
xAB $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 1}^2 \end{matrix}$ 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 $\begin{matrix} {r-1 \choose 1}{4 \choose 3} \end{matrix}$ 28 0.0016189 616.7 : 1 32 0.0018501 539.5 : 1
AAB $\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 1,008 0.0582794 16.2 : 1 1,344 0.0777058 11.9 : 1
xxA $\begin{matrix} {1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 1} \end{matrix}$ 28 0.0016189 616.7 : 1 32 0.0018501 539.5 : 1
xAA $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2} \end{matrix}$ 84 0.0048566 204.9 : 1 96 0.0055504 179.2 : 1
??* $\begin{matrix} {4r-2 \choose 2}{50-4r \choose 1} \end{matrix}$ 7,830 0.4527058 1.2 : 1 7,854 0.4540934 1.2 : 1
?** $\begin{matrix} {4r-2 \choose 1}{50-4r \choose 2} \end{matrix}$ 4,590 0.2653793 2.8 : 1 3,094 0.1788853 4.6 : 1
*** $\begin{matrix} {50-4r \choose 3} \end{matrix}$ 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 $\begin{matrix} {r-1 \choose 4}{4 \choose 1}^4 \end{matrix}$ 8,960 0.0460479 20.7 : 1 17,920 0.0920958 9.9 : 1
AABC $\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$ 10,080 0.0518039 18.3 : 1 16,128 0.0828862 11.1 : 1
ABC* $\begin{matrix} {r-1 \choose 3}{4 \choose 1}^3{50-4r \choose 1} \end{matrix}$ 40,320 0.2072155 3.8 : 1 50,176 0.2578682 2.9 : 1
xxAB $\begin{matrix} {1 \choose 1}{2 \choose 2}{r-1 \choose 2}{4 \choose 1}^2 \end{matrix}$ 336 0.0017268 578.1 : 1 448 0.0023024 433.3 : 1
xABC $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 3}{4 \choose 1}^3 \end{matrix}$ 4,480 0.0230239 42.4 : 1 7,168 0.0368383 26.1 : 1
xAAB $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 2,016 0.0103608 95.5 : 1 2,688 0.0138144 71.4 : 1
xAB* $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 1}^2{50-4r \choose 1} \end{matrix}$ 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 $\begin{matrix} {r-1 \choose 1}{4 \choose 4} \end{matrix}$ 7 0.0000360 27,796.1 : 1 8 0.0000411 24,321.5 : 1
AAAB $\begin{matrix} {r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 672 0.0034536 288.6 : 1 896 0.0046048 216.2 : 1
AABB $\begin{matrix} {r-1 \choose 2}{4 \choose 2}^2 \end{matrix}$ 756 0.0038853 256.4 : 1 1,008 0.0051804 192.0 : 1
AAA* $\begin{matrix} {r-1 \choose 1}{4 \choose 3}{50-4r \choose 1} \end{matrix}$ 504 0.0025902 385.1 : 1 448 0.0023024 433.3 : 1
AAB* $\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1} \end{matrix}$ 18,144 0.0932470 9.7 : 1 18,816 0.0967006 9.3 : 1
xxAA $\begin{matrix} {1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 2} \end{matrix}$ 42 0.0002158 4,631.9 : 1 48 0.0002467 4,052.8 : 1
xxA* $\begin{matrix} {1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 1}{50-4r \choose 1} \end{matrix}$ 504 0.0025902 385.1 : 1 448 0.0023024 433.3 : 1
xAAA $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 3} \end{matrix}$ 56 0.0002878 3,473.6 : 1 64 0.0003289 3,039.3 : 1
xAA* $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}{50-4r \choose 1} \end{matrix}$ 1,512 0.0077706 127.7 : 1 1,344 0.0069072 143.8 : 1
??** $\begin{matrix} {4r-2 \choose 2}{50-4r \choose 2} \end{matrix}$ 66,555 0.3420444 1.9 : 1 51,051 0.2623651 2.8 : 1
?*** $\begin{matrix} {4r-2 \choose 1}{50-4r \choose 3} \end{matrix}$ 24,480 0.1258094 6.9 : 1 12,376 0.0636037 14.7 : 1
**** $\begin{matrix} {50-4r \choose 4} \end{matrix}$ 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 $\begin{matrix} {r-1 \choose 5}{4 \choose 1}^5 \end{matrix}$ 21,504 0.0125585 78.6 : 1 57,344 0.0334894 28.9 : 1
AABCD $\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$ 53,760 0.0313963 30.9 : 1 107,520 0.0627926 14.9 : 1
ABCD* $\begin{matrix} {r-1 \choose 4}{4 \choose 1}^4{50-4r \choose 1} \end{matrix}$ 161,280 0.0941889 9.6 : 1 250,880 0.1465160 5.8 : 1
AABBC $\begin{matrix} {r-1 \choose 2}{4 \choose 2}^2{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 15,120 0.0088302 112.2 : 1 24,192 0.0141283 69.8 : 1
AAABC $\begin{matrix} {r-1 \choose 1}{4 \choose 3}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$ 6,720 0.0039245 253.8 : 1 10,752 0.0062793 158.3 : 1
AABC* $\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^2{50-4r \choose 1} \end{matrix}$ 181,440 0.1059625 8.4 : 1 225,792 0.1318644 6.6 : 1
ABC** $\begin{matrix} {r-1 \choose 3}{4 \choose 1}^3{50-4r \choose 2} \end{matrix}$ 342,720 0.2001514 4.0 : 1 326,144 0.1904709 4.3 : 1
xABCD $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 4}{4 \choose 1}^4 \end{matrix}$ 17,920 0.0104654 94.6 : 1 35,840 0.0209309 46.8 : 1
xABC* $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 3}{4 \choose 1}^3{50-4r \choose 1} \end{matrix}$ 80,640 0.0470944 20.2 : 1 100,352 0.0586064 16.1 : 1
xAABC $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$ 20,160 0.0117736 83.9 : 1 32,256 0.0188378 52.1 : 1
xAAB* $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1} \end{matrix}$ 36,288 0.0211925 46.2 : 1 37,632 0.0219774 44.5 : 1
xAAAB $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 1,344 0.0007849 1,273.0 : 1 1,792 0.0010465 954.5 : 1
xAABB $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 2}^2 \end{matrix}$ 1,512 0.0008830 1,131.5 : 1 2,016 0.0011774 848.4 : 1
xAB** $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 1}^2{50-4r \choose 2} \end{matrix}$ 102,816 0.0600454 15.7 : 1 81,536 0.0476177 20.0 : 1
xxABC $\begin{matrix} {1 \choose 1}{2 \choose 2}{r-1 \choose 3}{4 \choose 1}^3 \end{matrix}$ 2,240 0.0013082 763.4 : 1 3,584 0.0020931 476.8 : 1
xxAB* $\begin{matrix} {1 \choose 1}{2 \choose 2}{r-1 \choose 2}{4 \choose 1}^2{50-4r \choose 1} \end{matrix}$ 6,048 0.0035321 282.1 : 1 6,272 0.0036629 272.0 : 1
xxAAB $\begin{matrix} {1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 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 $\begin{matrix} {r-1 \choose 1}{4 \choose 4}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 168 0.0000981 10,191.3 : 1 224 0.0001308 7,643.2 : 1
AAAA* $\begin{matrix} {r-1 \choose 1}{4 \choose 4}{50-4r \choose 1} \end{matrix}$ 126 0.0000736 13,588.7 : 1 112 0.0000654 15,287.4 : 1
AAABB $\begin{matrix} {r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 2} \end{matrix}$ 1,008 0.0005887 1,697.7 : 1 1,344 0.0007849 1,273.0 : 1
AAAB* $\begin{matrix} {r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1} \end{matrix}$ 12,096 0.0070642 140.6 : 1 12,544 0.0073258 135.5 : 1
AABB* $\begin{matrix} {r-1 \choose 2}{4 \choose 2}^2{50-4r \choose 1} \end{matrix}$ 13,608 0.0079472 124.8 : 1 14,112 0.0082415 120.3 : 1
AAA** $\begin{matrix} {r-1 \choose 1}{4 \choose 3}{50-4r \choose 2} \end{matrix}$ 4,284 0.0025019 398.7 : 1 2,912 0.0017006 587.0 : 1
AAB** $\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 2} \end{matrix}$ 154,224 0.0900681 10.1 : 1 122,304 0.0714266 13.0 : 1
xxAAA $\begin{matrix} {1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 3} \end{matrix}$ 28 0.0000164 61,152.7 : 1 32 0.0000187 53,508.5 : 1
xxAA* $\begin{matrix} {1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 2}{50-4r \choose 1} \end{matrix}$ 756 0.0004415 2,264.0 : 1 672 0.0003925 2,547.1 : 1
xxA** $\begin{matrix} {1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 1}{50-4r \choose 2} \end{matrix}$ 4,284 0.0025019 398.7 : 1 2,912 0.0017006 587.0 : 1
xAAAA $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 4} \end{matrix}$ 14 0.0000082 122,306.4 : 1 16 0.0000093 107,018.0 : 1
xAAA* $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 3}{50-4r \choose 1} \end{matrix}$ 1,008 0.0005887 1,697.7 : 1 896 0.0005233 1,910.1 : 1
xAA** $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}{50-4r \choose 2} \end{matrix}$ 12,852 0.0075057 132.2 : 1 8,736 0.0051019 195.0 : 1
??*** $\begin{matrix} {4r-2 \choose 2}{50-4r \choose 3} \end{matrix}$ 354,960 0.2072996 3.8 : 1 204,204 0.1192569 7.4 : 1
?**** $\begin{matrix} {4r-2 \choose 1}{50-4r \choose 4} \end{matrix}$ 91,800 0.0536120 17.7 : 1 34,034 0.0198761 49.3 : 1
***** $\begin{matrix} {50-4r \choose 5} \end{matrix}$ 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 $52 - 4r - 1 = 51 - 4r$ 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 make Board Derivation 8-high qualifier (r = 8) 9-high qualifier (r = 9)
Combos Probability Odds Combos Probability Odds
Miss low ABC $\begin{matrix} {r-1 \choose 3}{4 \choose 1}^3 \end{matrix}$ 2,240 0.1295097 6.7 : 1 3,584 0.2072155 3.8 : 1
xAB $\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 2}{4 \choose 1}^2 \end{matrix}$ 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 $\begin{matrix} {r-1 \choose 1}{4 \choose 3} \end{matrix}$ 28 0.0016189 616.7 : 1 32 0.0018501 539.5 : 1
AAB $\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 1,008 0.0582794 16.2 : 1 1,344 0.0777058 11.9 : 1
xAA $\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 2} \end{matrix}$ 42 0.0024283 410.8 : 1 48 0.0027752 359.3 : 1
??* $\begin{matrix} {4r-3 \choose 2}{51-4r \choose 1} \end{matrix}$ 7,714 0.4459991 1.2 : 1 7,920 0.4579093 1.2 : 1
?** $\begin{matrix} {4r-3 \choose 1}{51-4r \choose 2} \end{matrix}$ 4,959 0.2867137 2.5 : 1 3,465 0.2003353 4.0 : 1
*** $\begin{matrix} {51-4r \choose 3} \end{matrix}$ 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 $\begin{matrix} {r-1 \choose 4}{4 \choose 1}^4 \end{matrix}$ 8,960 0.0460479 20.7 : 1 17,920 0.0920958 9.9 : 1
AABC $\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$ 10,080 0.0518039 18.3 : 1 16,128 0.0828862 11.1 : 1
ABC* $\begin{matrix} {r-1 \choose 3}{4 \choose 1}^3{51-4r \choose 1} \end{matrix}$ 42,560 0.2187275 3.6 : 1 53,760 0.2762874 2.6 : 1
xABC $\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 3}{4 \choose 1}^3 \end{matrix}$ 2,240 0.0115120 85.9 : 1 3,584 0.0184192 53.3 : 1
xAAB $\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 1,008 0.0051804 192.0 : 1 1,344 0.0069072 143.8 : 1
xAB* $\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 2}{4 \choose 1}^2{51-4r \choose 1} \end{matrix}$ 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 $\begin{matrix} {r-1 \choose 1}{4 \choose 4} \end{matrix}$ 7 0.0000360 27,796.1 : 1 8 0.0000411 24,321.5 : 1
AAAB $\begin{matrix} {r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 672 0.0034536 288.6 : 1 896 0.0046048 216.2 : 1
AABB $\begin{matrix} {r-1 \choose 2}{4 \choose 2}^2 \end{matrix}$ 756 0.0038853 256.4 : 1 1,008 0.0051804 192.0 : 1
AAA* $\begin{matrix} {r-1 \choose 1}{4 \choose 3}{51-4r \choose 1} \end{matrix}$ 532 0.0027341 364.8 : 1 480 0.0024669 404.4 : 1
AAB* $\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$ 19,152 0.0984274 9.2 : 1 20,160 0.1036078 8.7 : 1
xAAA $\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 3} \end{matrix}$ 28 0.0001439 6,948.3 : 1 32 0.0001645 6,079.6 : 1
xAA* $\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 2}{51-4r \choose 1} \end{matrix}$ 798 0.0041011 242.8 : 1 720 0.0037003 269.3 : 1
??** $\begin{matrix} {4r-3 \choose 2}{51-4r \choose 2} \end{matrix}$ 69,426 0.3567993 1.8 : 1 55,440 0.2849214 2.5 : 1
?*** $\begin{matrix} {4r-3 \choose 1}{51-4r \choose 3} \end{matrix}$ 28,101 0.1444187 5.9 : 1 15,015 0.0771662 12.0 : 1
**** $\begin{matrix} {51-4r \choose 4} \end{matrix}$ 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 $\begin{matrix} {r-1 \choose 5}{4 \choose 1}^5 \end{matrix}$ 21,504 0.0125585 78.6 : 1 57,344 0.0334894 28.9 : 1
AABCD $\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$ 53,760 0.0313963 30.9 : 1 107,520 0.0627926 14.9 : 1
ABCD* $\begin{matrix} {r-1 \choose 4}{4 \choose 1}^4{51-4r \choose 1} \end{matrix}$ 170,240 0.0994216 9.1 : 1 268,800 0.1569815 5.4 : 1
AABBC $\begin{matrix} {r-1 \choose 2}{4 \choose 2}^2{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 15,120 0.0088302 112.2 : 1 24,192 0.0141283 69.8 : 1
AAABC $\begin{matrix} {r-1 \choose 1}{4 \choose 3}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$ 6,720 0.0039245 253.8 : 1 10,752 0.0062793 158.3 : 1
AABC* $\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^2{51-4r \choose 1} \end{matrix}$ 191,520 0.1118493 7.9 : 1 241,920 0.1412833 6.1 : 1
ABC** $\begin{matrix} {r-1 \choose 3}{4 \choose 1}^3{51-4r \choose 2} \end{matrix}$ 383,040 0.2236986 3.5 : 1 376,320 0.2197741 3.6 : 1
xABCD $\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 4}{4 \choose 1}^4 \end{matrix}$ 8,960 0.0052327 190.1 : 1 17,920 0.0104654 94.6 : 1
xABC* $\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 3}{4 \choose 1}^3{51-4r \choose 1} \end{matrix}$ 42,560 0.0248554 39.2 : 1 53,760 0.0313963 30.9 : 1
xAABC $\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$ 10,080 0.0058868 168.9 : 1 16,128 0.0094189 105.2 : 1
xAAB* $\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$ 19,152 0.0111849 88.4 : 1 20,160 0.0117736 83.9 : 1
xAAAB $\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 672 0.0003925 2,547.1 : 1 896 0.0005233 1,910.1 : 1
xAABB $\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 2}{4 \choose 2}^2 \end{matrix}$ 756 0.0004415 2,264.0 : 1 1,008 0.0005887 1,697.7 : 1
xAB** $\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 2}{4 \choose 1}^2{51-4r \choose 2} \end{matrix}$ 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 $\begin{matrix} {r-1 \choose 1}{4 \choose 4}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 168 0.0000981 10,191.3 : 1 224 0.0001308 7,643.2 : 1
AAAA* $\begin{matrix} {r-1 \choose 1}{4 \choose 4}{51-4r \choose 1} \end{matrix}$ 133 0.0000777 12,873.5 : 1 120 0.0000701 14,268.2 : 1
AAABB $\begin{matrix} {r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 2} \end{matrix}$ 1,008 0.0005887 1,697.7 : 1 1,344 0.0007849 1,273.0 : 1
AAAB* $\begin{matrix} {r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$ 12,768 0.0074566 133.1 : 1 13,440 0.0078491 126.4 : 1
AABB* $\begin{matrix} {r-1 \choose 2}{4 \choose 2}^2{51-4r \choose 1} \end{matrix}$ 14,364 0.0083887 118.2 : 1 15,120 0.0088302 112.2 : 1
AAA** $\begin{matrix} {r-1 \choose 1}{4 \choose 3}{51-4r \choose 2} \end{matrix}$ 4,788 0.0027962 356.6 : 1 3,360 0.0019623 508.6 : 1
AAB** $\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 2} \end{matrix}$ 172,368 0.1006644 8.9 : 1 141,120 0.0824153 11.1 : 1
xAAAA $\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 4} \end{matrix}$ 7 0.0000041 244,613.9 : 1 8 0.0000047 214,037.0 : 1
xAAA* $\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 3}{51-4r \choose 1} \end{matrix}$ 532 0.0003107 3,217.6 : 1 480 0.0002803 3,566.3 : 1
xAA** $\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 2}{51-4r \choose 2} \end{matrix}$ 7,182 0.0041943 237.4 : 1 5,040 0.0029434 338.7 : 1
??*** $\begin{matrix} {4r-3 \choose 2}{51-4r \choose 3} \end{matrix}$ 393,414 0.2297571 3.4 : 1 240,240 0.1403022 6.1 : 1
?**** $\begin{matrix} {4r-3 \choose 1}{51-4r \choose 4} \end{matrix}$ 112,404 0.0656449 14.2 : 1 45,045 0.0263067 37.0 : 1
***** $\begin{matrix} {51-4r \choose 5} \end{matrix}$ 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 $52 - 4r - 0 = 52 - 4r$ 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 make Board Derivation 8-high qualifier (r = 8) 9-high qualifier (r = 9)
Combos Probability Odds Combos Probability Odds
Miss low ABC $\begin{matrix} {r-1 \choose 3}{4 \choose 1}^3 \end{matrix}$ 2,240 0.1295097 6.7 : 1 3,584 0.2072155 3.8 : 1
No low AAA $\begin{matrix} {r-1 \choose 1}{4 \choose 3} \end{matrix}$ 28 0.0016189 616.7 : 1 32 0.0018501 539.5 : 1
AAB $\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 1,008 0.0582794 16.2 : 1 1,344 0.0777058 11.9 : 1
??* $\begin{matrix} {4r-4 \choose 2}{52-4r \choose 1} \end{matrix}$ 7,560 0.4370953 1.3 : 1 7,936 0.4588344 1.2 : 1
?** $\begin{matrix} {4r-4 \choose 1}{52-4r \choose 2} \end{matrix}$ 5,320 0.3075856 2.3 : 1 3,840 0.2220167 3.5 : 1
*** $\begin{matrix} {52-4r \choose 3} \end{matrix}$ 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 $\begin{matrix} {r-1 \choose 4}{4 \choose 1}^4 \end{matrix}$ 8,960 0.0460479 20.7 : 1 17,920 0.0920958 9.9 : 1
AABC $\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$ 10,080 0.0518039 18.3 : 1 16,128 0.0828862 11.1 : 1
ABC* $\begin{matrix} {r-1 \choose 3}{4 \choose 1}^3{52-4r \choose 1} \end{matrix}$ 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 $\begin{matrix} {r-1 \choose 1}{4 \choose 4} \end{matrix}$ 7 0.0000360 27,796.1 : 1 8 0.0000411 24,321.5 : 1
AAAB $\begin{matrix} {r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 672 0.0034536 288.6 : 1 896 0.0046048 216.2 : 1
AABB $\begin{matrix} {r-1 \choose 2}{4 \choose 2}^2 \end{matrix}$ 756 0.0038853 256.4 : 1 1,008 0.0051804 192.0 : 1
AAA* $\begin{matrix} {r-1 \choose 1}{4 \choose 3}{52-4r \choose 1} \end{matrix}$ 560 0.0028780 346.5 : 1 512 0.0026313 379.0 : 1
AAB* $\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 20,160 0.1036078 8.7 : 1 21,504 0.1105150 8.0 : 1
??** $\begin{matrix} {4r-4 \choose 2}{52-4r \choose 2} \end{matrix}$ 71,820 0.3691027 1.7 : 1 59,520 0.3058896 2.3 : 1
?*** $\begin{matrix} {4r-4 \choose 1}{52-4r \choose 3} \end{matrix}$ 31,920 0.1640456 5.1 : 1 17,920 0.0920958 9.9 : 1
**** $\begin{matrix} {52-4r \choose 4} \end{matrix}$ 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 $\begin{matrix} {r-1 \choose 5}{4 \choose 1}^5 \end{matrix}$ 21,504 0.0125585 78.6 : 1 57,344 0.0334894 28.9 : 1
AABCD $\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$ 53,760 0.0313963 30.9 : 1 107,520 0.0627926 14.9 : 1
ABCD* $\begin{matrix} {r-1 \choose 4}{4 \choose 1}^4{52-4r \choose 1} \end{matrix}$ 179,200 0.1046543 8.6 : 1 286,720 0.1674469 5.0 : 1
AABBC $\begin{matrix} {r-1 \choose 2}{4 \choose 2}^2{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 15,120 0.0088302 112.2 : 1 24,192 0.0141283 69.8 : 1
AAABC $\begin{matrix} {r-1 \choose 1}{4 \choose 3}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$ 6,720 0.0039245 253.8 : 1 10,752 0.0062793 158.3 : 1
AABC* $\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$ 201,600 0.1177361 7.5 : 1 258,048 0.1507022 5.6 : 1
ABC** $\begin{matrix} {r-1 \choose 3}{4 \choose 1}^3{52-4r \choose 2} \end{matrix}$ 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 $\begin{matrix} {r-1 \choose 1}{4 \choose 4}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 168 0.0000981 10,191.3 : 1 224 0.0001308 7,643.2 : 1
AAAA* $\begin{matrix} {r-1 \choose 1}{4 \choose 4}{52-4r \choose 1} \end{matrix}$ 140 0.0000818 12,229.7 : 1 128 0.0000748 13,376.4 : 1
AAABB $\begin{matrix} {r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 2} \end{matrix}$ 1,008 0.0005887 1,697.7 : 1 1,344 0.0007849 1,273.0 : 1
AAAB* $\begin{matrix} {r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 13,440 0.0078491 126.4 : 1 14,336 0.0083723 118.4 : 1
AABB* $\begin{matrix} {r-1 \choose 2}{4 \choose 2}^2{52-4r \choose 1} \end{matrix}$ 15,120 0.0088302 112.2 : 1 16,128 0.0094189 105.2 : 1
AAA** $\begin{matrix} {r-1 \choose 1}{4 \choose 3}{52-4r \choose 2} \end{matrix}$ 5,320 0.0031069 320.9 : 1 3,840 0.0022426 444.9 : 1
AAB** $\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 2} \end{matrix}$ 191,520 0.1118493 7.9 : 1 161,280 0.0941889 9.6 : 1
??*** $\begin{matrix} {4r-4 \choose 2}{52-4r \choose 3} \end{matrix}$ 430,920 0.2516609 3.0 : 1 277,760 0.1622142 5.2 : 1
?**** $\begin{matrix} {4r-4 \choose 1}{52-4r \choose 4} \end{matrix}$ 135,660 0.0792266 11.6 : 1 58,240 0.0340127 28.4 : 1
***** $\begin{matrix} {52-4r \choose 5} \end{matrix}$ 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 $52 - 4r - 2 = 50 - 4r$ 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 make Board Derivation 8-high qualifier (r = 8) 9-high qualifier (r = 9)
Combos Probability Odds Combos Probability Odds
Make low ABC $\begin{matrix} {r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$ 1,280 0.0740056 12.5 : 1 2,240 0.1295097 6.7 : 1
Miss low xAB $\begin{matrix} {2 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$ 1,440 0.0832562 11.0 : 1 2,016 0.1165587 7.6 : 1
xyA $\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 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 $\begin{matrix} {r-2 \choose 1}{4 \choose 3} \end{matrix}$ 24 0.0013876 719.7 : 1 28 0.0016189 616.7 : 1
AAB $\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 720 0.0416281 23.0 : 1 1,008 0.0582794 16.2 : 1
xxx $\begin{matrix} {2 \choose 1}{3 \choose 3} \end{matrix}$ 2 0.0001156 8,647.0 : 1 2 0.0001156 8,647.0 : 1
xxy $\begin{matrix} {2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1} \end{matrix}$ 18 0.0010407 959.9 : 1 18 0.0010407 959.9 : 1
xxA $\begin{matrix} {2 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 144 0.0083256 119.1 : 1 168 0.0097132 102.0 : 1
xAA $\begin{matrix} {2 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2} \end{matrix}$ 216 0.0124884 79.1 : 1 252 0.0145698 67.6 : 1
??* $\begin{matrix} {4r-2 \choose 2}{50-4r \choose 1} \end{matrix}$ 7,830 0.4527058 1.2 : 1 7,854 0.4540934 1.2 : 1
?** $\begin{matrix} {4r-2 \choose 1}{50-4r \choose 2} \end{matrix}$ 4,590 0.2653793 2.8 : 1 3,094 0.1788853 4.6 : 1
*** $\begin{matrix} {50-4r \choose 3} \end{matrix}$ 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 $\begin{matrix} {r-2 \choose 4}{4 \choose 1}^4 \end{matrix}$ 3,840 0.0197348 49.7 : 1 8,960 0.0460479 20.7 : 1
AABC $\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$ 5,760 0.0296022 32.8 : 1 10,080 0.0518039 18.3 : 1
ABC* $\begin{matrix} {r-2 \choose 3}{4 \choose 1}^3{50-4r \choose 1} \end{matrix}$ 23,040 0.1184089 7.4 : 1 31,360 0.1611676 5.2 : 1
xABC $\begin{matrix} {2 \choose 1}{3 \choose 1}{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$ 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 $\begin{matrix} {2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 432 0.0022202 449.4 : 1 504 0.0025902 385.1 : 1
xxAB $\begin{matrix} {2 \choose 1}{3 \choose 2}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$ 1,440 0.0074006 134.1 : 1 2,016 0.0103608 95.5 : 1
xyAA $\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-2 \choose 1}{4 \choose 2} \end{matrix}$ 324 0.0016651 599.6 : 1 378 0.0019426 513.8 : 1
xAAB $\begin{matrix} {2 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 4,320 0.0222017 44.0 : 1 6,048 0.0310823 31.2 : 1
xyAB $\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$ 2,160 0.0111008 89.1 : 1 3,024 0.0155412 63.3 : 1
xyA* $\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1} \end{matrix}$ 3,888 0.0199815 49.0 : 1 3,528 0.0181314 54.2 : 1
xAB* $\begin{matrix} {2 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^2{50-4r \choose 1} \end{matrix}$ 25,920 0.1332100 6.5 : 1 28,224 0.1450509 5.9 : 1
Total 38,484 0.1977798 4.1 : 1 43,722 0.2246994 3.5 : 1
No low AAAA $\begin{matrix} {r-2 \choose 1}{4 \choose 4} \end{matrix}$ 6 0.0000308 32,429.0 : 1 7 0.0000360 27,796.1 : 1
AAAB $\begin{matrix} {r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 480 0.0024669 404.4 : 1 672 0.0034536 288.6 : 1
AABB $\begin{matrix} {r-2 \choose 2}{4 \choose 2}^2 \end{matrix}$ 540 0.0027752 359.3 : 1 756 0.0038853 256.4 : 1
AAA* $\begin{matrix} {r-2 \choose 1}{4 \choose 3}{50-4r \choose 1} \end{matrix}$ 432 0.0022202 449.4 : 1 392 0.0020146 495.4 : 1
AAB* $\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}{50-4r \choose 1} \end{matrix}$ 14,112 0.0725254 12.8 : 1 12,960 0.0666050 14.0 : 1
xxxy $\begin{matrix} {2 \choose 1}{3 \choose 3}{1 \choose 1}{3 \choose 1} \end{matrix}$ 6 0.0000308 32,429.0 : 1 6 0.0000308 32,429.0 : 1
xxxA $\begin{matrix} {2 \choose 1}{3 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 56 0.0002878 3,473.6 : 1 48 0.0002467 4,052.8 : 1
xxx* $\begin{matrix} {2 \choose 1}{3 \choose 3}{50-4r \choose 1} \end{matrix}$ 28 0.0001439 6,948.3 : 1 36 0.0001850 5,404.0 : 1
xxyy $\begin{matrix} {2 \choose 2}{3 \choose 2}^2 \end{matrix}$ 9 0.0000463 21,619.0 : 1 9 0.0000463 21,619.0 : 1
xxy* $\begin{matrix} {2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{50-4r \choose 1} \end{matrix}$ 324 0.0016651 599.6 : 1 252 0.0012951 771.1 : 1
xxAA $\begin{matrix} {2 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2} \end{matrix}$ 216 0.0011101 899.8 : 1 252 0.0012951 771.1 : 1
xxA* $\begin{matrix} {2 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1} \end{matrix}$ 2,592 0.0133210 74.1 : 1 2,352 0.0120876 81.7 : 1
xAAA $\begin{matrix} {2 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 3} \end{matrix}$ 144 0.0007401 1,350.3 : 1 168 0.0008634 1,157.2 : 1
xAA* $\begin{matrix} {2 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{50-4r \choose 1} \end{matrix}$ 3,888 0.0199815 49.0 : 1 3,528 0.0181314 54.2 : 1
??** $\begin{matrix} {4r-2 \choose 2}{50-4r \choose 2} \end{matrix}$ 66,555 0.3420444 1.9 : 1 51,051 0.2623651 2.8 : 1
?*** $\begin{matrix} {4r-2 \choose 1}{50-4r \choose 3} \end{matrix}$ 24,480 0.1258094 6.9 : 1 12,376 0.0636037 14.7 : 1
**** $\begin{matrix} {50-4r \choose 4} \end{matrix}$ 3,060 0.0157262 62.6 : 1 1,001 0.0051444 193.4 : 1
Total 115,776 0.5950046 0.7 : 1 87,018 0.4472094 1.2 : 1
Derivations for low hand shape xy** (two low ranks 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
Make low ABCDE $\begin{matrix} {r-2 \choose 5}{4 \choose 1}^5 \end{matrix}$ 6,144 0.0035881 277.7 : 1 21,504 0.0125585 78.6 : 1
AABCD $\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 3}{4 \choose 1}^3 \end{matrix}$ 23,040 0.0134556 73.3 : 1 53,760 0.0313963 30.9 : 1
ABCD* $\begin{matrix} {r-2 \choose 4}{4 \choose 1}^4{50-4r \choose 1} \end{matrix}$ 69,120 0.0403667 23.8 : 1 125,440 0.0732580 12.7 : 1
AABBC $\begin{matrix} {r-2 \choose 2}{4 \choose 2}^2{r-4 \choose 1}{4 \choose 1} \end{matrix}$ 8,640 0.0050458 197.2 : 1 15,120 0.0088302 112.2 : 1
AABC* $\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^2{50-4r \choose 1} \end{matrix}$ 103,680 0.0605500 15.5 : 1 141,120 0.0824153 11.1 : 1
AAABC $\begin{matrix} {r-2 \choose 1}{4 \choose 3}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$ 3,840 0.0022426 444.9 : 1 6,720 0.0039245 253.8 : 1
ABC** $\begin{matrix} {r-2 \choose 3}{4 \choose 1}^3{50-4r \choose 2} \end{matrix}$ 195,840 0.1143722 7.7 : 1 203,840 0.1190443 7.4 : 1
xABCD $\begin{matrix} {2 \choose 1}{3 \choose 1}{r-2 \choose 4}{4 \choose 1}^4 \end{matrix}$ 23,040 0.0134556 73.3 : 1 53,760 0.0313963 30.9 : 1
xABC* $\begin{matrix} {2 \choose 1}{3 \choose 1}{r-2 \choose 3}{4 \choose 1}^3{50-4r \choose 1} \end{matrix}$ 138,240 0.0807333 11.4 : 1 188,160 0.1098870 8.1 : 1
xAABC $\begin{matrix} {2 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$ 34,560 0.0201833 48.5 : 1 60,480 0.0353208 27.3 : 1
xxABC $\begin{matrix} {2 \choose 1}{3 \choose 2}{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$ 7,680 0.0044852 222.0 : 1 13,440 0.0078491 126.4 : 1
xyABC $\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$ 11,520 0.0067278 147.6 : 1 20,160 0.0117736 83.9 : 1
Total 625,344 0.3652062 1.7 : 1 903,504 0.5276540 0.9 : 1
Miss low xxxyA $\begin{matrix} {2 \choose 1}{3 \choose 3}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 144 0.0000841 11,890.0 : 1 168 0.0000981 10,191.3 : 1
xxxAB $\begin{matrix} {2 \choose 1}{3 \choose 3}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$ 480 0.0002803 3,566.3 : 1 672 0.0003925 2,547.1 : 1
xxyyA $\begin{matrix} {2 \choose 2}{3 \choose 2}^2{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 216 0.0001261 7,926.3 : 1 252 0.0001472 6,793.9 : 1
xxyAB $\begin{matrix} {2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$ 4,320 0.0025229 395.4 : 1 6,048 0.0035321 282.1 : 1
xxyAA $\begin{matrix} {2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2} \end{matrix}$ 648 0.0003784 2,641.4 : 1 756 0.0004415 2,264.0 : 1
xxyA* $\begin{matrix} {2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1} \end{matrix}$ 7,776 0.0045412 219.2 : 1 7,056 0.0041208 241.7 : 1
xxAAB $\begin{matrix} {2 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 4,320 0.0025229 395.4 : 1 6,048 0.0035321 282.1 : 1
xxAB* $\begin{matrix} {2 \choose 1}{3 \choose 2}{r-2 \choose 2}{4 \choose 1}^2{50-4r \choose 1} \end{matrix}$ 25,920 0.0151375 65.1 : 1 28,224 0.0164831 59.7 : 1
xyAAA $\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-2 \choose 1}{4 \choose 3} \end{matrix}$ 216 0.0001261 7,926.3 : 1 252 0.0001472 6,793.9 : 1
xyAAB $\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 6,480 0.0037844 263.2 : 1 9,072 0.0052981 187.7 : 1
xyAA* $\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-2 \choose 1}{4 \choose 2}{50-4r \choose 1} \end{matrix}$ 5,832 0.0034059 292.6 : 1 5,292 0.0030906 322.6 : 1
xyAB* $\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-2 \choose 2}{4 \choose 1}^2{50-4r \choose 1} \end{matrix}$ 38,880 0.0227062 43.0 : 1 42,336 0.0247246 39.4 : 1
xyA** $\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-2 \choose 1}{4 \choose 1}{50-4r \choose 2} \end{matrix}$ 33,048 0.0193003 50.8 : 1 22,932 0.0133925 73.7 : 1
xAAAB $\begin{matrix} {2 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 2,880 0.0016819 593.6 : 1 4,032 0.0023547 423.7 : 1
xAABB $\begin{matrix} {2 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 2}^2 \end{matrix}$ 3,240 0.0018922 527.5 : 1 4,536 0.0026491 376.5 : 1
xAAB* $\begin{matrix} {2 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1} \end{matrix}$ 77,760 0.0454125 21.0 : 1 84,672 0.0494492 19.2 : 1
xAB** $\begin{matrix} {2 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 1}^2{50-4r \choose 2} \end{matrix}$ 220,320 0.1286687 6.8 : 1 183,456 0.1071399 8.3 : 1
Total 432,480 0.2525720 3.0 : 1 405,804 0.2369930 3.2 : 1
No low AAAAB $\begin{matrix} {r-2 \choose 1}{4 \choose 4}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 120 0.0000701 14,268.2 : 1 168 0.0000981 10,191.3 : 1
AAAA* $\begin{matrix} {r-2 \choose 1}{4 \choose 4}{50-4r \choose 1} \end{matrix}$ 108 0.0000631 15,853.7 : 1 98 0.0000572 17,471.5 : 1
AAABB $\begin{matrix} {r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 2} \end{matrix}$ 720 0.0004205 2,377.2 : 1 1,008 0.0005887 1,697.7 : 1
AAAB* $\begin{matrix} {r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 1}{50-4r \choose 1} \end{matrix}$ 8,640 0.0050458 197.2 : 1 9,408 0.0054944 181.0 : 1
AABB* $\begin{matrix} {r-2 \choose 2}{4 \choose 2}^2{50-4r \choose 1} \end{matrix}$ 9,720 0.0056766 175.2 : 1 10,584 0.0061811 160.8 : 1
AAA** $\begin{matrix} {r-2 \choose 1}{4 \choose 3}{50-4r \choose 2} \end{matrix}$ 3,672 0.0021445 465.3 : 1 2,548 0.0014881 671.0 : 1
AAB** $\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 2} \end{matrix}$ 110,160 0.0643344 14.5 : 1 91,728 0.0535699 17.7 : 1
xxxyy $\begin{matrix} {2 \choose 1}{3 \choose 3}{1 \choose 1}{3 \choose 2} \end{matrix}$ 6 0.0000035 285,383.0 : 1 6 0.0000035 285,383.0 : 1
xxxy* $\begin{matrix} {2 \choose 1}{3 \choose 3}{1 \choose 1}{3 \choose 1}{50-4r \choose 1} \end{matrix}$ 108 0.0000631 15,853.7 : 1 84 0.0000491 20,383.6 : 1
xxxAA $\begin{matrix} {2 \choose 1}{3 \choose 3}{r-2 \choose 1}{4 \choose 2} \end{matrix}$ 72 0.0000420 23,781.0 : 1 84 0.0000491 20,383.6 : 1
xxxA* $\begin{matrix} {2 \choose 1}{3 \choose 3}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1} \end{matrix}$ 864 0.0005046 1,980.8 : 1 784 0.0004579 2,183.1 : 1
xxx** $\begin{matrix} {2 \choose 1}{3 \choose 3}{50-4r \choose 2} \end{matrix}$ 306 0.0001787 5,594.8 : 1 182 0.0001063 9,407.3 : 1
xxyy* $\begin{matrix} {2 \choose 2}{3 \choose 2}^2{50-4r \choose 1} \end{matrix}$ 162 0.0000946 10,568.8 : 1 126 0.0000736 13,588.7 : 1
xxy** $\begin{matrix} {2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{50-4r \choose 2} \end{matrix}$ 2,754 0.0016084 620.8 : 1 1,638 0.0009566 1,044.4 : 1
xxAAA $\begin{matrix} {2 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 3} \end{matrix}$ 144 0.0000841 11,890.0 : 1 168 0.0000981 10,191.3 : 1
xxAA* $\begin{matrix} {2 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2}{50-4r \choose 1} \end{matrix}$ 3,888 0.0022706 439.4 : 1 3,528 0.0020604 484.3 : 1
xxA** $\begin{matrix} {2 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 2} \end{matrix}$ 22,032 0.0128669 76.7 : 1 15,288 0.0089283 111.0 : 1
xAAAA $\begin{matrix} {2 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 4} \end{matrix}$ 36 0.0000210 47,563.0 : 1 42 0.0000245 40,768.1 : 1
xAAA* $\begin{matrix} {2 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 3}{50-4r \choose 1} \end{matrix}$ 2,592 0.0015137 659.6 : 1 2,352 0.0013736 727.0 : 1
xAA** $\begin{matrix} {2 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{50-4r \choose 2} \end{matrix}$ 33,048 0.0193003 50.8 : 1 22,932 0.0133925 73.7 : 1
??*** $\begin{matrix} {4r-2 \choose 2}{50-4r \choose 3} \end{matrix}$ 354,960 0.2072996 3.8 : 1 204,204 0.1192569 7.4 : 1
?**** $\begin{matrix} {4r-2 \choose 1}{50-4r \choose 4} \end{matrix}$ 91,800 0.0536120 17.7 : 1 34,034 0.0198761 49.3 : 1
***** $\begin{matrix} {50-4r \choose 5} \end{matrix}$ 8,568 0.0050038 198.8 : 1 2,002 0.0011692 854.3 : 1
Total 654,480 0.3822218 1.6 : 1 402,996 0.2353531 3.2 : 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 make Board Derivation 8-high qualifier (r = 8) 9-high qualifier (r = 9)
Combos Probability Odds Combos Probability Odds
Make low ABC $\begin{matrix} {r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$ 1,280 0.0740056 12.5 : 1 2,240 0.1295097 6.7 : 1
Miss low xAB $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$ 480 0.0277521 35.0 : 1 672 0.0388529 24.7 : 1
yAB $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$ 720 0.0416281 23.0 : 1 1,008 0.0582794 16.2 : 1
xyA $\begin{matrix} {1 \choose 1}{2 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 144 0.0083256 119.1 : 1 168 0.0097132 102.0 : 1
Total 1,344 0.0777058 11.9 : 1 1,848 0.1068455 8.4 : 1
No low AAA $\begin{matrix} {r-2 \choose 1}{4 \choose 3} \end{matrix}$ 24 0.0013876 719.7 : 1 28 0.0016189 616.7 : 1
AAB $\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 720 0.0416281 23.0 : 1 1,008 0.0582794 16.2 : 1
xxy $\begin{matrix} {1 \choose 1}{2 \choose 2}{1 \choose 1}{3 \choose 1} \end{matrix}$ 3 0.0001735 5,764.3 : 1 3 0.0001735 5,764.3 : 1
xxA $\begin{matrix} {1 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 24 0.0013876 719.7 : 1 28 0.0016189 616.7 : 1
xyy $\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 2} \end{matrix}$ 6 0.0003469 2,881.7 : 1 6 0.0003469 2,881.7 : 1
xAA $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 2} \end{matrix}$ 72 0.0041628 239.2 : 1 84 0.0048566 204.9 : 1
yyy $\begin{matrix} {1 \choose 1}{3 \choose 3} \end{matrix}$ 1 0.0000578 17,295.0 : 1 1 0.0000578 17,295.0 : 1
yyA $\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 108 0.0062442 159.1 : 1 126 0.0072849 136.3 : 1
yAA $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2} \end{matrix}$ 216 0.0124884 79.1 : 1 252 0.0145698 67.6 : 1
??* $\begin{matrix} {4r-3 \choose 2}{51-4r \choose 1} \end{matrix}$ 7,714 0.4459991 1.2 : 1 7,920 0.4579093 1.2 : 1
?** $\begin{matrix} {4r-3 \choose 1}{51-4r \choose 2} \end{matrix}$ 4,959 0.2867137 2.5 : 1 3,465 0.2003353 4.0 : 1
*** $\begin{matrix} {51-4r \choose 3} \end{matrix}$ 969 0.0560245 16.8 : 1 455 0.0263067 37.0 : 1
Total 14,672 0.8482886 0.2 : 1 13,208 0.7636448 0.3 : 1
Derivations for low hand shape xxy* (a low pair, a low card 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
Make low ABCD $\begin{matrix} {r-2 \choose 4}{4 \choose 1}^4 \end{matrix}$ 3,840 0.0197348 49.7 : 1 8,960 0.0460479 20.7 : 1
AABC $\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$ 5,760 0.0296022 32.8 : 1 10,080 0.0518039 18.3 : 1
ABC* $\begin{matrix} {r-2 \choose 3}{4 \choose 1}^3{51-4r \choose 1} \end{matrix}$ 24,320 0.1249872 7.0 : 1 33,600 0.1726796 4.8 : 1
xABC $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$ 2,560 0.0131565 75.0 : 1 4,480 0.0230239 42.4 : 1
yABC $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$ 3,840 0.0197348 49.7 : 1 6,720 0.0345359 28.0 : 1
Total 40,320 0.2072155 3.8 : 1 63,840 0.3280913 2.0 : 1
Miss low xxyA $\begin{matrix} {1 \choose 1}{2 \choose 2}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 72 0.0003700 2,701.5 : 1 84 0.0004317 2,315.4 : 1
xxAB $\begin{matrix} {1 \choose 1}{2 \choose 2}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$ 240 0.0012334 809.8 : 1 336 0.0017268 578.1 : 1
xyyA $\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 144 0.0007401 1,350.3 : 1 168 0.0008634 1,157.2 : 1
xyAA $\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2} \end{matrix}$ 216 0.0124884 79.1 : 1 252 0.0145698 67.6 : 1
xAAB $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 1,440 0.0074006 134.1 : 1 2,016 0.0103608 95.5 : 1
xyAB $\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$ 1,440 0.0074006 134.1 : 1 2,016 0.0103608 95.5 : 1
xyA* $\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$ 2,736 0.0140611 70.1 : 1 2,520 0.0129510 76.2 : 1
xAB* $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-2 \choose 2}{4 \choose 1}^2{51-4r \choose 1} \end{matrix}$ 9,120 0.0468702 20.3 : 1 10,080 0.0518039 18.3 : 1
yyAB $\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$ 720 0.0037003 269.3 : 1 1,008 0.0051804 192.0 : 1
yAAB $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 2,160 0.0111008 89.1 : 1 3,024 0.0155412 63.3 : 1
yAB* $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^2{51-4r \choose 1} \end{matrix}$ 13,680 0.0703053 13.2 : 1 15,120 0.0777058 11.9 : 1
Total 31,968 0.1642923 5.1 : 1 36,624 0.1882208 4.3 : 1
No low AAAA $\begin{matrix} {r-2 \choose 1}{4 \choose 4} \end{matrix}$ 6 0.0000308 32,429.0 : 1 7 0.0000360 27,796.1 : 1
AAAB $\begin{matrix} {r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 480 0.0024669 404.4 : 1 672 0.0034536 288.6 : 1
AABB $\begin{matrix} {r-2 \choose 2}{4 \choose 2}^2 \end{matrix}$ 540 0.0027752 359.3 : 1 756 0.0038853 256.4 : 1
AAA* $\begin{matrix} {r-2 \choose 1}{4 \choose 3}{51-4r \choose 1} \end{matrix}$ 456 0.0023435 425.7 : 1 420 0.0021585 462.3 : 1
AAB* $\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$ 13,680 0.0703053 13.2 : 1 15,120 0.0777058 11.9 : 1
xxyy $\begin{matrix} {1 \choose 1}{2 \choose 2}{1 \choose 1}{3 \choose 2} \end{matrix}$ 3 0.0000154 64,859.0 : 1 3 0.0000154 64,859.0 : 1
xxy* $\begin{matrix} {1 \choose 1}{2 \choose 2}{1 \choose 1}{3 \choose 1}{51-4r \choose 1} \end{matrix}$ 57 0.0002929 3,412.7 : 1 45 0.0002313 4,323.0 : 1
xxAA $\begin{matrix} {1 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 2} \end{matrix}$ 36 0.0001850 5,404.0 : 1 42 0.0002158 4,631.9 : 1
xxA* $\begin{matrix} {1 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1} \end{matrix}$ 456 0.0023435 425.7 : 1 420 0.0021585 462.3 : 1
xyyy $\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 3} \end{matrix}$ 2 0.0000103 97,289.0 : 1 2 0.0000103 97,289.0 : 1
xyy* $\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 2}{51-4r \choose 1} \end{matrix}$ 114 0.0005859 1,705.8 : 1 90 0.0004625 2,161.0 : 1
xAAA $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 3} \end{matrix}$ 48 0.0002467 4,052.8 : 1 56 0.0002878 3,473.6 : 1
xAA* $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 2}{51-4r \choose 1} \end{matrix}$ 1,368 0.0070305 141.2 : 1 1,260 0.0064755 153.4 : 1
yyyA $\begin{matrix} {1 \choose 1}{3 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 24 0.0001233 8,106.5 : 1 28 0.0001439 6,948.3 : 1
yyy* $\begin{matrix} {1 \choose 1}{3 \choose 3}{51-4r \choose 1} \end{matrix}$ 19 0.0000976 10,240.1 : 1 15 0.0000771 12,971.0 : 1
yyAA $\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2} \end{matrix}$ 108 0.0005550 1,800.7 : 1 126 0.0006475 1,543.3 : 1
yyA* $\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$ 1,368 0.0070305 141.2 : 1 1,260 0.0064755 153.4 : 1
yAAA $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 3} \end{matrix}$ 72 0.0003700 2,701.5 : 1 84 0.0004317 2,315.4 : 1
yAA* $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{51-4r \choose 1} \end{matrix}$ 2,052 0.0105458 93.8 : 1 1,890 0.0097132 102.0 : 1
??** $\begin{matrix} {4r-3 \choose 2}{51-4r \choose 2} \end{matrix}$ 69,426 0.3567993 1.8 : 1 55,440 0.2849214 2.5 : 1
?*** $\begin{matrix} {4r-3 \choose 1}{51-4r \choose 3} \end{matrix}$ 28,101 0.1444187 5.9 : 1 15,015 0.0771662 12.0 : 1
**** $\begin{matrix} {51-4r \choose 4} \end{matrix}$ 3,876 0.0199198 49.2 : 1 1,365 0.0070151 141.5 : 1
Total 122,292 0.6284921 0.6 : 1 94,116 0.4836879 1.1 : 1
Derivations for low hand shape xxy* (a low pair, a low card 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
Make low ABCDE $\begin{matrix} {r-2 \choose 5}{4 \choose 1}^5 \end{matrix}$ 6,144 0.0035881 277.7 : 1 21,504 0.0125585 78.6 : 1
AABCD $\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 3}{4 \choose 1}^3 \end{matrix}$ 23,040 0.0134556 73.3 : 1 53,760 0.0313963 30.9 : 1
ABCD* $\begin{matrix} {r-2 \choose 4}{4 \choose 1}^4{51-4r \choose 1} \end{matrix}$ 72,960 0.0426093 22.5 : 1 134,400 0.0784907 11.7 : 1
AABBC $\begin{matrix} {r-2 \choose 2}{4 \choose 2}^2{r-4 \choose 1}{4 \choose 1} \end{matrix}$ 8,640 0.0050458 197.2 : 1 15,120 0.0088302 112.2 : 1
AABC* $\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^2{51-4r \choose 1} \end{matrix}$ 109,440 0.0639139 14.6 : 1 151,200 0.0883021 10.3 : 1
AAABC $\begin{matrix} {r-2 \choose 1}{4 \choose 3}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$ 3,840 0.0022426 444.9 : 1 6,720 0.0039245 253.8 : 1
ABC** $\begin{matrix} {r-2 \choose 3}{4 \choose 1}^3{51-4r \choose 2} \end{matrix}$ 218,880 0.1278278 6.8 : 1 235,200 0.1373588 6.3 : 1
xABCD $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-2 \choose 4}{4 \choose 1}^4 \end{matrix}$ 7,680 0.0044852 222.0 : 1 17,920 0.0104654 94.6 : 1
xABC* $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-2 \choose 3}{4 \choose 1}^3{51-4r \choose 1} \end{matrix}$ 48,640 0.0284062 34.2 : 1 67,200 0.0392454 24.5 : 1
xAABC $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$ 11,520 0.0067278 147.6 : 1 20,160 0.0117736 83.9 : 1
xxABC $\begin{matrix} {1 \choose 1}{2 \choose 2}{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$ 1,280 0.0007475 1,336.7 : 1 2,240 0.0013082 763.4 : 1
xyABC $\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$ 7,680 0.0044852 222.0 : 1 13,440 0.0078491 126.4 : 1
yyABC $\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$ 3,840 0.0022426 444.9 : 1 6,720 0.0039245 253.8 : 1
yABCD $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 4}{4 \choose 1}^4 \end{matrix}$ 11,520 0.0067278 147.6 : 1 26,880 0.0156981 62.7 : 1
yABC* $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 3}{4 \choose 1}^3{51-4r \choose 1} \end{matrix}$ 72,960 0.0426093 22.5 : 1 100,800 0.0588681 16.0 : 1
yAABC $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$ 17,280 0.0100917 98.1 : 1 30,240 0.0176604 55.6 : 1
Total 625,344 0.3652062 1.7 : 1 903,504 0.5276540 0.9 : 1
Miss low xxyyA $\begin{matrix} {1 \choose 1}{2 \choose 2}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 72 0.0000420 23,781.0 : 1 84 0.0000491 20,383.6 : 1
xxyAB $\begin{matrix} {1 \choose 1}{2 \choose 2}{1 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$ 720 0.0004205 2,377.2 : 1 1,008 0.0005887 1,697.7 : 1
xxyAA $\begin{matrix} {1 \choose 1}{2 \choose 2}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2} \end{matrix}$ 108 0.0000631 15,853.7 : 1 126 0.0000736 13,588.7 : 1
xxyA* $\begin{matrix} {1 \choose 1}{2 \choose 2}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1} \end{matrix}$ 1,368 0.0007989 1,250.7 : 1 1,260 0.0007359 1,358.0 : 1
xxAAB $\begin{matrix} {1 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 720 0.0004205 2,377.2 : 1 1,008 0.0005887 1,697.7 : 1
xxAB* $\begin{matrix} {1 \choose 1}{2 \choose 2}{r-2 \choose 2}{4 \choose 1}^2{51-4r \choose 1} \end{matrix}$ 4,560 0.0026631 374.5  : 1 5,040 0.0029434 338.7 : 1
xyyyA $\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 48 0.0000280 35,672.0 : 1 56 0.0000327 30,575.9 : 1
xyyAA $\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 3} \end{matrix}$ 216 0.0001261 7,926.3 : 1 252 0.0001472 6,793.9 : 1
xyyAB $\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 1,440 0.0008410 1,188.1 : 1 2,016 0.0011774 848.4 : 1
xyyA* $\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2}{51-4r \choose 1} \end{matrix}$ 2,736 0.0015978 624.8 : 1 2,520 0.0014717 678.5 : 1
xyAAA $\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 3} \end{matrix}$ 144 0.0000841 11,890.0 : 1 168 0.0000981 10,191.3 : 1
xyAAB $\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 4,320 0.0025229 395.4 : 1 6,048 0.0035321 282.1 : 1
xyAA* $\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{51-4r \choose 1} \end{matrix}$ 4,104 0.0023968 416.2 : 1 3,780 0.0022076 452.0 : 1
xyAB* $\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^2{51-4r \choose 1} \end{matrix}$ 27,360 0.0159785 61.6 : 1 30,240 0.0176604 55.6 : 1
xyA** $\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 2} \end{matrix}$ 24,624 0.0143806 68.5 : 1 17,640 0.0103019 96.1 : 1
xAAAB $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 960 0.0005606 1,782.7 : 1 1,344 0.0007849 1,273.0 : 1
xAABB $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 2}^2 \end{matrix}$ 1,080 0.0006307 1,584.5 : 1 1,512 0.0008830 1,131.5 : 1
xAAB* $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$ 27,360 0.0159785 61.6 : 1 30,240 0.0176604 55.6 : 1
xAB** $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 1}^2{51-4r \choose 2} \end{matrix}$ 82,080 0.0479354 19.9 : 1 70,560 0.0412076 23.3 : 1
yyyAB $\begin{matrix} {1 \choose 1}{3 \choose 3}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$ 240 0.0001402 7,133.6 : 1 336 0.0001962 5,095.1 : 1
yyAAB $\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 2,160 0.0012615 791.7 : 1 3,024 0.0017660 565.2 : 1
yyAB* $\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 2}{4 \choose 1}^2{51-4r \choose 1} \end{matrix}$ 13,680 0.0079892 124.2 : 1 15,120 0.0088302 112.2 : 1
yAAAB $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 1,440 0.0008410 1,188.1 : 1 2,016 0.0011774 848.4 : 1
yAABB $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 2}^2 \end{matrix}$ 1,620 0.0009461 1,056.0 : 1 2,268 0.0013245 754.0 : 1
yAAB* $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$ 41,040 0.0239677 40.7 : 1 45,360 0.0264906 36.7 : 1
yAB** $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 1}^2{51-4r \choose 2} \end{matrix}$ 123,120 0.0719031 12.9 : 1 105,840 0.0618115 15.2 : 1
Total 367,320 0.2145180 3.7 : 1 348,866 0.2037407 3.9 : 1
No low AAAAB $\begin{matrix} {r-2 \choose 1}{4 \choose 4}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 120 0.0000701 15,019.2 : 1 168 0.0000613 16,306.7 : 1
AAAA* $\begin{matrix} {r-2 \choose 1}{4 \choose 4}{51-4r \choose 1} \end{matrix}$ 114 0.0000666 15,853.7 : 1 105 0.0000572 17,471.5 : 1
AAABB $\begin{matrix} {r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 2} \end{matrix}$ 720 0.0004205 2,377.2 : 1 1,008 0.0005887 1,697.7 : 1
AAAB* $\begin{matrix} {r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$ 10,080 0.0058868 168.9 : 1 9,120 0.0053262 186.8 : 1
AABB* $\begin{matrix} {r-2 \choose 2}{4 \choose 2}^2{51-4r \choose 1} \end{matrix}$ 11,340 0.0066227 150.0 : 1 10,260 0.0059919 165.9 : 1
AAA** $\begin{matrix} {r-2 \choose 1}{4 \choose 3}{51-4r \choose 2} \end{matrix}$ 2,940 0.0017170 581.4 : 1 4,104 0.0023968 416.2 : 1
AAB** $\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 2} \end{matrix}$ 105,840 0.0618115 15.2 : 1 123,120 0.0719031 12.9 : 1
xxyyy $\begin{matrix} {1 \choose 1}{2 \choose 2}{1 \choose 1}{3 \choose 3} \end{matrix}$ 1 0.0000006 1,712,303.0 : 1 1 0.0000006 1,712,303.0 : 1
xxyy* $\begin{matrix} {1 \choose 1}{2 \choose 2}{1 \choose 1}{3 \choose 2}{51-4r \choose 1} \end{matrix}$ 57 0.0000333 30,039.4 : 1 45 0.0000263 38,050.2 : 1
xxy** $\begin{matrix} {1 \choose 1}{2 \choose 2}{1 \choose 1}{3 \choose 1}{51-4r \choose 2} \end{matrix}$ 513 0.0002996 3,336.8 : 1 315 0.0001840 5,434.9 : 1
xxAAA $\begin{matrix} {1 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 3} \end{matrix}$ 24 0.0000140 71,345.0 : 1 28 0.0000164 61,152.7 : 1
xxAA* $\begin{matrix} {1 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 2}{51-4r \choose 1} \end{matrix}$ 684 0.0003995 2,502.4 : 1 630 0.0003679 2,716.9 : 1
xxA** $\begin{matrix} {1 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 2} \end{matrix}$ 4,104 0.0023968 416.2 : 1 2,940 0.0017170 581.4 : 1
xyyy* $\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 3}{51-4r \choose 1} \end{matrix}$ 38 0.0000222 45,059.6 : 1 30 0.0000175 57,075.8 : 1
xyy** $\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 2}{51-4r \choose 2} \end{matrix}$ 1,026 0.0005992 1,667.9 : 1 630 0.0003679 2,716.9 : 1
xAAAA $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 4} \end{matrix}$ 12 0.0000070 142,691.0 : 1 14 0.0000082 122,306.4 : 1
xAAA* $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 3}{51-4r \choose 1} \end{matrix}$ 912 0.0005326 1,876.5 : 1 840 0.0004906 2,037.5 : 1
xAA** $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 2}{51-4r \choose 2} \end{matrix}$ 12,312 0.0071903 138.1 : 1 8,820 0.0051510 193.1 : 1
yyyAA $\begin{matrix} {1 \choose 1}{3 \choose 3}{r-2 \choose 1}{4 \choose 2} \end{matrix}$ 36 0.0000210 47,563.0 : 1 42 0.0000245 40,768.1 : 1
yyyA* $\begin{matrix} {1 \choose 1}{3 \choose 3}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$ 456 0.0002663 3,754.1 : 1 420 0.0002453 4,075.9 : 1
yyy** $\begin{matrix} {1 \choose 1}{3 \choose 3}{51-4r \choose 2} \end{matrix}$ 171 0.0000999 10,012.5 : 1 105 0.0000613 16,306.7 : 1
yyAAA $\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 3} \end{matrix}$ 72 0.0000420 23,781.0 : 1 84 0.0000491 20,383.6 : 1
yyAA* $\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2}{51-4r \choose 1} \end{matrix}$ 2,052 0.0011984 833.5 : 1 1,890 0.0011038 905.0 : 1
yyA** $\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 2} \end{matrix}$ 12,312 0.0071903 138.1 : 1 8,820 0.0051510 193.1 : 1
yAAAA $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 4} \end{matrix}$ 18 0.0000105 95,127.0 : 1 21 0.0000123 81,537.3 : 1
yAAA* $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 3}{51-4r \choose 1} \end{matrix}$ 1,368 0.0007989 1,250.7 : 1 1,260 0.0007359 1,358.0 : 1
yAA** $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{51-4r \choose 2} \end{matrix}$ 18,468 0.0107855 91.7 : 1 13,230 0.0077264 128.4 : 1
??*** $\begin{matrix} {4r-3 \choose 2}{51-4r \choose 3} \end{matrix}$ 393,414 0.2297571 3.4 : 1 240,240 0.1403022 6.1 : 1
?**** $\begin{matrix} {4r-3 \choose 1}{51-4r \choose 4} \end{matrix}$ 112,404 0.0656449 14.2 : 1 45,045 0.0263067 37.0 : 1
***** $\begin{matrix} {51-4r \choose 5} \end{matrix}$ 11,628 0.0067909 146.3 : 1 3,003 0.0017538 569.2 : 1
Total 719,640 0.4202758 1.4 : 1 459,934 0.2686053 2.7 : 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 make Board Derivation 8-high qualifier (r = 8) 9-high qualifier (r = 9)
Combos Probability Odds Combos Probability Odds
Make low ABC $\begin{matrix} {r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$ 1,280 0.0740056 12.5 : 1 2,240 0.1295097 6.7 : 1
Miss low xAB $\begin{matrix} {1 \choose 1}{1 \choose 1}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$ 240 0.0138760 71.1 : 1 336 0.0194265 50.5 : 1
yAB $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$ 720 0.0416281 23.0 : 1 1,008 0.0582794 16.2 : 1
xyA $\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 72 0.0041628 239.2 : 1 84 0.0048566 204.9 : 1
Total 1,032 0.0596670 15.8 : 1 1,428 0.0825624 11.1 : 1
No low AAA $\begin{matrix} {r-2 \choose 1}{4 \choose 3} \end{matrix}$ 24 0.0013876 719.7 : 1 28 0.0016189 616.7 : 1
AAB $\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 720 0.0416281 23.0 : 1 1,008 0.0582794 16.2 : 1
xyy $\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 2} \end{matrix}$ 3 0.0001735 5,764.3 : 1 3 0.0001735 5,764.3 : 1
xAA $\begin{matrix} {1 \choose 1}{1 \choose 1}{r-2 \choose 1}{4 \choose 2} \end{matrix}$ 36 0.0020814 479.4 : 1 42 0.0024283 410.8 : 1
yyy $\begin{matrix} {1 \choose 1}{3 \choose 3} \end{matrix}$ 1 0.0000578 17,295.0 : 1 1 0.0000578 17,295.0 : 1
yyA $\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 108 0.0062442 159.1 : 1 126 0.0072849 136.3 : 1
yAA $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2} \end{matrix}$ 216 0.0124884 79.1 : 1 252 0.0145698 67.6 : 1
??* $\begin{matrix} {4r-4 \choose 2}{52-4r \choose 1} \end{matrix}$ 7,560 0.4370953 1.3 : 1 7,936 0.4588344 1.2 : 1
?** $\begin{matrix} {4r-4 \choose 1}{52-4r \choose 2} \end{matrix}$ 5,320 0.3075856 2.3 : 1 3,840 0.2220167 3.5 : 1
*** $\begin{matrix} {52-4r \choose 3} \end{matrix}$ 1,140 0.0659112 14.2 : 1 560 0.0323774 29.9 : 1
Total 14,984 0.8663275 0.2 : 1 13,628 0.7879278 0.3 : 1
Derivations for low hand shape xxxy (a low three of a kind and a low 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
Make low ABCD $\begin{matrix} {r-2 \choose 4}{4 \choose 1}^4 \end{matrix}$ 8,960 0.0460479 20.7 : 1 17,920 0.0920958 9.9 : 1
AABC $\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$ 10,080 0.0518039 18.3 : 1 16,128 0.0828862 11.1 : 1
ABC* $\begin{matrix} {r-2 \choose 3}{4 \choose 1}^3{52-4r \choose 1} \end{matrix}$ 25,600 0.1315654 6.6 : 1 35,840 0.1841916 4.4 : 1
xABC $\begin{matrix} {1 \choose 1}{1 \choose 1}{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$ 1,280 0.0065783 151.0 : 1 2,240 0.0115120 85.9 : 1
yABC $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$ 3,840 0.0197348 49.7 : 1 6,720 0.0345359 28.0 : 1
Total 40,320 0.2072155 3.8 : 1 63,840 0.3280913 2.0 : 1
Miss low xyyA $\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 72 0.0003700 2,701.5 : 1 84 0.0004317 2,315.4 : 1
xyAA $\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2} \end{matrix}$ 108 0.0005550 1,800.7 : 1 126 0.0006475 1,543.3 : 1
xAAB $\begin{matrix} {1 \choose 1}{1 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 720 0.0037003 269.3 : 1 1,008 0.0051804 192.0 : 1
xyAB $\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$ 720 0.0037003 269.3 : 1 1,008 0.0051804 192.0 : 1
xyA* $\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 1,440 0.0074006 134.1 : 1 1,344 0.0069072 143.8 : 1
xAB* $\begin{matrix} {1 \choose 1}{1 \choose 1}{r-2 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$ 4,800 0.0246685 39.5 : 1 5,376 0.0276287 35.2 : 1
yyAB $\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$ 720 0.0037003 269.3 : 1 1,008 0.0051804 192.0 : 1
yAAB $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 2,160 0.0111008 89.1 : 1 3,024 0.0155412 63.3 : 1
yAB* $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$ 14,400 0.0740056 12.5 : 1 16,128 0.0828862 11.1 : 1
Total 25,140 0.1292014 6.7 : 1 29,106 0.1495837 5.7 : 1
No low AAAA $\begin{matrix} {r-2 \choose 1}{4 \choose 4} \end{matrix}$ 6 0.0000308 32,429.0 : 1 7 0.0000360 27,796.1 : 1
AAAB $\begin{matrix} {r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 480 0.0024669 404.4 : 1 672 0.0034536 288.6 : 1
AABB $\begin{matrix} {r-2 \choose 2}{4 \choose 2}^2 \end{matrix}$ 540 0.0027752 359.3 : 1 756 0.0038853 256.4 : 1
AAA* $\begin{matrix} {r-2 \choose 1}{4 \choose 3}{52-4r \choose 1} \end{matrix}$ 480 0.0024669 404.4 : 1 448 0.0023024 433.3 : 1
AAB* $\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 14,400 0.0740056 12.5 : 1 16,128 0.0828862 11.1 : 1
xyyy $\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 3} \end{matrix}$ 1 0.0000051 194,579.0 : 1 1 0.0000051 194,579.0 : 1
xyy* $\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 2}{52-4r \choose 1} \end{matrix}$ 60 0.0003084 3,242.0 : 1 48 0.0002467 4,052.8 : 1
xAAA $\begin{matrix} {1 \choose 1}{1 \choose 1}{r-2 \choose 1}{4 \choose 3} \end{matrix}$ 24 0.0001233 8,106.5 : 1 28 0.0001439 6,948.3 : 1
xAA* $\begin{matrix} {1 \choose 1}{1 \choose 1}{r-2 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$ 720 0.0037003 269.3 : 1 672 0.0034536 288.6 : 1
yyyA $\begin{matrix} {1 \choose 1}{3 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 24 0.0001233 8,106.5 : 1 28 0.0001439 6,948.3 : 1
yyy* $\begin{matrix} {1 \choose 1}{3 \choose 3}{52-4r \choose 1} \end{matrix}$ 20 0.0001028 9,728.0 : 1 16 0.0000822 12,160.3 : 1
yyAA $\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2} \end{matrix}$ 108 0.0005550 1,800.7 : 1 126 0.0006475 1,543.3 : 1
yyA* $\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 1,440 0.0074006 134.1 : 1 1,344 0.0069072 143.8 : 1
yAAA $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 3} \end{matrix}$ 72 0.0003700 2,701.5 : 1 84 0.0004317 2,315.4 : 1
yAA* $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$ 2,160 0.0111008 89.1 : 1 2,016 0.0103608 95.5 : 1
??** $\begin{matrix} {4r-4 \choose 2}{52-4r \choose 2} \end{matrix}$ 71,820 0.3691027 1.7 : 1 59,520 0.3058896 2.3 : 1
?*** $\begin{matrix} {4r-4 \choose 1}{52-4r \choose 3} \end{matrix}$ 31,920 0.1640456 5.1 : 1 17,920 0.0920958 9.9 : 1
**** $\begin{matrix} {52-4r \choose 4} \end{matrix}$ 4,845 0.0248998 39.2 : 1 1,820 0.0093535 105.9 : 1
Total 129,120 0.6635831 0.5 : 1 101,634 0.5223250 0.9 : 1
Derivations for low hand shape xxxy (a low three of a kind and a low 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
Make low ABCDE $\begin{matrix} {r-2 \choose 5}{4 \choose 1}^5 \end{matrix}$ 6,144 0.0035881 277.7 : 1 21,504 0.0125585 78.6 : 1
AABCD $\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 3}{4 \choose 1}^3 \end{matrix}$ 23,040 0.0134556 73.3 : 1 53,760 0.0313963 30.9 : 1
ABCD* $\begin{matrix} {r-2 \choose 4}{4 \choose 1}^4{52-4r \choose 1} \end{matrix}$ 76,800 0.0448518 21.3 : 1 143,360 0.0837235 10.9 : 1
AABBC $\begin{matrix} {r-2 \choose 2}{4 \choose 2}^2{r-4 \choose 1}{4 \choose 1} \end{matrix}$ 8,640 0.0050458 197.2 : 1 15,120 0.0088302 112.2 : 1
AABC* $\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$ 115,200 0.0672778 13.9 : 1 161,280 0.0941889 9.6 : 1
AAABC $\begin{matrix} {r-2 \choose 1}{4 \choose 3}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$ 3,840 0.0022426 444.9 : 1 6,720 0.0039245 253.8 : 1
ABC** $\begin{matrix} {r-2 \choose 3}{4 \choose 1}^3{52-4r \choose 2} \end{matrix}$ 243,200 0.1420309 6.0 : 1 268,800 0.1569815 5.4 : 1
xABCD $\begin{matrix} {1 \choose 1}{1 \choose 1}{r-2 \choose 4}{4 \choose 1}^4 \end{matrix}$ 3,840 0.0022426 444.9 : 1 8,960 0.0052327 190.1 : 1
xABC* $\begin{matrix} {1 \choose 1}{1 \choose 1}{r-2 \choose 3}{4 \choose 1}^3{52-4r \choose 1} \end{matrix}$ 25,600 0.0149506 65.9 : 1 35,840 0.0209309 46.8 : 1
xAABC $\begin{matrix} {1 \choose 1}{1 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$ 5,760 0.0033639 296.3 : 1 10,080 0.0058868 168.9 : 1
xyABC $\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$ 3,840 0.0022426 444.9 : 1 6,720 0.0039245 253.8 : 1
yyABC $\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$ 3,840 0.0022426 444.9 : 1 6,720 0.0039245 253.8 : 1
yABCD $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 4}{4 \choose 1}^4 \end{matrix}$ 11,520 0.0067278 147.6 : 1 26,880 0.0156981 62.7 : 1
yABC* $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 3}{4 \choose 1}^3{52-4r \choose 1} \end{matrix}$ 76,800 0.0448518 21.3 : 1 107,520 0.0627926 14.9 : 1
yAABC $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$ 17,280 0.0100917 98.1 : 1 30,240 0.0176604 55.6 : 1
Total 625,344 0.3652062 1.7 : 1 903,504 0.5276540 0.9 : 1
Miss low xyyyA $\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 24 0.0000140 71,345.0 : 1 28 0.0000164 61,152.7 : 1
xyyAA $\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 3} \end{matrix}$ 108 0.0000631 15,853.7 : 1 126 0.0000736 13,588.7 : 1
xyyAB $\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 720 0.0004205 2,377.2 : 1 1,008 0.0005887 1,697.7 : 1
xyyA* $\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$ 1,440 0.0008410 1,188.1 : 1 1,344 0.0007849 1,273.0 : 1
xyAAA $\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 3} \end{matrix}$ 72 0.0000420 23,781.0 : 1 84 0.0000491 20,383.6 : 1
xyAAB $\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 2,160 0.0012615 791.7 : 1 3,024 0.0017660 565.2 : 1
xyAA* $\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$ 2,160 0.0012615 791.7 : 1 2,016 0.0011774 848.4 : 1
xyAB* $\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$ 14,400 0.0084097 117.9 : 1 16,128 0.0094189 105.2 : 1
xyA** $\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 2} \end{matrix}$ 13,680 0.0079892 124.2 : 1 10,080 0.0058868 168.9 : 1
xAAAB $\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 480 0.0002803 3,566.3 : 1 672 0.0003925 2,547.1 : 1
xAABB $\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 2}{4 \choose 2}^2 \end{matrix}$ 540 0.0003154 3,169.9 : 1 756 0.0004415 2,264.0 : 1
xAAB* $\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 14,400 0.0084097 117.9 : 1 16,128 0.0094189 105.2 : 1
xAB** $\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 2}{4 \choose 1}^2{52-4r \choose 2} \end{matrix}$ 45,600 0.0266308 36.6 : 1 40,320 0.0235472 41.5 : 1
yyyAB $\begin{matrix} {1 \choose 1}{3 \choose 3}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$ 240 0.0001402 7,133.6 : 1 336 0.0001962 5,095.1 : 1
yyAAB $\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 2,160 0.0012615 791.7 : 1 3,024 0.0017660 565.2 : 1
yyAB* $\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$ 14,400 0.0084097 117.9 : 1 16,128 0.0094189 105.2 : 1
yAAAB $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 1,440 0.0008410 1,188.1 : 1 2,016 0.0011774 848.4 : 1
yAABB $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 2}^2 \end{matrix}$ 1,620 0.0009461 1,056.0 : 1 2,268 0.0013245 754.0 : 1
yAAB* $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 43,200 0.0252292 38.6 : 1 48,384 0.0282567 34.4 : 1
yAB** $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 1}^2{52-4r \choose 2} \end{matrix}$ 136,800 0.0798924 11.5 : 1 120,960 0.0706417 13.2 : 1
Total 295,644 0.1726586 4.8 : 1 284,830 0.1663431 5.0 : 1
No low AAAAB $\begin{matrix} {r-2 \choose 1}{4 \choose 4}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 120 0.0000701 15,019.2 : 1 168 0.0000613 16,306.7 : 1
AAAA* $\begin{matrix} {r-2 \choose 1}{4 \choose 4}{52-4r \choose 1} \end{matrix}$ 120 0.0000701 14,268.2 : 1 112 0.0000654 15,287.4 : 1
AAABB $\begin{matrix} {r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 2} \end{matrix}$ 720 0.0004205 2,377.2 : 1 1,008 0.0005887 1,697.7 : 1
AAAB* $\begin{matrix} {r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 9,600 0.0056065 177.4 : 1 10,752 0.0062793 158.3 : 1
AABB* $\begin{matrix} {r-2 \choose 2}{4 \choose 2}^2{52-4r \choose 1} \end{matrix}$ 10,800 0.0063073 157.5 : 1 12,096 0.0070642 140.6 : 1
AAA** $\begin{matrix} {r-2 \choose 1}{4 \choose 3}{52-4r \choose 2} \end{matrix}$ 4,560 0.0026631 374.5 : 1 3,360 0.0019623 508.6 : 1
AAB** $\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 2} \end{matrix}$ 136,800 0.0798924 11.5 : 1 120,960 0.0706417 13.2 : 1
xyyy* $\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 3}{52-4r \choose 1} \end{matrix}$ 20 0.0000117 85,614.2 : 1 16 0.0000093 107,018.0 : 1
xyy** $\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 2}{52-4r \choose 2} \end{matrix}$ 570 0.0003329 3,003.0 : 1 360 0.0002102 4,755.4 : 1
xAAAA $\begin{matrix} {1 \choose 1}{1 \choose 1}{r-2 \choose 1}{4 \choose 4} \end{matrix}$ 6 0.0000035 285,383.0 : 1 7 0.0000041 244,613.9 : 1
xAAA* $\begin{matrix} {1 \choose 1}{1 \choose 1}{r-2 \choose 1}{4 \choose 3}{52-4r \choose 1} \end{matrix}$ 480 0.0002803 3,566.3 : 1 448 0.0002616 3,821.1 : 1
xAA** $\begin{matrix} {1 \choose 1}{1 \choose 1}{r-2 \choose 1}{4 \choose 2}{52-4r \choose 2} \end{matrix}$ 6,840 0.0039946 249.3 : 1 5,040 0.0029434 338.7 : 1
yyyAA $\begin{matrix} {1 \choose 1}{3 \choose 3}{r-2 \choose 1}{4 \choose 2} \end{matrix}$ 36 0.0000210 47,563.0 : 1 42 0.0000245 40,768.1 : 1
yyyA* $\begin{matrix} {1 \choose 1}{3 \choose 3}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 480 0.0002803 3,566.3 : 1 448 0.0002616 3,821.1 : 1
yyy** $\begin{matrix} {1 \choose 1}{3 \choose 3}{52-4r \choose 2} \end{matrix}$ 190 0.0001110 9,011.1 : 1 120 0.0000701 14,268.2 : 1
yyAAA $\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 3} \end{matrix}$ 72 0.0000420 23,781.0 : 1 84 0.0000491 20,383.6 : 1
yyAA* $\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$ 2,160 0.0012615 791.7 : 1 2,016 0.0011774 848.4 : 1
yyA** $\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 2} \end{matrix}$ 13,680 0.0079892 124.2 : 1 10,080 0.0058868 168.9 : 1
yAAAA $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 4} \end{matrix}$ 18 0.0000105 95,127.0 : 1 21 0.0000123 81,537.3 : 1
yAAA* $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 3}{52-4r \choose 1} \end{matrix}$ 1,440 0.0008410 1,188.1 : 1 1,344 0.0007849 1,273.0 : 1
yAA** $\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{52-4r \choose 2} \end{matrix}$ 20,520 0.0119839 82.4 : 1 15,120 0.0088302 112.2 : 1
??*** $\begin{matrix} {4r-4 \choose 2}{52-4r \choose 3} \end{matrix}$ 430,920 0.2516609 3.0 : 1 277,760 0.1622142 5.2 : 1
?**** $\begin{matrix} {4r-4 \choose 1}{52-4r \choose 4} \end{matrix}$ 135,660 0.0792266 11.6 : 1 58,240 0.0340127 28.4 : 1
***** $\begin{matrix} {52-4r \choose 5} \end{matrix}$ 15,504 0.0090545 109.4 : 1 4,368 0.0025509 391.0 : 1
Total 791,316 0.4621352 1.2 : 1 523,970 0.3060029 2.3 : 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 make Board Derivation 8-high qualifier (r = 8) 9-high qualifier (r = 9)
Combos Probability Odds Combos Probability Odds
Make low ABC $\begin{matrix} {r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$ 1,280 0.0740056 12.5 : 1 2,240 0.1295097 6.7 : 1
Miss low xAB $\begin{matrix} {2 \choose 1}{2 \choose 1}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$ 960 0.0555042 17.0 : 1 1,344 0.0777058 11.9 : 1
xyA $\begin{matrix} {2 \choose 2}{2 \choose 1}^2{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 96 0.0055504 179.2 : 1 112 0.0064755 153.4 : 1
Total 1,056 0.0610546 15.4 : 1 1,456 0.0841813 10.9 : 1
No low AAA $\begin{matrix} {r-2 \choose 1}{4 \choose 3} \end{matrix}$ 24 0.0013876 719.7 : 1 28 0.0016189 616.7 : 1
AAB $\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 720 0.0416281 23.0 : 1 1,008 0.0582794 16.2 : 1
xxy $\begin{matrix} {2 \choose 1}{2 \choose 2}{1 \choose 1}{2 \choose 1} \end{matrix}$ 4 0.0002313 4,323.0 : 1 4 0.0002313 4,323.0 : 1
xxA $\begin{matrix} {2 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 48 0.0027752 359.3 : 1 56 0.0032377 307.9 : 1
xAA $\begin{matrix} {2 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 2} \end{matrix}$ 144 0.0083256 119.1 : 1 168 0.0097132 102.0 : 1
??* $\begin{matrix} {4r-4 \choose 2}{52-4r \choose 1} \end{matrix}$ 7,560 0.4370953 1.3 : 1 7,936 0.4588344 1.2 : 1
?** $\begin{matrix} {4r-4 \choose 1}{52-4r \choose 2} \end{matrix}$ 5,320 0.3075856 2.3 : 1 3,840 0.2220167 3.5 : 1
*** $\begin{matrix} {52-4r \choose 3} \end{matrix}$ 1,140 0.0659112 14.2 : 1 560 0.0323774 29.9 : 1
Total 14,960 0.8649399 0.2 : 1 13,600 0.7863090 0.3 : 1
Derivations for low hand shape xxyy (two low pair) 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 $\begin{matrix} {r-2 \choose 4}{4 \choose 1}^4 \end{matrix}$ 3,840 0.0197348 49.7 : 1 8,960 0.0460479 20.7 : 1
AABC $\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$ 5,760 0.0296022 32.8 : 1 10,080 0.0518039 18.3 : 1
ABC* $\begin{matrix} {r-2 \choose 3}{4 \choose 1}^3{52-4r \choose 1} \end{matrix}$ 25,600 0.1315654 6.6 : 1 35,840 0.1841916 4.4 : 1
xABC $\begin{matrix} {2 \choose 1}{2 \choose 1}{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$ 5,120 0.0263131 37.0 : 1 8,960 0.0460479 20.7 : 1
Total 40,320 0.2072155 3.8 : 1 63,840 0.3280913 2.0 : 1
Miss low xxyA $\begin{matrix} {2 \choose 1}{2 \choose 2}{1 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 96 0.0004934 2,025.9 : 1 112 0.0005756 1,736.3 : 1
xxAB $\begin{matrix} {2 \choose 1}{2 \choose 2}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$ 480 0.0024669 404.4 : 1 672 0.0034536 288.6 : 1
xyAA $\begin{matrix} {2 \choose 2}{2 \choose 1}^2{r-2 \choose 1}{4 \choose 2} \end{matrix}$ 144 0.0007401 1,350.3 : 1 168 0.0008634 1,157.2 : 1
xAAB $\begin{matrix} {2 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 2,880 0.0148011 66.6 : 1 4,032 0.0207216 47.3 : 1
xyAB $\begin{matrix} {2 \choose 2}{2 \choose 1}^2{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$ 960 0.0049337 201.7 : 1 1,344 0.0069072 143.8 : 1
xyA* $\begin{matrix} {2 \choose 2}{2 \choose 1}^2{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 1,920 0.0098674 100.3 : 1 1,792 0.0092096 107.6 : 1
xAB* $\begin{matrix} {2 \choose 1}{2 \choose 1}{r-2 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$ 19,200 0.0986741 9.1 : 1 21,504 0.1105150 8.0 : 1
Total 25,680 0.1319766 6.6 : 1 29,624 0.1522459 5.6 : 1
No low AAAA $\begin{matrix} {r-2 \choose 1}{4 \choose 4} \end{matrix}$ 6 0.0000308 32,429.0 : 1 7 0.0000360 27,796.1 : 1
AAAB $\begin{matrix} {r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 480 0.0024669 404.4 : 1 672 0.0034536 288.6 : 1
AABB $\begin{matrix} {r-2 \choose 2}{4 \choose 2}^2 \end{matrix}$ 540 0.0027752 359.3 : 1 756 0.0038853 256.4 : 1
AAA* $\begin{matrix} {r-2 \choose 1}{4 \choose 3}{52-4r \choose 1} \end{matrix}$ 480 0.0024669 404.4 : 1 448 0.0023024 433.3 : 1
AAB* $\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 14,400 0.0740056 12.5 : 1 16,128 0.0828862 11.1 : 1
xxyy $\begin{matrix} {2 \choose 2}{2 \choose 2}^2 \end{matrix}$ 1 0.0000051 194,579.0 : 1 1 0.0000051 194,579.0 : 1
xxy* $\begin{matrix} {2 \choose 1}{2 \choose 2}{1 \choose 1}{2 \choose 1}{52-4r \choose 1} \end{matrix}$ 80 0.0004111 2,431.3 : 1 64 0.0003289 3,039.3 : 1
xxAA $\begin{matrix} {2 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 2} \end{matrix}$ 72 0.0003700 2,701.5 : 1 84 0.0004317 2,315.4 : 1
xxA* $\begin{matrix} {2 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 960 0.0049337 201.7 : 1 896 0.0046048 216.2 : 1
xAAA $\begin{matrix} {2 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 3} \end{matrix}$ 96 0.0004934 2,025.9 : 1 112 0.0005756 1,736.3 : 1
xAA* $\begin{matrix} {2 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$ 2,880 0.0148011 66.6 : 1 2,688 0.0138144 71.4 : 1
??** $\begin{matrix} {4r-4 \choose 2}{52-4r \choose 2} \end{matrix}$ 71,820 0.3691027 1.7 : 1 59,520 0.3058896 2.3 : 1
?*** $\begin{matrix} {4r-4 \choose 1}{52-4r \choose 3} \end{matrix}$ 31,920 0.1640456 5.1 : 1 17,920 0.0920958 9.9 : 1
**** $\begin{matrix} {52-4r \choose 4} \end{matrix}$ 4,845 0.0248998 39.2 : 1 1,820 0.0093535 105.9 : 1
Total 128,580 0.6608079 0.5 : 1 101,116 0.5196629 0.9 : 1
Derivations for low hand shape xxyy (two low pair) on the river
Hand to make Board Derivation 8-high qualifier (r = 8) 9-high qualifier (r = 9)
Combos Probability Odds Combos Probability Odds
Make low ABCDE $\begin{matrix} {r-2 \choose 5}{4 \choose 1}^5 \end{matrix}$ 6,144 0.0035881 277.7 : 1 21,504 0.0125585 78.6 : 1
AABCD $\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 3}{4 \choose 1}^3 \end{matrix}$ 23,040 0.0134556 73.3 : 1 53,760 0.0313963 30.9 : 1
ABCD* $\begin{matrix} {r-2 \choose 4}{4 \choose 1}^4{52-4r \choose 1} \end{matrix}$ 76,800 0.0448518 21.3 : 1 143,360 0.0837235 10.9 : 1
AABBC $\begin{matrix} {r-2 \choose 2}{4 \choose 2}^2{r-4 \choose 1}{4 \choose 1} \end{matrix}$ 8,640 0.0050458 197.2 : 1 15,120 0.0088302 112.2 : 1
AABC* $\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$ 115,200 0.0672778 13.9 : 1 161,280 0.0941889 9.6 : 1
AAABC $\begin{matrix} {r-2 \choose 1}{4 \choose 3}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$ 3,840 0.0022426 444.9 : 1 6,720 0.0039245 253.8 : 1
ABC** $\begin{matrix} {r-2 \choose 3}{4 \choose 1}^3{52-4r \choose 2} \end{matrix}$ 243,200 0.1420309 6.0 : 1 268,800 0.1569815 5.4 : 1
xABCD $\begin{matrix} {2 \choose 1}{2 \choose 1}{r-2 \choose 4}{4 \choose 1}^4 \end{matrix}$ 15,360 0.0089704 110.5 : 1 35,840 0.0209309 46.8 : 1
xABC* $\begin{matrix} {2 \choose 1}{2 \choose 1}{r-2 \choose 3}{4 \choose 1}^3{52-4r \choose 1} \end{matrix}$ 102,400 0.0598025 15.7 : 1 143,360 0.0837235 10.9 : 1
xAABC $\begin{matrix} {2 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$ 23,040 0.0134556 73.3 : 1 40,320 0.0235472 41.5 : 1
xxABC $\begin{matrix} {2 \choose 1}{2 \choose 2}{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$ 2,560 0.0014951 667.9 : 1 4,480 0.0026164 381.2 : 1
xyABC $\begin{matrix} {2 \choose 2}{2 \choose 1}^2{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$ 5,120 0.0029901 333.4 : 1 8,960 0.0052327 190.1 : 1
Total 625,344 0.3652062 1.7 : 1 903,504 0.5276540 0.9 : 1
Miss low xxyyA $\begin{matrix} {2 \choose 2}{2 \choose 2}^2{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 24 0.0000140 71,345.0 : 1 28 0.0000164 61,152.7 : 1
xxyAB $\begin{matrix} {2 \choose 1}{2 \choose 2}{1 \choose 1}{2 \choose 1}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$ 960 0.0005606 1,782.7 : 1 1,344 0.0007849 1,273.0 : 1
xxyAA $\begin{matrix} {2 \choose 1}{2 \choose 2}{1 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 2} \end{matrix}$ 144 0.0000841 11,890.0 : 1 168 0.0000981 10,191.3 : 1
xxyA* $\begin{matrix} {2 \choose 1}{2 \choose 2}{1 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 1,920 0.0011213 890.8 : 1 1,792 0.0010465 954.5 : 1
xxAAB $\begin{matrix} {2 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 1,440 0.0008410 1,188.1 : 1 2,016 0.0011774 848.4 : 1
xxAB* $\begin{matrix} {2 \choose 1}{2 \choose 2}{r-2 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$ 9,600 0.0056065 177.4 : 1 10,752 0.0062793 158.3 : 1
xyAAA $\begin{matrix} {2 \choose 2}{2 \choose 1}^2{r-2 \choose 1}{4 \choose 3} \end{matrix}$ 96 0.0000561 17,835.5 : 1 112 0.0000654 15,287.4 : 1
xyAAB $\begin{matrix} {2 \choose 2}{2 \choose 1}^2{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 2,880 0.0016819 593.6 : 1 4,032 0.0023547 423.7 : 1
xyAA* $\begin{matrix} {2 \choose 2}{2 \choose 1}^2{r-2 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$ 2,880 0.0016819 593.6 : 1 2,688 0.0015698 636.0 : 1
xyAB* $\begin{matrix} {2 \choose 2}{2 \choose 1}^2{r-2 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$ 19,200 0.0112130 88.2 : 1 21,504 0.0125585 78.6 : 1
xyA** $\begin{matrix} {2 \choose 2}{2 \choose 1}^2{r-2 \choose 1}{4 \choose 1}{52-4r \choose 2} \end{matrix}$ 18,240 0.0106523 92.9 : 1 13,440 0.0078491 126.4 : 1
xAAAB $\begin{matrix} {2 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$ 1,920 0.0011213 890.8 : 1 2,688 0.0015698 636.0 : 1
xAABB $\begin{matrix} {2 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 2}^2 \end{matrix}$ 2,160 0.0012615 791.7 : 1 3,024 0.0017660 565.2 : 1
xAAB* $\begin{matrix} {2 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 57,600 0.0336389 28.7 : 1 64,512 0.0376756 25.5 : 1
xAB** $\begin{matrix} {2 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 1}^2{52-4r \choose 2} \end{matrix}$ 182,400 0.1065231 8.4 : 1 161,280 0.0941889 9.6 : 1
Total 301,464 0.1760575 4.7 : 1 289,380 0.1690004 4.9 : 1
No low AAAAB $\begin{matrix} {r-2 \choose 1}{4 \choose 4}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 120 0.0000701 14,268.2 : 1 168 0.0000981 10,191.3 : 1
AAAA* $\begin{matrix} {r-2 \choose 1}{4 \choose 4}{52-4r \choose 1} \end{matrix}$ 120 0.0000701 14,268.2 : 1 112 0.0000654 15,287.4 : 1
AAABB $\begin{matrix} {r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 2} \end{matrix}$ 720 0.0004205 2,377.2 : 1 1,008 0.0005887 1,697.7 : 1
AAAB* $\begin{matrix} {r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 9,600 0.0056065 177.4 : 1 10,752 0.0062793 158.3 : 1
AABB* $\begin{matrix} {r-2 \choose 2}{4 \choose 2}^2{52-4r \choose 1} \end{matrix}$ 10,800 0.0063073 157.5 : 1 12,096 0.0070642 140.6 : 1
AAA** $\begin{matrix} {r-2 \choose 1}{4 \choose 3}{52-4r \choose 2} \end{matrix}$ 4,560 0.0026631 374.5 : 1 3,360 0.0019623 508.6 : 1
AAB** $\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 2} \end{matrix}$ 136,800 0.0798924 11.5 : 1 120,960 0.0706417 13.2 : 1
xxyy* $\begin{matrix} {2 \choose 2}{2 \choose 2}^2{52-4r \choose 1} \end{matrix}$ 20 0.0000117 85,614.2 : 1 16 0.0000093 107,018.0 : 1
xxy** $\begin{matrix} {2 \choose 1}{2 \choose 2}{1 \choose 1}{2 \choose 1}{52-4r \choose 2} \end{matrix}$ 760 0.0004438 2,252.0 : 1 480 0.0002803 3,566.3 : 1
xxAAA $\begin{matrix} {2 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 3} \end{matrix}$ 48 0.0000280 35,672.0 : 1 56 0.0000327 30,575.9 : 1
xxAA* $\begin{matrix} {2 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$ 1,440 0.0008410 1,188.1 : 1 1,344 0.0007849 1,273.0 : 1
xxA** $\begin{matrix} {2 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 2} \end{matrix}$ 9,120 0.0053262 186.8 : 1 6,720 0.0039245 253.8 : 1
xAAAA $\begin{matrix} {2 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 4} \end{matrix}$ 24 0.0000140 71,345.0 : 1 28 0.0000164 61,152.7 : 1
xAAA* $\begin{matrix} {2 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 3}{52-4r \choose 1} \end{matrix}$ 1,920 0.0011213 890.8 : 1 1,792 0.0010465 954.5 : 1
xAA** $\begin{matrix} {2 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 2}{52-4r \choose 2} \end{matrix}$ 27,360 0.0159785 61.6 : 1 20,160 0.0117736 83.9 : 1
??*** $\begin{matrix} {4r-4 \choose 2}{52-4r \choose 3} \end{matrix}$ 430,920 0.2516609 3.0 : 1 277,760 0.1622142 5.2 : 1
?**** $\begin{matrix} {4r-4 \choose 1}{52-4r \choose 4} \end{matrix}$ 135,660 0.0792266 11.6 : 1 58,240 0.0340127 28.4 : 1
***** $\begin{matrix} {52-4r \choose 5} \end{matrix}$ 15,504 0.0090545 109.4 : 1 4,368 0.0025509 391.0 : 1
Total 785,496 0.4587363 1.2 : 1 519,420 0.3033457 2.3 : 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 make Board Derivation 8-high qualifier (r = 8) 9-high qualifier (r = 9)
Combos Probability Odds Combos Probability Odds
Make low ABC $\begin{matrix} {r-3 \choose 3}{4 \choose 1}^3 \end{matrix}$ 640 0.0370028 26.0 : 1 1,280 0.0740056 12.5 : 1
xAB $\begin{matrix} {3 \choose 1}{3 \choose 1}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$ 1,440 0.0832562 11.0 : 1 2,160 0.1248844 7.0 : 1
Total 2,080 0.1202590 7.3 : 1 3,440 0.1988899 4.0 : 1
Miss low xyA $\begin{matrix} {3 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 540 0.0312211 31.0 : 1 648 0.0374653 25.7 : 1
xyz $\begin{matrix} {3 \choose 3}{3 \choose 1}^3 \end{matrix}$ 27 0.0015611 639.6 : 1 27 0.0015611 639.6 : 1
Total 567 0.0327821 29.5 : 1 675 0.0390264 24.6 : 1
No low AAA $\begin{matrix} {r-3 \choose 1}{4 \choose 3} \end{matrix}$ 20 0.0011563 863.8 : 1 24 0.0013876 719.7 : 1
AAB $\begin{matrix} {r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1} \end{matrix}$ 480 0.0277521 35.0 : 1 720 0.0416281 23.0 : 1
xxx $\begin{matrix} {3 \choose 1}{3 \choose 3} \end{matrix}$ 3 0.0001735 5,764.3 : 1 3 0.0001735 5,764.3 : 1
xxy $\begin{matrix} {3 \choose 1}{3 \choose 2}{2 \choose 1}{3 \choose 1} \end{matrix}$ 54 0.0031221 319.3 : 1 54 0.0031221 319.3 : 1
xxA $\begin{matrix} {3 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 180 0.0104070 95.1 : 1 216 0.0124884 79.1 : 1
xAA $\begin{matrix} {3 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2} \end{matrix}$ 270 0.0156105 63.1 : 1 324 0.0187327 52.4 : 1
??* $\begin{matrix} {4r-3 \choose 2}{51-4r \choose 1} \end{matrix}$ 7,714 0.4459991 1.2 : 1 7,920 0.4579093 1.2 : 1
?** $\begin{matrix} {4r-3 \choose 1}{51-4r \choose 2} \end{matrix}$ 4,959 0.2867137 2.5 : 1 3,465 0.2003353 4.0 : 1
*** $\begin{matrix} {51-4r \choose 3} \end{matrix}$ 969 0.0560245 16.8 : 1 455 0.0263067 37.0 : 1
Total 14,649 0.8469588 0.2 : 1 13,181 0.7620837 0.3 : 1
Derivations for low hand shape xyz* (three low ranks 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
Make low ABCD $\begin{matrix} {r-3 \choose 4}{4 \choose 1}^4 \end{matrix}$ 1,280 0.0065783 151.0 : 1 3,840 0.0197348 49.7 : 1
AABC $\begin{matrix} {r-3 \choose 1}{4 \choose 2}{r-4 \choose 2}{4 \choose 1}^2 \end{matrix}$ 2,880 0.0148011 66.6 : 1 5,760 0.0296022 32.8 : 1
ABC* $\begin{matrix} {r-3 \choose 3}{4 \choose 1}^3{51-4r \choose 1} \end{matrix}$ 12,160 0.0624936 15.0 : 1 19,200 0.0986741 9.1 : 1
xABC $\begin{matrix} {3 \choose 1}{3 \choose 1}{r-3 \choose 3}{4 \choose 1}^3 \end{matrix}$ 5,760 0.0296022 32.8 : 1 11,520 0.0592044 15.9 : 1
xxAB $\begin{matrix} {3 \choose 1}{3 \choose 2}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$ 1,440 0.0074006 134.1 : 1 2,160 0.0111008 89.1 : 1
xAAB $\begin{matrix} {3 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1} \end{matrix}$ 4,320 0.0222017 44.0 : 1 6,480 0.0333025 29.0 : 1
xyAB $\begin{matrix} {3 \choose 2}{3 \choose 1}^2{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$ 4,320 0.0222017 44.0 : 1 6,480 0.0333025 29.0 : 1
xAB* $\begin{matrix} {3 \choose 1}{3 \choose 1}{r-3 \choose 2}{4 \choose 1}^2{51-4r \choose 1} \end{matrix}$ 27,360 0.1406105 6.1 : 1 32,400 0.1665125 5.0 : 1
Total 59,520 0.3058896 2.3 : 1 87,840 0.4514339 1.2 : 1
Miss low xxyA $\begin{matrix} {3 \choose 1}{3 \choose 2}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 1,080 0.0055504 179.2 : 1 1,296 0.0066605 149.1 : 1
xyAA $\begin{matrix} {3 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 2} \end{matrix}$ 810 0.0041628 239.2 : 1 972 0.0049954 199.2 : 1
xyA* $\begin{matrix} {3 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$ 10,260 0.0527290 18.0 : 1 9,720 0.0499537 19.0 : 1
xxyz $\begin{matrix} {3 \choose 1}{3 \choose 2}{2 \choose 2}{3 \choose 1}^2 \end{matrix}$ 81 0.0004163 2,401.2 : 1 81 0.0004163 2,401.2 : 1
xyzA $\begin{matrix} {3 \choose 3}{3 \choose 1}^3{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 540 0.0027752 359.3 : 1 648 0.0033302 299.3 : 1
xyz* $\begin{matrix} {3 \choose 3}{3 \choose 1}^3{51-4r \choose 1} \end{matrix}$ 513 0.0026364 378.3 : 1 405 0.0020814 479.4 : 1
Total 13,284 0.0682701 13.6 : 1 13,122 0.0674376 13.8 : 1
No low AAAA $\begin{matrix} {r-3 \choose 1}{4 \choose 4} \end{matrix}$ 5 0.0000257 38,915.0 : 1 6 0.0000308 32,429.0 : 1
AAAB $\begin{matrix} {r-3 \choose 1}{4 \choose 3}{r-4 \choose 1}{4 \choose 1} \end{matrix}$ 320 0.0016446 607.1 : 1 480 0.0024669 404.4 : 1
AABB $\begin{matrix} {r-3 \choose 2}{4 \choose 2}^2 \end{matrix}$ 360 0.0018501 539.5 : 1 540 0.0027752 359.3 : 1
AAA* $\begin{matrix} {r-3 \choose 1}{4 \choose 3}{51-4r \choose 1} \end{matrix}$ 380 0.0019529 511.1 : 1 360 0.0018501 539.5 : 1
AAB* $\begin{matrix} {r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$ 9,120 0.0468702 20.3 : 1 10,800 0.0555042 17.0 : 1
xxxy $\begin{matrix} {3 \choose 1}{3 \choose 3}{2 \choose 1}{3 \choose 1} \end{matrix}$ 18 0.0000925 10,809.0 : 1 18 0.0000925 10,809.0 : 1
xxxA $\begin{matrix} {3 \choose 1}{3 \choose 3}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 60 0.0003084 3,242.0 : 1 72 0.0003700 2,701.5 : 1
xxx* $\begin{matrix} {3 \choose 1}{3 \choose 3}{51-4r \choose 1} \end{matrix}$ 57 0.0002929 3,412.7 : 1 45 0.0002313 4,323.0 : 1
xxyy $\begin{matrix} {3 \choose 2}{3 \choose 2}^2 \end{matrix}$ 27 0.0001388 7,205.7 : 1 27 0.0001388 7,205.7 : 1
xxy* $\begin{matrix} {3 \choose 1}{3 \choose 2}{2 \choose 1}{3 \choose 1}{51-4r \choose 1} \end{matrix}$ 1,026 0.0052729 188.6 : 1 810 0.0041628 239.2 : 1
xxAA $\begin{matrix} {3 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 2} \end{matrix}$ 270 0.0013876 719.7 : 1 324 0.0016651 599.6 : 1
xxA* $\begin{matrix} {3 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$ 3,420 0.0175763 55.9 : 1 3,240 0.0166512 59.1 : 1
xAAA $\begin{matrix} {3 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 3} \end{matrix}$ 180 0.0009251 1,080.0 : 1 216 0.0011101 899.8 : 1
xAA* $\begin{matrix} {3 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}{51-4r \choose 1} \end{matrix}$ 5,130 0.0263645 36.9 : 1 4,860 0.0249769 39.0 : 1
??** $\begin{matrix} {4r-3 \choose 2}{51-4r \choose 2} \end{matrix}$ 69,426 0.3567993 1.8 : 1 55,440 0.2849214 2.5 : 1
?*** $\begin{matrix} {4r-3 \choose 1}{51-4r \choose 3} \end{matrix}$ 28,101 0.1444187 5.9 : 1 15,015 0.0771662 12.0 : 1
**** $\begin{matrix} {51-4r \choose 4} \end{matrix}$ 3,876 0.0199198 49.2 : 1 1,365 0.0070151 141.5 : 1
Total 121,776 0.6258403 0.6 : 1 93,618 0.4811286 1.1 : 1
Derivations for low hand shape xyz* (three low ranks 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
Make low ABCDE $\begin{matrix} {r-3 \choose 5}{4 \choose 1}^5 \end{matrix}$ 1,024 0.0005980 1,671.2 : 1 6,144 0.0035881 277.7 : 1
AABCD $\begin{matrix} {r-3 \choose 1}{4 \choose 2}{r-4 \choose 3}{4 \choose 1}^3 \end{matrix}$ 7,680 0.0044852 222.0 : 1 23,040 0.0134556 73.3 : 1
ABCD* $\begin{matrix} {r-3 \choose 4}{4 \choose 1}^4{51-4r \choose 1} \end{matrix}$ 24,320 0.0142031 69.4 : 1 57,600 0.0336389 28.7 : 1
AABBC $\begin{matrix} {r-3 \choose 2}{4 \choose 2}^2{r-4 \choose 1}{4 \choose 1} \end{matrix}$ 4,320 0.0025229 395.4 : 1 8,640 0.0050458 197.2 : 1
AABC* $\begin{matrix} {r-3 \choose 1}{4 \choose 2}{r-4 \choose 2}{4 \choose 1}^2{51-4r \choose 1} \end{matrix}$ 54,720 0.0319569 30.3 : 1 86,400 0.0504583 18.8 : 1
AAABC $\begin{matrix} {r-3 \choose 1}{4 \choose 3}{r-4 \choose 2}{4 \choose 1}^2 \end{matrix}$ 1,920 0.0011213 890.8 : 1 3,840 0.0022426 444.9 : 1
ABC** $\begin{matrix} {r-3 \choose 3}{4 \choose 1}^3{51-4r \choose 2} \end{matrix}$ 109,440 0.0639139 14.6 : 1 134,400 0.0784907 11.7 : 1
xABCD $\begin{matrix} {3 \choose 1}{3 \choose 1}{r-3 \choose 4}{4 \choose 1}^4 \end{matrix}$ 11,520 0.0067278 147.6 : 1 34,560 0.0201833 48.5 : 1
xABC* $\begin{matrix} {3 \choose 1}{3 \choose 1}{r-3 \choose 3}{4 \choose 1}^3{51-4r \choose 1} \end{matrix}$ 109,440 0.0639139 14.6 : 1 172,800 0.1009167 8.9 : 1
xAABC $\begin{matrix} {3 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 2}{4 \choose 1}^2 \end{matrix}$ 25,920 0.0151375 65.1 : 1 51,840 0.0302750 32.0 : 1
xxABC $\begin{matrix} {3 \choose 1}{3 \choose 2}{r-3 \choose 3}{4 \choose 1}^3 \end{matrix}$ 5,760 0.0033639 296.3 : 1 11,520 0.0067278 147.6 : 1
xyABC $\begin{matrix} {3 \choose 2}{3 \choose 1}^2{r-3 \choose 3}{4 \choose 1}^3 \end{matrix}$ 17,280 0.0100917 98.1 : 1 34,560 0.0201833 48.5 : 1
xxxAB $\begin{matrix} {3 \choose 1}{3 \choose 3}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$ 480 0.0002803 3,566.3 : 1 720 0.0004205 2,377.2 : 1
xxyAB $\begin{matrix} {3 \choose 1}{3 \choose 2}{2 \choose 1}{3 \choose 1}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$ 8,640 0.0050458 197.2 : 1 12,960 0.0075687 131.1 : 1
xxAAB $\begin{matrix} {3 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1} \end{matrix}$ 4,320 0.0025229 395.4 : 1 6,480 0.0037844 263.2 : 1
xxAB* $\begin{matrix} {3 \choose 1}{3 \choose 2}{r-3 \choose 2}{4 \choose 1}^2{51-4r \choose 1} \end{matrix}$ 27,360 0.0159785 61.6 : 1 32,400 0.0189219 51.8 : 1
xyAAB $\begin{matrix} {3 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1} \end{matrix}$ 12,960 0.0075687 131.1 : 1 19,440 0.0113531 87.1 : 1
xyAB* $\begin{matrix} {3 \choose 2}{3 \choose 1}^2{r-3 \choose 2}{4 \choose 1}^2{51-4r \choose 1} \end{matrix}$ 82,080 0.0479354 19.9 : 1 97,200 0.0567656 16.6 : 1
xAAAB $\begin{matrix} {3 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 2,880 0.0016819 593.6 : 1 4,320 0.0025229 395.4 : 1
xAABB $\begin{matrix} {3 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 2}^2 \end{matrix}$ 3,240 0.0018922 527.5 : 1 4,860 0.0028383 351.3 : 1
xAAB* $\begin{matrix} {3 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$ 82,080 0.0479354 19.9 : 1 97,200 0.0567656 16.6 : 1
xAB** $\begin{matrix} {3 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 1}^2{51-4r \choose 2} \end{matrix}$ 246,240 0.1438062 6.0 : 1 226,800 0.1324531 6.5 : 1
xyzAB $\begin{matrix} {3 \choose 3}{3 \choose 1}^3{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$ 4,320 0.0025229 395.4 : 1 6,480 0.0037844 263.2 : 1
Total 847,944 0.4952065 1.0 : 1 1,134,204 0.6623847 0.5 : 1
Miss low xxxyA $\begin{matrix} {3 \choose 1}{3 \choose 3}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 360 0.0002102 4,755.4 : 1 432 0.0002523 3,962.7 : 1
xxyyA $\begin{matrix} {3 \choose 2}{3 \choose 2}^2{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 540 0.0003154 3,169.9 : 1 648 0.0003784 2,641.4 : 1
xxyAA $\begin{matrix} {3 \choose 1}{3 \choose 2}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2} \end{matrix}$ 1,620 0.0009461 1,056.0 : 1 1,944 0.0011353 879.8 : 1
xxyA* $\begin{matrix} {3 \choose 1}{3 \choose 2}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$ 20,520 0.0119839 82.4 : 1 19,440 0.0113531 87.1 : 1
xyAAA $\begin{matrix} {3 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 3} \end{matrix}$ 540 0.0003154 3,169.9 : 1 648 0.0003784 2,641.4 : 1
xyAA* $\begin{matrix} {3 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 2}{51-4r \choose 1} \end{matrix}$ 15,390 0.0089879 110.3 : 1 14,580 0.0085148 116.4 : 1
xyA** $\begin{matrix} {3 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 1}{51-4r \choose 2} \end{matrix}$ 92,340 0.0539273 17.5 : 1 68,040 0.0397359 24.2 : 1
xxxyz $\begin{matrix} {3 \choose 1}{3 \choose 3}{2 \choose 2}{3 \choose 1}^2 \end{matrix}$ 27 0.0000158 63,417.7 : 1 27 0.0000158 63,417.7 : 1
xxyyz $\begin{matrix} {3 \choose 2}{3 \choose 2}^2{1 \choose 1}{3 \choose 1} \end{matrix}$ 81 0.0000473 21,138.6 : 1 81 0.0000473 21,138.6 : 1
xxyzA $\begin{matrix} {3 \choose 1}{3 \choose 2}{2 \choose 2}{3 \choose 1}^2 \end{matrix}$ 1,620 0.0009461 1,056.0 : 1 1,944 0.0011353 879.8 : 1
xxyz* $\begin{matrix} {3 \choose 1}{3 \choose 2}{2 \choose 2}{3 \choose 1}^2{51-4r \choose 1} \end{matrix}$ 1,539 0.0008988 1,111.6 : 1 1,215 0.0007096 1,408.3 : 1
xyzAA $\begin{matrix} {3 \choose 3}{3 \choose 1}^3{r-3 \choose 1}{4 \choose 2} \end{matrix}$ 810 0.0004730 2,113.0 : 1 972 0.0005677 1,760.6 : 1
xyzA* $\begin{matrix} {3 \choose 3}{3 \choose 1}^3{r-3 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$ 10,260 0.0059919 165.9 : 1 9,720 0.0056766 175.2 : 1
xyz** $\begin{matrix} {3 \choose 3}{3 \choose 1}^3{51-4r \choose 2} \end{matrix}$ 4,617 0.0026964 369.9 : 1 2,835 0.0016557 603.0 : 1
Total 150,264 0.0877554 10.4 : 1 122,526 0.0715562 13.0 : 1
No low AAAAB $\begin{matrix} {r-3 \choose 1}{4 \choose 4}{r-4 \choose 1}{4 \choose 1} \end{matrix}$ 80 0.0000467 21,402.8 : 1 120 0.0000701 14,268.2 : 1
AAAA* $\begin{matrix} {r-3 \choose 1}{4 \choose 4}{51-4r \choose 1} \end{matrix}$ 95 0.0000555 18,023.3 : 1 90 0.0000526 19,024.6 : 1
AAABB $\begin{matrix} {r-3 \choose 1}{4 \choose 3}{r-4 \choose 1}{4 \choose 2} \end{matrix}$ 480 0.0002803 3,566.3 : 1 720 0.0004205 2,377.2 : 1
AAAB* $\begin{matrix} {r-3 \choose 1}{4 \choose 3}{r-4 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$ 6,080 0.0035508 280.6 : 1 7,200 0.0042049 236.8 : 1
AABB* $\begin{matrix} {r-3 \choose 2}{4 \choose 2}^2{51-4r \choose 1} \end{matrix}$ 6,840 0.0039946 249.3 : 1 8,100 0.0047305 210.4 : 1
AAA** $\begin{matrix} {r-3 \choose 1}{4 \choose 3}{51-4r \choose 2} \end{matrix}$ 3,420 0.0019973 499.7 : 1 2,520 0.0014717 678.5 : 1
AAB** $\begin{matrix} {r-3 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}{51-4r \choose 2} \end{matrix}$ 82,080 0.0479354 19.9 : 1 75,600 0.0441510 21.6 : 1
xxxyy $\begin{matrix} {3 \choose 1}{3 \choose 3}{2 \choose 1}{3 \choose 2} \end{matrix}$ 18 0.0000105 95,127.0 : 1 18 0.0000105 95,127.0 : 1
xxxy* $\begin{matrix} {3 \choose 1}{3 \choose 3}{2 \choose 1}{3 \choose 1}{51-4r \choose 1} \end{matrix}$ 342 0.0001997 5,005.7 : 1 270 0.0001577 6,340.9 : 1
xxxAA $\begin{matrix} {3 \choose 1}{3 \choose 3}{r-3 \choose 1}{4 \choose 2} \end{matrix}$ 90 0.0000526 19,024.6 : 1 108 0.0000631 15,853.7 : 1
xxxA* $\begin{matrix} {3 \choose 1}{3 \choose 3}{r-3 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$ 1,140 0.0006658 1,501.0 : 1 1,080 0.0006307 1,584.5 : 1
xxx** $\begin{matrix} {3 \choose 1}{3 \choose 3}{51-4r \choose 2} \end{matrix}$ 513 0.0002996 3,336.8 : 1 315 0.0001840 5,434.9 : 1
xxyy* $\begin{matrix} {3 \choose 2}{3 \choose 2}^2{51-4r \choose 1} \end{matrix}$ 513 0.0002996 3,336.8 : 1 405 0.0002365 4,226.9 : 1
xxy** $\begin{matrix} {3 \choose 1}{3 \choose 2}{2 \choose 1}{3 \choose 1}{51-4r \choose 2} \end{matrix}$ 9,234 0.0053927 184.4 : 1 5,670 0.0033113 301.0 : 1
xxAAA $\begin{matrix} {3 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 3} \end{matrix}$ 180 0.0001051 9,511.8 : 1 216 0.0001261 7,926.3 : 1
xxAA* $\begin{matrix} {3 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 2}{51-4r \choose 1} \end{matrix}$ 5,130 0.0029960 332.8 : 1 4,860 0.0028383 351.3 : 1
xxA** $\begin{matrix} {3 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 1}{51-4r \choose 2} \end{matrix}$ 30,780 0.0179758 54.6 : 1 22,680 0.0132453 74.5 : 1
xAAAA $\begin{matrix} {3 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 4} \end{matrix}$ 45 0.0000263 38,050.2 : 1 54 0.0000315 31,708.3 : 1
xAAA* $\begin{matrix} {3 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 3}{51-4r \choose 1} \end{matrix}$ 3,420 0.0019973 499.7 : 1 3,240 0.0018922 527.5 : 1
xAA** $\begin{matrix} {3 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}{51-4r \choose 2} \end{matrix}$ 46,170 0.0269637 36.1 : 1 34,020 0.0198680 49.3 : 1
??*** $\begin{matrix} {4r-3 \choose 2}{51-4r \choose 3} \end{matrix}$ 393,414 0.2297571 3.4 : 1 240,240 0.1403022 6.1 : 1
?**** $\begin{matrix} {4r-3 \choose 1}{51-4r \choose 4} \end{matrix}$ 112,404 0.0656449 14.2 : 1 45,045 0.0263067 37.0 : 1
***** $\begin{matrix} {51-4r \choose 5} \end{matrix}$ 11,628 0.0067909 146.3 : 1 3,003 0.0017538 569.2 : 1
Total 714,096 0.4170381 1.4 : 1 455,574 0.2660591 2.8 : 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 make Board Derivation 8-high qualifier (r = 8) 9-high qualifier (r = 9)
Combos Probability Odds Combos Probability Odds
Make low ABC $\begin{matrix} {r-3 \choose 3}{4 \choose 1}^3 \end{matrix}$ 640 0.0370028 26.0 : 1 1,280 0.0740056 12.5 : 1
xAB $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$ 320 0.0185014 53.1 : 1 480 0.0277521 35.0 : 1
yAB $\begin{matrix} {2 \choose 1}{3 \choose 1}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$ 960 0.0555042 17.0 : 1 1,440 0.0832562 11.0 : 1
Total 1,920 0.1110083 8.0 : 1 3,200 0.1850139 4.4 : 1
Miss low xyA $\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 240 0.0138760 71.1 : 1 288 0.0166512 95.1 : 1
yzA $\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 180 0.0104070 59.1 : 1 216 0.0124884 79.1 : 1
xyz $\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 2}{3 \choose 1}^2 \end{matrix}$ 18 0.0010407 959.9 : 1 18 0.0010407 959.9 : 1
Total 438 0.0253238 38.5 : 1 522 0.0301804 32.1 : 1
No low AAA $\begin{matrix} {r-3 \choose 1}{4 \choose 3} \end{matrix}$ 20 0.0011563 863.8 : 1 24 0.0013876 719.7 : 1
AAB $\begin{matrix} {r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1} \end{matrix}$ 480 0.0277521 35.0 : 1 720 0.0416281 23.0 : 1
xxy $\begin{matrix} {1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 1} \end{matrix}$ 6 0.0003469 2,881.7 : 1 6 0.0003469 2,881.7 : 1
xxA $\begin{matrix} {1 \choose 1}{2 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 20 0.0011563 863.8 : 1 24 0.0013876 719.7 : 1
xyy $\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 2} \end{matrix}$ 12 0.0006938 1,440.3 : 1 12 0.0006938 1,440.3 : 1
xAA $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-3 \choose 1}{4 \choose 2} \end{matrix}$ 60 0.0034690 287.3 : 1 72 0.0041628 239.2 : 1
yyy $\begin{matrix} {2 \choose 1}{3 \choose 3} \end{matrix}$ 2 0.0001156 8,647.0 : 1 2 0.0001156 8,647.0 : 1
yyA $\begin{matrix} {2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 120 0.0069380 143.1 : 1 144 0.0083256 119.1 : 1
yAA $\begin{matrix} {2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2} \end{matrix}$ 180 0.0104070 59.1 : 1 216 0.0124884 79.1 : 1
yyz $\begin{matrix} {2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1} \end{matrix}$ 18 0.0010407 959.9 : 1 18 0.0010407 959.9 : 1
??* $\begin{matrix} {4r-4 \choose 2}{52-4r \choose 1} \end{matrix}$ 7,560 0.4370953 1.3 : 1 7,936 0.4588344 1.2 : 1
?** $\begin{matrix} {4r-4 \choose 1}{52-4r \choose 2} \end{matrix}$ 5,320 0.3075856 2.3 : 1 3,840 0.2220167 3.5 : 1
*** $\begin{matrix} {52-4r \choose 3} \end{matrix}$ 1,140 0.0659112 14.2 : 1 560 0.0323774 29.9 : 1
Total 14,938 0.8636679 0.2 : 1 13,574 0.7848057 0.3 : 1
Derivations for low hand shape xxyz (a low pair and two low ranks) 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 $\begin{matrix} {r-3 \choose 4}{4 \choose 1}^4 \end{matrix}$ 1,280 0.0065783 151.0 : 1 3,840 0.0197348 49.7 : 1
AABC $\begin{matrix} {r-3 \choose 1}{4 \choose 2}{r-4 \choose 2}{4 \choose 1}^2 \end{matrix}$ 2,880 0.0148011 66.6 : 1 5,760 0.0296022 32.8 : 1
ABC* $\begin{matrix} {r-3 \choose 3}{4 \choose 1}^3{52-4r \choose 1} \end{matrix}$ 12,800 0.0657827 14.2 : 1 20,480 0.1052523 8.5 : 1
xABC $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-3 \choose 3}{4 \choose 1}^3 \end{matrix}$ 1,280 0.0065783 151.0 : 1 2,560 0.0131565 75.0 : 1
yABC $\begin{matrix} {2 \choose 1}{3 \choose 1}{r-3 \choose 3}{4 \choose 1}^3 \end{matrix}$ 6,720 0.0345359 28.0 : 1 7,680 0.0394696 24.3 : 1
xxAB $\begin{matrix} {1 \choose 1}{2 \choose 2}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$ 160 0.0008223 1,215.1 : 1 240 0.0012334 809.8 : 1
xAAB $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1} \end{matrix}$ 960 0.0049337 201.7 : 1 1,440 0.0074006 134.1 : 1
xyAB $\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 1}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$ 1,920 0.0098674 100.3 : 1 2,880 0.0148011 66.6 : 1
xAB* $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-3 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$ 6,400 0.0328914 29.4 : 1 7,680 0.0394696 24.3 : 1
yyAB $\begin{matrix} {2 \choose 1}{3 \choose 2}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$ 960 0.0049337 201.7 : 1 1,440 0.0074006 134.1 : 1
yAAB $\begin{matrix} {2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1} \end{matrix}$ 2,880 0.0148011 66.6 : 1 4,320 0.0222017 44.0 : 1
yAB* $\begin{matrix} {2 \choose 1}{3 \choose 1}{r-3 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$ 19,200 0.0986741 9.1 : 1 23,040 0.1184089 7.4 : 1
yzAB $\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$ 1,440 0.0074006 134.1 : 1 2,160 0.0111008 89.1 : 1
Total 56,000 0.2877994 2.5 : 1 83,520 0.4292322 1.3 : 1
Miss low xxyA $\begin{matrix} {1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 120 0.0006167 1,620.5 : 1 144 0.0007401 1,350.3 : 1
xyyA $\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 240 0.0012334 809.8 : 1 288 0.0014801 674.6 : 1
xyAA $\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2} \end{matrix}$ 360 0.0018501 539.5 : 1 432 0.0022202 449.4 : 1
xyA* $\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 4,800 0.0246685 39.5 : 1 4,608 0.0236818 41.2 : 1
yyzA $\begin{matrix} {2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 360 0.0018501 539.5 : 1 432 0.0022202 449.4 : 1
yzAA $\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 2} \end{matrix}$ 270 0.0013876 719.7 : 1 324 0.0016651 599.6 : 1
yzA* $\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 3,600 0.0185014 53.1 : 1 3,456 0.0177613 55.3 : 1
xxyz $\begin{matrix} {1 \choose 1}{2 \choose 2}{2 \choose 2}{3 \choose 1}^2 \end{matrix}$ 9 0.0000463 21,619.0 : 1 9 0.0000463 21,619.0 : 1
xyyz $\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1} \end{matrix}$ 36 0.0001850 5,404.0 : 1 36 0.0001850 5,404.0 : 1
xyzA $\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 360 0.0018501 539.5 : 1 432 0.0022202 449.4 : 1
xyz* $\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 2}{3 \choose 1}^2{51-4r \choose 1} \end{matrix}$ 360 0.0018501 539.5 : 1 288 0.0014801 674.6 : 1
Total 10,515 0.0540395 17.5 : 1 10,449 0.0537003 17.6 : 1
No low AAAA $\begin{matrix} {r-3 \choose 1}{4 \choose 4} \end{matrix}$ 5 0.0000257 38,915.0 : 1 6 0.0000308 32,429.0 : 1
AAAB $\begin{matrix} {r-3 \choose 1}{4 \choose 3}{r-4 \choose 1}{4 \choose 1} \end{matrix}$ 320 0.0016446 607.1 : 1 480 0.0024669 404.4 : 1
AABB $\begin{matrix} {r-3 \choose 2}{4 \choose 2}^2 \end{matrix}$ 360 0.0018501 539.5 : 1 540 0.0027752 359.3 : 1
AAA* $\begin{matrix} {r-3 \choose 1}{4 \choose 3}{52-4r \choose 1} \end{matrix}$ 400 0.0020557 485.5 : 1 384 0.0019735 505.7 : 1
AAB* $\begin{matrix} {r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 9,600 0.0493370 19.3 : 1 11,520 0.0592044 15.9 : 1
xxyy $\begin{matrix} {1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 2} \end{matrix}$ 6 0.0000308 32,429.0 : 1 6 0.0000308 32,429.0 : 1
xxy* $\begin{matrix} {1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 1}{52-4r \choose 1} \end{matrix}$ 120 0.0006167 1,620.5 : 1 96 0.0004934 2,025.9 : 1
xxAA $\begin{matrix} {1 \choose 1}{2 \choose 2}{r-3 \choose 1}{4 \choose 2} \end{matrix}$ 30 0.0001542 6,485.0 : 1 36 0.0001850 5,404.0 : 1
xxA* $\begin{matrix} {1 \choose 1}{2 \choose 2}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 400 0.0020557 485.5 : 1 384 0.0019735 505.7 : 1
xyyy $\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 3} \end{matrix}$ 4 0.0000206 48,644.0 : 1 4 0.0000206 48,644.0 : 1
xyy* $\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 2}{52-4r \choose 1} \end{matrix}$ 240 0.0012334 809.8 : 1 192 0.0009867 1,012.4 : 1
xAAA $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-3 \choose 1}{4 \choose 3} \end{matrix}$ 40 0.0002056 4,863.5 : 1 48 0.0002467 4,052.8 : 1
xAA* $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-3 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$ 1,200 0.0061671 161.2 : 1 1,152 0.0059204 167.9 : 1
yyyA $\begin{matrix} {2 \choose 1}{3 \choose 3}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 40 0.0002056 4,863.5 : 1 48 0.0002467 4,052.8 : 1
yyy* $\begin{matrix} {2 \choose 1}{3 \choose 3}{52-4r \choose 1} \end{matrix}$ 40 0.0002056 4,863.5 : 1 32 0.0001645 6,079.6 : 1
yyAA $\begin{matrix} {2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 2} \end{matrix}$ 180 0.0009251 1,080.0 : 1 216 0.0011101 899.8 : 1
yyA* $\begin{matrix} {2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 2,400 0.0123343 80.1 : 1 2,304 0.0118409 83.5 : 1
yAAA $\begin{matrix} {2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 3} \end{matrix}$ 120 0.0006167 1,620.5 : 1 144 0.0007401 1,350.3 : 1
yAA* $\begin{matrix} {2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$ 3,600 0.0185014 53.1 : 1 3,456 0.0177613 55.3 : 1
yyyz $\begin{matrix} {2 \choose 1}{3 \choose 3}{1 \choose 1}{3 \choose 1} \end{matrix}$ 6 0.0000308 32,429.0 : 1 6 0.0000308 32,429.0 : 1
yyzz $\begin{matrix} {2 \choose 2}{3 \choose 2}^2 \end{matrix}$ 9 0.0000463 21,619.0 : 1 9 0.0000463 21,619.0 : 1
yyz* $\begin{matrix} {2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{52-4r \choose 1} \end{matrix}$ 360 0.0018501 539.5 : 1 288 0.0014801 674.6 : 1
??** $\begin{matrix} {4r-4 \choose 2}{52-4r \choose 2} \end{matrix}$ 71,820 0.3691027 1.7 : 1 59,520 0.3058896 2.3 : 1
?*** $\begin{matrix} {4r-4 \choose 1}{52-4r \choose 3} \end{matrix}$ 31,920 0.1640456 5.1 : 1 17,920 0.0920958 9.9 : 1
**** $\begin{matrix} {52-4r \choose 4} \end{matrix}$ 4,845 0.0248998 39.2 : 1 1,820 0.0093535 105.9 : 1
Total 128,065 0.6581612 0.5 : 1 100,611 0.5170675 0.9 : 1
Derivations for low hand shape xxyz (a low pair and two low ranks) on the river
Hand to make Board Derivation 8-high qualifier (r = 8) 9-high qualifier (r = 9)
Combos Probability Odds Combos Probability Odds
Make low ABCDE $\begin{matrix} {r-3 \choose 5}{4 \choose 1}^5 \end{matrix}$ 1,024 0.0005980 1,671.2 : 1 6,144 0.0035881 277.7 : 1
AABCD $\begin{matrix} {r-3 \choose 1}{4 \choose 2}{r-4 \choose 3}{4 \choose 1}^3 \end{matrix}$ 7,680 0.0044852 30.9 : 1 23,040 0.0134556 73.3 : 1
ABCD* $\begin{matrix} {r-3 \choose 4}{4 \choose 1}^4{52-4r \choose 1} \end{matrix}$ 25,600 0.0149506 222.0 : 1 61,440 0.0358815 26.9 : 1
AABBC $\begin{matrix} {r-3 \choose 2}{4 \choose 2}^2{r-4 \choose 1}{4 \choose 1} \end{matrix}$ 4,320 0.0025229 395.4 : 1 8,640 0.0050458 197.2 : 1
AABC* $\begin{matrix} {r-3 \choose 1}{4 \choose 2}{r-4 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$ 57,600 0.0336389 28.7 : 1 92,160 0.0538222 17.6 : 1
AAABC $\begin{matrix} {r-3 \choose 1}{4 \choose 3}{r-4 \choose 2}{4 \choose 1}^2 \end{matrix}$ 1,920 0.0011213 890.8 : 1 3,840 0.0022426 444.9 : 1
ABC** $\begin{matrix} {r-3 \choose 3}{4 \choose 1}^3{52-4r \choose 2} \end{matrix}$ 121,600 0.0710154 13.1 : 1 153,600 0.0897037 10.1 : 1
xABCD $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-3 \choose 4}{4 \choose 1}^4 \end{matrix}$ 2,560 0.0014951 667.9 : 1 7,680 0.0044852 222.0 : 1
xABC* $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-3 \choose 3}{4 \choose 1}^3{52-4r \choose 1} \end{matrix}$ 25,600 0.0149506 65.9 : 1 40,960 0.0239210 40.8 : 1
xAABC $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 2}{4 \choose 1}^2 \end{matrix}$ 5,760 0.0033639 296.3 : 1 11,520 0.0067278 147.6 : 1
xxABC $\begin{matrix} {1 \choose 1}{2 \choose 2}{r-3 \choose 3}{4 \choose 1}^3 \end{matrix}$ 640 0.0003738 2,674.5 : 1 1,280 0.0007475 1,336.7 : 1
xyABC $\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 1}{r-3 \choose 3}{4 \choose 1}^3 \end{matrix}$ 7,680 0.0044852 222.0 : 1 15,360 0.0089704 110.5 : 1
yyABC $\begin{matrix} {2 \choose 1}{3 \choose 2}{r-3 \choose 3}{4 \choose 1}^3 \end{matrix}$ 3,840 0.0022426 444.9 : 1 7,680 0.0044852 222.0 : 1
yABCD $\begin{matrix} {2 \choose 1}{3 \choose 1}{r-3 \choose 4}{4 \choose 1}^4 \end{matrix}$ 7,680 0.0044852 222.0 : 1 23,040 0.0134556 73.3 : 1
yABC* $\begin{matrix} {2 \choose 1}{3 \choose 1}{r-3 \choose 3}{4 \choose 1}^3{52-4r \choose 1} \end{matrix}$ 76,800 0.0448518 21.3 : 1 122,880 0.0717630 12.9 : 1
yAABC $\begin{matrix} {2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 2}{4 \choose 1}^2 \end{matrix}$ 17,280 0.0100917 98.1 : 1 34,560 0.0201833 48.5 : 1
xxyAB $\begin{matrix} {1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 1}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$ 960 0.0005606 1,782.7 : 1 1,440 0.0008410 1,188.1 : 1
xxAAB $\begin{matrix} {1 \choose 1}{2 \choose 2}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1} \end{matrix}$ 480 0.0002803 3,566.3 : 1 720 0.0004205 2,377.2 : 1
xxAB* $\begin{matrix} {1 \choose 1}{2 \choose 2}{r-3 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$ 3,200 0.0018688 534.1 : 1 3,840 0.0022426 444.9 : 1
xyyAB $\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1} \end{matrix}$ 1,920 0.0011213 890.8 : 1 2,880 0.0016819 593.6 : 1
xyAAB $\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1} \end{matrix}$ 5,760 0.0033639 296.3 : 1 8,640 0.0050458 197.2 : 1
xyAB* $\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 1}{r-3 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$ 38,400 0.0224259 43.6 : 1 46,080 0.0269111 36.2 : 1
xAAAB $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 640 0.0003738 2,674.5 : 1 960 0.0005606 1,782.7 : 1
xAABB $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 2}^2 \end{matrix}$ 720 0.0004205 2,377.2 : 1 1,080 0.0006307 1,584.5 : 1
xAAB* $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 19,200 0.0112130 88.2 : 1 23,040 0.0134556 73.3 : 1
xAB** $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 1}^2{52-4r \choose 2} \end{matrix}$ 60,800 0.0355077 27.2 : 1 57,600 0.0336389 28.7 : 1
yyyAB $\begin{matrix} {2 \choose 1}{3 \choose 3}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$ 320 0.0001869 5,350.0 : 1 480 0.0002803 3,566.3 : 1
yyAAB $\begin{matrix} {2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1} \end{matrix}$ 2,880 0.0016819 593.6 : 1 4,320 0.0025229 395.4 : 1
yyAB* $\begin{matrix} {2 \choose 1}{3 \choose 2}{r-3 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$ 19,200 0.0112130 88.2 : 1 23,040 0.0134556 73.3 : 1
yAAAB $\begin{matrix} {2 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 1,920 0.0011213 890.8 : 1 2,880 0.0016819 593.6 : 1
yAABB $\begin{matrix} {2 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 2}^2 \end{matrix}$ 1,620 0.0009461 1,056.0 : 1 3,240 0.0018922 527.5 : 1
yAAB* $\begin{matrix} {2 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 57,600 0.0336389 28.7 : 1 69,120 0.0403667 23.8 : 1
yAB** $\begin{matrix} {2 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 1}^2{52-4r \choose 2} \end{matrix}$ 182,400 0.1065231 8.4 : 1 172,800 0.1009167 8.9 : 1
yyzAB $\begin{matrix} {2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{r-3 \choose 3}{4 \choose 1}^3 \end{matrix}$ 2,880 0.0016819 593.6 : 1 4,320 0.0025229 395.4 : 1
yzAAB $\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1} \end{matrix}$ 4,320 0.0025229 395.4 : 1 6,480 0.0037844 263.2 : 1
yzABC $\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-3 \choose 3}{4 \choose 1}^3 \end{matrix}$ 5,760 0.0033639 296.3 : 1 11,520 0.0067278 147.6 : 1
yzAB* $\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-3 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$ 28,800 0.0168194 58.5 : 1 34,560 0.0201833 48.5 : 1
xyzAB $\begin{matrix} {1 \choose 1}{2 \choose 2}{2 \choose 2}{3 \choose 1}^2{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$ 2,880 0.0016819 593.6 : 1 4,320 0.0025229 395.4 : 1
Total 810,784 0.4735047 1.1 : 1 1,097,184 0.6407647 0.6 : 1
Miss low xxyyA $\begin{matrix} {1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 120 0.0000701 14,268.2 : 1 144 0.0000841 11,890.0 : 1
xxyAA $\begin{matrix} {1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2} \end{matrix}$ 180 0.0001051 9,511.8 : 1 216 0.0001261 7,926.3 : 1
xxyA* $\begin{matrix} {1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 2,400 0.0014016 712.5 : 1 2,304 0.0013456 742.2 : 1
xyyyA $\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 3}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 80 0.0000467 21,402.8 : 1 96 0.0000561 17,835.5 : 1
xyyAA $\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 3} \end{matrix}$ 360 0.0002102 4,755.4 : 1 432 0.0002523 3,962.7 : 1
xyyA* $\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$ 4,800 0.0028032 355.7 : 1 4,608 0.0026911 370.6 : 1
xyAAA $\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 3} \end{matrix}$ 240 0.0001402 7,133.6 : 1 288 0.0001682 5,944.5 : 1
xyAA* $\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$ 7,200 0.0042049 236.8 : 1 6,912 0.0040367 246.7 : 1
xyA** $\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 2} \end{matrix}$ 45,600 0.0266308 36.6 : 1 34,560 0.0201833 48.5 : 1
yyyzA $\begin{matrix} {2 \choose 1}{3 \choose 3}{1 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 120 0.0000701 14,268.2 : 1 144 0.0000841 11,890.0 : 1
yyzzA $\begin{matrix} {2 \choose 2}{3 \choose 2}^2{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 180 0.0001051 9,511.8 : 1 216 0.0001261 7,926.3 : 1
yyzAA $\begin{matrix} {2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2} \end{matrix}$ 540 0.0003154 3,169.9 : 1 648 0.0003784 2,641.4 : 1
yyzA* $\begin{matrix} {2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 7,200 0.0042049 236.8 : 1 6,912 0.0040367 246.7 : 1
yzAAA $\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 3} \end{matrix}$ 180 0.0001051 9,511.8 : 1 216 0.0001261 7,926.3 : 1
yzAA* $\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$ 5,400 0.0031536 316.1 : 1 5,184 0.0030275 329.3 : 1
yzA** $\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 1}{52-4r \choose 2} \end{matrix}$ 34,200 0.0199731 49.1 : 1 25,920 0.0151375 65.1 : 1
xxyyz $\begin{matrix} {1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1} \end{matrix}$ 18 0.0000105 95,127.0 : 1 18 0.0000105 95,127.0 : 1
xxyzA $\begin{matrix} {1 \choose 1}{2 \choose 2}{2 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 180 0.0001051 9,511.8 : 1 216 0.0001261 7,926.3 : 1
xxyz* $\begin{matrix} {1 \choose 1}{2 \choose 2}{2 \choose 2}{3 \choose 1}^2{52-4r \choose 1} \end{matrix}$ 180 0.0001051 9,511.8 : 1 144 0.0000841 11,890.0 : 1
xyyyz $\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 3}{1 \choose 1}{3 \choose 1} \end{matrix}$ 12 0.0000070 14,2691 : 1 12 0.0000070 14,2691 : 1
xyyzz $\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 2}{3 \choose 2}^2 \end{matrix}$ 18 0.0000105 95,127.0 : 1 18 0.0000105 95,127.0 : 1
xyyzA $\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1} \end{matrix}$ 720 0.0004205 2,377.2 : 1 864 0.0005046 1,980.8 : 1
xyyz* $\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{52-4r \choose 1} \end{matrix}$ 720 0.0004205 2,377.2 : 1 576 0.0003364 2,971.8 : 1
xyzAA $\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 2} \end{matrix}$ 540 0.0003154 3,169.9 : 1 648 0.0003784 2,641.4 : 1
xyzA* $\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 7,200 0.0042049 236.8 : 1 6,912 0.0040367 246.7 : 1
xyz** $\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 2}{3 \choose 1}^2{52-4r \choose 2} \end{matrix}$ 3,420 0.0019973 499.7 : 1 2,160 0.0012615 791.7 : 1
Total 121,808 0.0711369 13.1 : 1 100,368 0.0586158 16.1 : 1
No low AAAAB $\begin{matrix} {r-3 \choose 1}{4 \choose 4}{r-4 \choose 1}{4 \choose 1} \end{matrix}$ 80 0.0000467 21,402.8 : 1 120 0.0000701 15,019.2 : 1
AAAA* $\begin{matrix} {r-3 \choose 1}{4 \choose 4}{52-4r \choose 1} \end{matrix}$ 100 0.0000584 17,122.0 : 1 96 0.0000561 17,835.5 : 1
AAABB $\begin{matrix} {r-3 \choose 1}{4 \choose 3}{r-4 \choose 1}{4 \choose 2} \end{matrix}$ 480 0.0002803 3,566.3 : 1 720 0.0004205 2,377.2 : 1
AAAB* $\begin{matrix} {r-3 \choose 1}{4 \choose 3}{r-4 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 6,400 0.0037377 266.5 : 1 7,680 0.0044852 30.9 : 1
AABB* $\begin{matrix} {r-3 \choose 2}{4 \choose 2}^2{52-4r \choose 1} \end{matrix}$ 7,200 0.0042049 236.8 : 1 8,640 0.0050458 197.2 : 1
AAA** $\begin{matrix} {r-3 \choose 1}{4 \choose 3}{52-4r \choose 2} \end{matrix}$ 3,800 0.0022192 449.6 : 1 2,880 0.0016819 593.6 : 1
AAB** $\begin{matrix} {r-3 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 2} \end{matrix}$ 91,200 0.0532616 17.8 : 1 86,400 0.0504583 18.8 : 1
xxyyy $\begin{matrix} {1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 3} \end{matrix}$ 2 0.0000012 856151.0 : 1 2 0.0000012 856151.0 : 1
xxyy* $\begin{matrix} {1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 2}{52-4r \choose 1} \end{matrix}$ 120 0.0000701 14,268.2 : 1 96 0.0000561 17,835.5 : 1
xxy** $\begin{matrix} {1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 1}{52-4r \choose 2} \end{matrix}$ 1,140 0.0006658 1,501.0 : 1 720 0.0004205 2,377.2 : 1
xxAAA $\begin{matrix} {1 \choose 1}{2 \choose 2}{r-3 \choose 1}{4 \choose 3} \end{matrix}$ 20 0.0000117 85,614.2 : 1 24 0.0000140 71,345.0 : 1
xxAA* $\begin{matrix} {1 \choose 1}{2 \choose 2}{r-3 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$ 600 0.0003504 2,852.8 : 1 576 0.0003364 2,971.8 : 1
xxA** $\begin{matrix} {1 \choose 1}{2 \choose 2}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 2} \end{matrix}$ 3,800 0.0022192 449.6 : 1 2,880 0.0016819 593.6 : 1
xyyy* $\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 3}{52-4r \choose 1} \end{matrix}$ 80 0.0000467 21,402.8 : 1 64 0.0000374 26,753.8 : 1
xyy** $\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 2}{52-4r \choose 2} \end{matrix}$ 2,280 0.0013315 750.0 : 1 1,440 0.0008410 1,188.1 : 1
xAAAA $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-3 \choose 1}{4 \choose 4} \end{matrix}$ 10 0.0000058 171,229.4 : 1 12 0.0000070 142,691.0 : 1
xAAA* $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-3 \choose 1}{4 \choose 3}{52-4r \choose 1} \end{matrix}$ 800 0.0004672 2,139.4 : 1 768 0.0004485 2,228.6 : 1
xAA** $\begin{matrix} {1 \choose 1}{2 \choose 1}{r-3 \choose 1}{4 \choose 2}{52-4r \choose 2} \end{matrix}$ 11,400 0.0066577 149.2 : 1 8,640 0.0050458 197.2 : 1
yyyAA $\begin{matrix} {2 \choose 1}{3 \choose 3}{r-3 \choose 1}{4 \choose 2} \end{matrix}$ 60 0.0000350 28,537.4 : 1 72 0.0000420 23,781.0 : 1
yyyA* $\begin{matrix} {2 \choose 1}{3 \choose 3}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 800 0.0004672 2,139.4 : 1 768 0.0004485 2,228.6 : 1
yyy** $\begin{matrix} {2 \choose 1}{3 \choose 3}{52-4r \choose 2} \end{matrix}$ 380 0.0002219 4,505.1 : 1 240 0.0001402 7,133.6 : 1
yyAAA $\begin{matrix} {2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 3} \end{matrix}$ 120 0.0000701 14,268.2 : 1 144 0.0000841 11,890.0 : 1
yyAA* $\begin{matrix} {2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$ 3,600 0.0021024 474.6 : 1 3,456 0.0020183 494.5 : 1
yyA** $\begin{matrix} {2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 2} \end{matrix}$ 22,800 0.0133154 74.1 : 1 17,280 0.0100917 98.1 : 1
yAAAA $\begin{matrix} {2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 4} \end{matrix}$ 30 0.0000175 57,075.8 : 1 36 0.0000210 47,563.0 : 1
yAAA* $\begin{matrix} {2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 3}{52-4r \choose 1} \end{matrix}$ 2,400 0.0014016 712.5 : 1 2,304 0.0013456 742.2 : 1
yAA** $\begin{matrix} {2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}{52-4r \choose 2} \end{matrix}$ 34,200 0.0199731 49.1 : 1 25,920 0.0151375 65.1 : 1
yyyzz $\begin{matrix} {2 \choose 1}{3 \choose 3}{1 \choose 1}{3 \choose 2} \end{matrix}$ 6 0.0000035 285,383.0 : 1 6 0.0000035 285,383.0 : 1
yyyz* $\begin{matrix} {2 \choose 1}{3 \choose 3}{1 \choose 1}{3 \choose 1}{52-4r \choose 1} \end{matrix}$ 120 0.0000701 14,268.2 : 1 96 0.0000561 17,835.5 : 1
yyzz* $\begin{matrix} {2 \choose 2}{3 \choose 2}^2{52-4r \choose 1} \end{matrix}$ 180 0.0001051 9,511.8 : 1 144 0.0000841 11,890.0 : 1
yyz** $\begin{matrix} {2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{52-4r \choose 2} \end{matrix}$ 3,420 0.0019973 499.7 : 1 2,160 0.0012615 791.7 : 1
??*** $\begin{matrix} {4r-4 \choose 2}{52-4r \choose 3} \end{matrix}$ 430,920 0.2516609 3.0 : 1 277,760 0.1622142 5.2 : 1
?**** $\begin{matrix} {4r-4 \choose 1}{52-4r \choose 4} \end{matrix}$ 135,660 0.0792266 11.6 : 1 58,240 0.0340127 28.4 : 1
***** $\begin{matrix} {52-4r \choose 5} \end{matrix}$ 15,504 0.0090545 109.4 : 1 4,368 0.0025509 391.0 : 1
Total 779,712 0.4553584 1.2 : 1 514,752 0.3006195 2.3 : 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 make Board Derivation 8-high qualifier (r = 8) 9-high qualifier (r = 9)
Combos Probability Odds Combos Probability Odds
Make low ABC $\begin{matrix} {r-4 \choose 3}{4 \choose 1}^3 \end{matrix}$ 256 0.0148011 66.6 : 1 640 0.0370028 26.0 : 1
xAB $\begin{matrix} {4 \choose 1}{3 \choose 1}{r-4 \choose 2}{4 \choose 1}^2 \end{matrix}$ 1,152 0.0666050 14.0 : 1 1,920 0.1110083 8.0 : 1
xyA $\begin{matrix} {4 \choose 2}{3 \choose 1}^2{r-4 \choose 1}{4 \choose 1} \end{matrix}$ 864 0.0499537 19.0 : 1 1,080 0.0624422 15.0 : 1
Total 2,272 0.1313599 6.6 : 1 3,640 0.2104533 3.8 : 1
Miss low xyz $\begin{matrix} {4 \choose 3}{3 \choose 1}^3 \end{matrix}$ 108 0.0062442 159.1 : 1 108 0.0062442 159.1 : 1
No low AAA $\begin{matrix} {r-4 \choose 1}{4 \choose 3} \end{matrix}$ 16 0.0009251 1,080.0 : 1 20 0.0011563 863.8 : 1
AAB $\begin{matrix} {r-4 \choose 1}{4 \choose 2}{r-5 \choose 1}{4 \choose 1} \end{matrix}$ 288 0.0166512 59.1 : 1 480 0.0277521 35.0 : 1
xxx $\begin{matrix} {4 \choose 1}{3 \choose 3} \end{matrix}$ 4 0.0002313 4,323.0 : 1 4 0.0002313 4,323.0 : 1
xxy $\begin{matrix} {4 \choose 1}{3 \choose 2}{3 \choose 1}{3 \choose 1} \end{matrix}$ 108 0.0062442 159.1 : 1 108 0.0062442 159.1 : 1
xxA $\begin{matrix} {4 \choose 1}{3 \choose 2}{r-4 \choose 1}{4 \choose 1} \end{matrix}$ 192 0.0111008 89.1 : 1 240 0.0138760 71.1 : 1
xAA $\begin{matrix} {4 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 2} \end{matrix}$ 288 0.0166512 59.1 : 1 360 0.0208141 47.0 : 1
??* $\begin{matrix} {4r-4 \choose 2}{52-4r \choose 1} \end{matrix}$ 7,560 0.4370953 1.3 : 1 7,936 0.4588344 1.2 : 1
?** $\begin{matrix} {4r-4 \choose 1}{52-4r \choose 2} \end{matrix}$ 5,320 0.3075856 2.3 : 1 3,840 0.2220167 3.5 : 1
*** $\begin{matrix} {52-4r \choose 3} \end{matrix}$ 1,140 0.0659112 14.2 : 1 560 0.0323774 29.9 : 1
Total 14,916 0.8623959 0.2 : 1 13,548 0.7833025 0.3 : 1
Derivations for low hand shape xyzr (four low ranks) 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 $\begin{matrix} {r-4 \choose 4}{4 \choose 1}^4 \end{matrix}$ 256 0.0013157 759.1 : 1 1,280 0.0065783 151.0 : 1
AABC $\begin{matrix} {r-4 \choose 1}{4 \choose 2}{r-5 \choose 2}{4 \choose 1}^2 \end{matrix}$ 1,152 0.0059204 167.9 : 1 2,880 0.0148011 66.6 : 1
ABC* $\begin{matrix} {r-4 \choose 3}{4 \choose 1}^3{52-4r \choose 1} \end{matrix}$ 5,120 0.0263131 37.0 : 1 10,240 0.0526262 18.0 : 1
xABC $\begin{matrix} {4 \choose 1}{3 \choose 1}{r-4 \choose 3}{4 \choose 1}^3 \end{matrix}$ 3,072 0.0157879 62.3 : 1 7,680 0.0394696 24.3 : 1
xxAB $\begin{matrix} {4 \choose 1}{3 \choose 2}{r-4 \choose 2}{4 \choose 1}^2 \end{matrix}$ 1,152 0.0059204 167.9 : 1 1,920 0.0098674 100.3 : 1
xAAB $\begin{matrix} {4 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 2}{r-5 \choose 1}{4 \choose 1} \end{matrix}$ 3,456 0.0177613 55.3 : 1 5,760 0.0296022 32.8 : 1
xyAB $\begin{matrix} {4 \choose 2}{3 \choose 1}^2{r-4 \choose 2}{4 \choose 1}^2 \end{matrix}$ 5,184 0.0266420 36.5 : 1 8,640 0.0444033 21.5 : 1
xAB* $\begin{matrix} {4 \choose 1}{3 \choose 1}{r-4 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$ 23,040 0.1184089 7.4 : 1 30,720 0.1578785 5.3 : 1
xxyA $\begin{matrix} {4 \choose 1}{3 \choose 2}{3 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 1} \end{matrix}$ 1,728 0.0088807 111.6 : 1 2,160 0.0111008 89.1 : 1
xyAA $\begin{matrix} {4 \choose 2}{3 \choose 1}^2{r-4 \choose 1}{4 \choose 2} \end{matrix}$ 1,296 0.0066605 149.1 : 1 1,620 0.0083256 119.1 : 1
xyA* $\begin{matrix} {4 \choose 2}{3 \choose 1}^2{r-4 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 17,280 0.0888067 10.3 : 1 17,280 0.0888067 10.3 : 1
xyzA $\begin{matrix} {4 \choose 3}{3 \choose 1}^3{r-4 \choose 1}{4 \choose 1} \end{matrix}$ 1,728 0.0088807 111.6 : 1 2,160 0.0111008 89.1 : 1
Total 64,464 0.3312982 2.0 : 1 92,340 0.4745606 1.1 : 1
Miss low xxyz $\begin{matrix} {4 \choose 1}{3 \choose 2}{3 \choose 2}{3 \choose 1}^2 \end{matrix}$ 324 0.0016651 599.6 : 1 324 0.0016651 599.6 : 1
xyzr $\begin{matrix} {4 \choose 4}{3 \choose 1}^4 \end{matrix}$ 81 0.0004163 2,401.2 : 1 81 0.0004163 2,401.2 : 1
xyz* $\begin{matrix} {4 \choose 3}{3 \choose 1}^3{52-4r \choose 1} \end{matrix}$ 2,160 0.0111008 89.1 : 1 1,728 0.0088807 111.6 : 1
Total 2,565 0.131822 74.9 : 1 2,133 0.109621 90.2 : 1
No low AAAA $\begin{matrix} {r-4 \choose 1}{4 \choose 4} \end{matrix}$ 4 0.0000206 48,644.0 : 1 5 0.0000257 38,915.0 : 1
AAAB $\begin{matrix} {r-4 \choose 1}{4 \choose 3}{r-5 \choose 1}{4 \choose 1} \end{matrix}$ 192 0.0009867 1,012.4 : 1 320 0.0016446 607.1 : 1
AABB $\begin{matrix} {r-4 \choose 2}{4 \choose 2}^2 \end{matrix}$ 216 0.0011101 899.8 : 1 360 0.0018501 539.5 : 1
AAA* $\begin{matrix} {r-4 \choose 1}{4 \choose 3}{52-4r \choose 1} \end{matrix}$ 320 0.0016446 607.1 : 1 320 0.0016446 607.1 : 1
AAB* $\begin{matrix} {r-4 \choose 1}{4 \choose 2}{r-5 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 5,760 0.0296022 32.8 : 1 7,680 0.0394696 24.3 : 1
xxxy $\begin{matrix} {4 \choose 1}{3 \choose 3}{3 \choose 1}{3 \choose 1} \end{matrix}$ 36 0.0001850 5,404.0 : 1 36 0.0001850 5,404.0 : 1
xxxA $\begin{matrix} {4 \choose 1}{3 \choose 3}{r-4 \choose 1}{4 \choose 1} \end{matrix}$ 64 0.0003289 3,039.3 : 1 80 0.0004111 2,431.3 : 1
xxx* $\begin{matrix} {4 \choose 1}{3 \choose 3}{52-4r \choose 1} \end{matrix}$ 80 0.0004111 2,431.3 : 1 64 0.0003289 3,039.3 : 1
xxyy $\begin{matrix} {4 \choose 2}{3 \choose 2}^2 \end{matrix}$ 54 0.0002775 3,602.3 : 1 54 0.0002775 3,602.3 : 1
xxy* $\begin{matrix} {4 \choose 1}{3 \choose 2}{3 \choose 1}{3 \choose 1}{52-4r \choose 1} \end{matrix}$ 2,160 0.0111008 89.1 : 1 1,728 0.0088807 111.6 : 1
xxAA $\begin{matrix} {4 \choose 1}{3 \choose 2}{r-4 \choose 1}{4 \choose 2} \end{matrix}$ 270 0.0013876 719.7 : 1 288 0.0014801 674.6 : 1
xxA* $\begin{matrix} {4 \choose 1}{3 \choose 2}{r-4 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 3,840 0.0197348 49.7 : 1 3,840 0.0197348 49.7 : 1
xAAA $\begin{matrix} {4 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 3} \end{matrix}$ 192 0.0009867 1,012.4 : 1 240 0.0012334 809.8 : 1
xAA* $\begin{matrix} {4 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$ 5,760 0.0296022 32.8 : 1 5,760 0.0296022 32.8 : 1
??** $\begin{matrix} {4r-4 \choose 2}{52-4r \choose 2} \end{matrix}$ 71,820 0.3691027 1.7 : 1 59,520 0.3058896 2.3 : 1
?*** $\begin{matrix} {4r-4 \choose 1}{52-4r \choose 3} \end{matrix}$ 31,920 0.1640456 5.1 : 1 17,920 0.0920958 9.9 : 1
**** $\begin{matrix} {52-4r \choose 4} \end{matrix}$ 4,845 0.0248998 39.2 : 1 1,820 0.0093535 105.9 : 1
Total 127,551 0.6555196 0.5 : 1 100,107 0.5144773 0.9 : 1
Derivations for low hand shape xyzr (four low ranks) on the river
Hand to make Board Derivation 8-high qualifier (r = 8) 9-high qualifier (r = 9)
Combos Probability Odds Combos Probability Odds
Make low ABCDE $\begin{matrix} {r-4 \choose 5}{4 \choose 1}^5 \end{matrix}$ (see #1 below) 0 0.0000000 ∞ : 1 1,024 0.0005980 1,671.2 : 1
AABCD $\begin{matrix} {r-4 \choose 1}{4 \choose 2}{r-5 \choose 3}{4 \choose 1}^3 \end{matrix}$ 1,536 0.0008970 1,113.8 : 1 7,680 0.0044852 222.0 : 1
ABCD* $\begin{matrix} {r-4 \choose 4}{4 \choose 1}^4{52-4r \choose 1} \end{matrix}$ 5,120 0.0029901 333.4 : 1 20,480 0.0119605 82.6 : 1
AABBC $\begin{matrix} {r-4 \choose 2}{4 \choose 2}^2{r-5 \choose 1}{4 \choose 1} \end{matrix}$ 1,728 0.0010092 989.9 : 1 4,320 0.0025229 395.4 : 1
AABC* $\begin{matrix} {r-4 \choose 1}{4 \choose 2}{r-5 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$ 23,040 0.0134556 73.3 : 1 46,080 0.0269111 36.2 : 1
AAABC $\begin{matrix} {r-4 \choose 1}{4 \choose 3}{r-5 \choose 2}{4 \choose 1}^2 \end{matrix}$ 768 0.0004485 2,228.6 : 1 1,920 0.0011213 890.8 : 1
ABC** $\begin{matrix} {r-4 \choose 3}{4 \choose 1}^3{52-4r \choose 2} \end{matrix}$ 48,640 0.0284062 34.2 : 1 76,800 0.0448518 21.3 : 1
xABCD $\begin{matrix} {4 \choose 1}{3 \choose 1}{r-4 \choose 4}{4 \choose 1}^4 \end{matrix}$ 3,072 0.0017941 556.4 : 1 23,040 0.0134556 73.3 : 1
xABC* $\begin{matrix} {4 \choose 1}{3 \choose 1}{r-4 \choose 3}{4 \choose 1}^3{52-4r \choose 1} \end{matrix}$ 61,440 0.0358815 26.9 : 1 122,880 0.0717630 12.9 : 1
xAABC $\begin{matrix} {4 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 2}{r-5 \choose 2}{4 \choose 1}^2 \end{matrix}$ 13,824 0.0080733 122.9 : 1 34,560 0.0201833 48.5 : 1
xxABC $\begin{matrix} {4 \choose 1}{3 \choose 2}{r-4 \choose 3}{4 \choose 1}^3 \end{matrix}$ 3,072 0.0017941 556.4 : 1 7,680 0.0044852 222.0 : 1
xyABC $\begin{matrix} {4 \choose 2}{3 \choose 1}^2{r-4 \choose 3}{4 \choose 1}^3 \end{matrix}$ 13,824 0.0080733 122.9 : 1 34,560 0.0201833 48.5 : 1
xxxAB $\begin{matrix} {4 \choose 1}{3 \choose 3}{r-4 \choose 2}{4 \choose 1}^2 \end{matrix}$ 384 0.0002243 4,458.1 : 1 640 0.0003738 2,674.5 : 1
xxyAB $\begin{matrix} {4 \choose 1}{3 \choose 2}{3 \choose 1}{3 \choose 1}{r-4 \choose 2}{4 \choose 1}^2 \end{matrix}$ 10,368 0.0060550 164.2 : 1 17,280 0.0100917 98.1 : 1
xxAAB $\begin{matrix} {4 \choose 1}{3 \choose 2}{r-4 \choose 1}{4 \choose 2}{r-5 \choose 1}{4 \choose 1} \end{matrix}$ 3,456 0.0020183 494.5 : 1 5,760 0.0033639 296.3 : 1
xxAB* $\begin{matrix} {4 \choose 1}{3 \choose 2}{r-4 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$ 23,040 0.0134556 73.3 : 1 30,720 0.0179407 54.7 : 1
xyAAB $\begin{matrix} {4 \choose 2}{3 \choose 1}^2{r-4 \choose 1}{4 \choose 2}{r-5 \choose 1}{4 \choose 1} \end{matrix}$ 15,552 0.0090825 109.1 : 1 25,920 0.0151375 65.1 : 1
xyAB* $\begin{matrix} {4 \choose 2}{3 \choose 1}^2{r-4 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$ 103,680 0.0605500 15.5 : 1 138,240 0.0807333 11.4 : 1
xAAAB $\begin{matrix} {4 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-4 \choose 1}{4 \choose 1} \end{matrix}$ 2,304 0.0013456 742.2 : 1 3,840 0.0022426 444.9 : 1
xAABB $\begin{matrix} {4 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 2}^2 \end{matrix}$ 2,592 0.0015137 659.6 : 1 4,320 0.0025229 395.4 : 1
xAAB* $\begin{matrix} {4 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 69,120 0.0403667 23.8 : 1 92,160 0.0538222 17.6 : 1
xAB** $\begin{matrix} {4 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 1}^2{52-4r \choose 2} \end{matrix}$ 218,880 0.1278278 6.8 : 1 230,400 0.1345555 6.4 : 1
xyzAB $\begin{matrix} {4 \choose 3}{3 \choose 1}^3{r-4 \choose 2}{4 \choose 1}^2 \end{matrix}$ 10,368 0.0060550 164.2 : 1 17,280 0.0100917 98.1 : 1
xxxyA $\begin{matrix} {4 \choose 1}{3 \choose 3}{3 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 1} \end{matrix}$ 576 0.0003364 2,971.8 : 1 720 0.0004205 2,377.2 : 1
xxyyA $\begin{matrix} {4 \choose 2}{3 \choose 2}^2{r-4 \choose 1}{4 \choose 1} \end{matrix}$ 864 0.0005046 1,980.8 : 1 1,080 0.0006307 1,584.5 : 1
xxyAA $\begin{matrix} {4 \choose 1}{3 \choose 2}{3 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 2} \end{matrix}$ 1,620 0.0009461 1,056.0 : 1 2,592 0.0015137 659.6 : 1
xxyA* $\begin{matrix} {4 \choose 1}{3 \choose 2}{3 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 34,560 0.0201833 48.5 : 1 34,560 0.0201833 48.5 : 1
xyAAA $\begin{matrix} {4 \choose 2}{3 \choose 1}^2{r-4 \choose 1}{4 \choose 3} \end{matrix}$ 864 0.0005046 1,980.8 : 1 1,080 0.0006307 1,584.5 : 1
xyAA* $\begin{matrix} {4 \choose 2}{3 \choose 1}^2{r-4 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$ 25,920 0.0151375 65.1 : 1 25,920 0.0151375 65.1 : 1
xyA** $\begin{matrix} {4 \choose 2}{3 \choose 1}^2{r-4 \choose 1}{4 \choose 1}{52-4r \choose 2} \end{matrix}$ 164,160 0.0958708 9.4 : 1 129,600 0.0756875 12.2 : 1
xxyzA $\begin{matrix} {4 \choose 1}{3 \choose 2}{3 \choose 2}{3 \choose 1}^2 \end{matrix}$ 5,184 0.0030275 329.3 : 1 6,480 0.0037844 263.2 : 1
xyzAA $\begin{matrix} {4 \choose 3}{3 \choose 1}^3{r-4 \choose 1}{4 \choose 2} \end{matrix}$ 2,592 0.0015137 659.6 : 1 3,240 0.0018922 527.5 : 1
xyzA* $\begin{matrix} {4 \choose 3}{3 \choose 1}^3{r-4 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 34,560 0.0201833 48.5 : 1 34,560 0.0201833 48.5 : 1
xyzrA $\begin{matrix} {4 \choose 4}{3 \choose 1}^4{r-4 \choose 1}{4 \choose 1} \end{matrix}$ 1,296 0.0007569 1,320.2 : 1 1,620 0.0009461 1,056.0 : 1
Total 908,976 0.5308497 0.9 : 1 1,182,004 0.6903003 0.4 : 1
Miss low xxxyz $\begin{matrix} {4 \choose 1}{3 \choose 3}{3 \choose 2}{3 \choose 1}^2 \end{matrix}$ 108 0.0000631 15,853.7 : 1 108 0.0000631 15,853.7 : 1
xxyyz $\begin{matrix} {4 \choose 2}{3 \choose 2}^2{2 \choose 1}{3 \choose 1} \end{matrix}$ 324 0.0001892 5,283.9 : 1 324 0.0001892 5,283.9 : 1
xxyz* $\begin{matrix} {4 \choose 1}{3 \choose 2}{3 \choose 2}{3 \choose 1}^2{52-4r \choose 1} \end{matrix}$ 6,480 0.0037844 263.2 : 1 5,184 0.0030275 329.3 : 1
xyz** $\begin{matrix} {4 \choose 3}{3 \choose 1}^3{52-4r \choose 2} \end{matrix}$ 20,520 0.0119839 82.4 : 1 12,960 0.0075687 131.1 : 1
xxyzr $\begin{matrix} {4 \choose 1}{3 \choose 2}{3 \choose 3}{3 \choose 1}^3 \end{matrix}$ 324 0.0001892 5,283.9 : 1 324 0.0001892 5,283.9 : 1
xyzr* $\begin{matrix} {4 \choose 4}{3 \choose 1}^4{52-4r \choose 2} \end{matrix}$ 1,620 0.0009461 1,056.0 : 1 1,296 0.0007569 1,320.2 : 1
Total 29,376 0.0171558 57.3 : 1 20,196 0.0117946 83.8 : 1
No low AAAAB $\begin{matrix} {r-4 \choose 1}{4 \choose 4}{r-5 \choose 1}{4 \choose 1} \end{matrix}$ 48 0.0000280 35,672.0 : 1 80 0.0000467 21,402.8 : 1
AAAA* $\begin{matrix} {r-4 \choose 1}{4 \choose 4}{52-4r \choose 1} \end{matrix}$ 80 0.0000467 21,402.8 : 1 80 0.0000467 21,402.8 : 1
AAABB $\begin{matrix} {r-4 \choose 1}{4 \choose 3}{r-5 \choose 1}{4 \choose 2} \end{matrix}$ 288 0.0001682 5,944.5 : 1 480 0.0002803 3,566.3 : 1
AAAB* $\begin{matrix} {r-4 \choose 1}{4 \choose 3}{r-5 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 3,840 0.0022426 444.9 : 1 5,120 0.0029901 333.4 : 1
AABB* $\begin{matrix} {r-4 \choose 2}{4 \choose 2}^2{52-4r \choose 1} \end{matrix}$ 4,320 0.0025229 395.4 : 1 5,760 0.0033639 296.3 : 1
AAA** $\begin{matrix} {r-4 \choose 1}{4 \choose 3}{52-4r \choose 2} \end{matrix}$ 3,040 0.0017754 562.3 : 1 2,400 0.0014016 712.5 : 1
AAB** $\begin{matrix} {r-4 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1}{52-4r \choose 2} \end{matrix}$ 54,720 0.0319569 30.3 : 1 57,600 0.0336389 28.7 : 1
xxxyy $\begin{matrix} {4 \choose 1}{3 \choose 3}{3 \choose 1}{3 \choose 2} \end{matrix}$ 36 0.0000210 47,563.0 : 1 36 0.0000210 47,563.0 : 1
xxxy* $\begin{matrix} {4 \choose 1}{3 \choose 3}{3 \choose 1}{3 \choose 1}{52-4r \choose 1} \end{matrix}$ 720 0.0004205 2,377.2 : 1 576 0.0003364 2,971.8 : 1
xxxAA $\begin{matrix} {4 \choose 1}{3 \choose 3}{r-4 \choose 1}{4 \choose 2} \end{matrix}$ 96 0.0000561 17,835.5 : 1 120 0.0000701 14,268.2 : 1
xxxA* $\begin{matrix} {4 \choose 1}{3 \choose 3}{r-4 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$ 1,280 0.0007475 1,336.7 : 1 1,280 0.0007475 1,336.7 : 1
xxx** $\begin{matrix} {4 \choose 1}{3 \choose 3}{52-4r \choose 2} \end{matrix}$ 760 0.0004438 2,252.0 : 1 480 0.0002803 3,566.3 : 1
xxyy* $\begin{matrix} {4 \choose 2}{3 \choose 2}^2{52-4r \choose 1} \end{matrix}$ 1,080 0.0006307 1,584.5 : 1 864 0.0005046 1,980.8 : 1
xxy** $\begin{matrix} {4 \choose 1}{3 \choose 2}{3 \choose 1}{3 \choose 1}{52-4r \choose 2} \end{matrix}$ 20,520 0.0119839 82.4 : 1 12,960 0.0075687 131.1 : 1
xxAAA $\begin{matrix} {4 \choose 1}{3 \choose 2}{r-4 \choose 1}{4 \choose 3} \end{matrix}$ 192 0.0001121 8,917.3 : 1 240 0.0001402 7,133.6 : 1
xxAA* $\begin{matrix} {4 \choose 1}{3 \choose 2}{r-4 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$ 5,760 0.0033639 296.3 : 1 5,760 0.0033639 296.3 : 1
xxA** $\begin{matrix} {4 \choose 1}{3 \choose 2}{r-4 \choose 1}{4 \choose 1}{52-4r \choose 2} \end{matrix}$ 36,480 0.0213046 45.9 : 1 28,800 0.0168194 58.5 : 1
xAAAA $\begin{matrix} {4 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 4} \end{matrix}$ 48 0.0000280 35,672.0 : 1 60 0.0000350 28,537.4 : 1
xAAA* $\begin{matrix} {4 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 3}{52-4r \choose 1} \end{matrix}$ 3,840 0.0022426 444.9 : 1 3,840 0.0022426 444.9 : 1
xAA** $\begin{matrix} {4 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 2}{52-4r \choose 2} \end{matrix}$ 54,720 0.0319569 30.3 : 1 43,200 0.0252292 38.6 : 1
??*** $\begin{matrix} {4r-4 \choose 2}{52-4r \choose 3} \end{matrix}$ 430,920 0.2516609 3.0 : 1 277,760 0.1622142 5.2 : 1
?**** $\begin{matrix} {4r-4 \choose 1}{52-4r \choose 4} \end{matrix}$ 135,660 0.0792266 11.6 : 1 58,240 0.0340127 28.4 : 1
***** $\begin{matrix} {52-4r \choose 5} \end{matrix}$ 15,504 0.0090545 109.4 : 1 4,368 0.0025509 391.0 : 1
Total 773,952 0.4519945 1.2 : 1 510,104 0.2979050 2.4 : 1
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 $\begin{matrix} {8-5 \choose 4} = 0 \end{matrix}.$