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

The probability derivations for starting hands making a qualifying low hand in Omaha hold 'em Hi-Lo are separate for each of the starting hand low shapes. The derivations require identifying the individual cases that complete a low hand; make a low hand possible on the board, but don't make a low hand; and that create a board with fewer than three low ranks, making no low hand possible. The ranks on the board are indicated using lower case letters for matches with the starting hand and upper case letters to indicate qualifying low ranks that don't match the starting hand. So the low hand shape xxyz is any hand with a low pair of x with two additional low ranks y and z and the board xyA represents a flop that contains one x, one of the non-paired ranks y and one other low rank A. Note that since y and z have an identical relationship to the starting hand—each represents an unpaired rank—xyA and xzA represent the same set of boards and are interchangeable, so derivations for this hand choose one of the two choices represented by y. In addition to the upper and lower case letters, * is used to represent any rank higher than the qualifying low. So for the low hand shape xxy*, the board xx* represents a flop that contains two xs and a high rank (either the same high rank as xxy* or a different high rank). One common calculation regardless of the hand is that no low is possible when there are two or fewer low cards on the board. If is the number of high cards in the hand ( = 0, 1, 2, 3, 4) and is the low qualifier ( = 8, 9), then is the number of low cards remaining in the deck and 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: {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |- style="Background: #eee;;" ! colspan=2| Flop !! colspan=2| Turn !! colspan=2| River |- style="Background: #eee;;" ! Board !! Derivation !! Board !! Derivation !! Board !! Derivation |- | ??* | | ??** | | ??*** | |- | ?** | | ?*** | | ?**** | |- | *** | | **** | | ***** | |} 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 boards on the flop, boards on the turn, and boards at the river. The probabilities for the boards in each table total 1.0. Derivations for 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 high cards and low cards left in the deck. The following tables show the derivations when holding four high cards. {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape **** (four high cards) on the flop |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=1| Miss low ! ABC | |align=right| 3,584 || 0.2072155 ||align=right| 3.8 : 1 |align=right| 5,376 || 0.3108233 ||align=right| 2.2 : 1 |- ! rowspan=6| No low ! AAA | |align=right| 32 || 0.0018501 ||align=right| 539.5 : 1 |align=right| 36 || 0.0020814 ||align=right| 479.4 : 1 |- ! AAB | |align=right| 1,344 || 0.0777058 ||align=right| 11.9 : 1 |align=right| 1,728 || 0.0999075 ||align=right| 9.0 : 1 |- ! ??* | |align=right| 7,168 || 0.4144311 ||align=right| 1.4 : 1 |align=right| 6,912 || 0.3996300 ||align=right| 1.5 : 1 |- ! ?** | |align=right| 7,936 || 0.4588344 ||align=right| 1.2 : 1 |align=right| 7,560 || 0.4370953 ||align=right| 1.3 : 1 |- ! *** | |align=right| 560 || 0.0323774 ||align=right| 29.9 : 1 |align=right| 220 || 0.0127197 ||align=right| 77.6 : 1 |- | colspan=2| Total |align=right| 13,712 || 0.7927845 ||align=right| 0.3 : 1 |align=right| 11,920 || 0.6891767 ||align=right| 0.5 : 1 |} {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape **** (four high cards) on the turn |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=4| Miss low ! ABCD | |align=right| 17,920 || 0.0920958 ||align=right| 9.9 : 1 |align=right| 32,256 || 0.1657724 ||align=right| 5.0 : 1 |- ! AABC | |align=right| 16,128 || 0.0828862 ||align=right| 11.1 : 1 |align=right| 24,192 || 0.1243293 ||align=right| 7.0 : 1 |- ! ABC* | |align=right| 57,344 || 0.2947065 ||align=right| 2.4 : 1 |align=right| 64,512 || 0.3315449 ||align=right| 2.0 : 1 |- | colspan=2| Total |align=right| 91,392 || 0.7927845 ||align=right| 1.1 : 1 |align=right| 120,960 || 0.6216466 ||align=right| 0.6 : 1 |- ! rowspan=9| No low ! AAAA | |align=right| 8 || 0.0000411 ||align=right| 24,321.5 : 1 |align=right| 9 || 0.0000463 ||align=right| 21,619.0 : 1 |- ! AAAB | |align=right| 896 || 0.0046048 ||align=right| 216.2 : 1 |align=right| 1,152 || 0.0059204 ||align=right| 167.9 : 1 |- ! AABB | |align=right| 1,008 || 0.0051804 ||align=right| 192.0 : 1 |align=right| 1,296 || 0.0066605 ||align=right| 149.1 : 1 |- ! AAA* | |align=right| 512 || 0.0026313 ||align=right| 379.0 : 1 |align=right| 432 || 0.0022202 ||align=right| 449.4 : 1 |- ! AAB* | |align=right| 21,504 || 0.1105150 ||align=right| 8.0 : 1 |align=right| 20,736 || 0.1065680 ||align=right| 8.4 : 1 |- ! ??** | |align=right| 59,520 || 0.3058896 ||align=right| 2.3 : 1 |align=right| 41,580 || 0.2136910 ||align=right| 3.7 : 1 |- ! ?*** | |align=right| 17,920 || 0.0920958 ||align=right| 9.9 : 1 |align=right| 7,920 || 0.0407031 ||align=right| 23.6 : 1 |- ! **** | |align=right| 1,820 || 0.0093535 ||align=right| 105.9 : 1 |align=right| 495 || 0.0025439 ||align=right| 392.1 : 1 |- | colspan=2| Total |align=right| 103,188 || 0.5303114 ||align=right| 0.9 : 1 |align=right| 73,620 || 0.3783534 ||align=right| 1.6 : 1 |} {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape **** (four high cards) on the river |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=8| Miss low ! ABCDE | |align=right| 57,344 || 0.0334894 ||align=right| 28.9 : 1 |align=right| 129,024 || 0.0753511 ||align=right| 12.3 : 1 |- ! AABCD | |align=right| 107,520 || 0.0627926 ||align=right| 14.9 : 1 |align=right| 193,536 || 0.1130267 ||align=right| 7.8 : 1 |- ! ABCD* | |align=right| 286,720 || 0.1674469 ||align=right| 5.0 : 1 |align=right| 387,072 || 0.2260533 ||align=right| 3.4 : 1 |- ! AABBC | |align=right| 24,192 || 0.0141283 ||align=right| 69.8 : 1 |align=right| 36,288 || 0.0211925 ||align=right| 46.2 : 1 |- ! AAABC | |align=right| 10,752 || 0.0062793 ||align=right| 158.3 : 1 |align=right| 16,128 || 0.0094189 ||align=right| 105.2 : 1 |- ! AABC* | |align=right| 258,048 || 0.1507022 ||align=right| 5.6 : 1 |align=right| 290,304 || 0.1695400 ||align=right| 4.9 : 1 |- ! ABC** | |align=right| 430,080 || 0.2511704 ||align=right| 3.0 : 1 |align=right| 354,816 || 0.2072155 ||align=right| 3.8 : 1 |- | colspan=2| Total |align=right|1,174,656 || 0.6860090 ||align=right| 0.5 : 1 |align=right|1,407,168 || 0.8217980 ||align=right| 0.2 : 1 |- ! rowspan=11| No low ! AAAAB | |align=right| 224 || 0.0001308 ||align=right| 7,643.2 : 1 |align=right| 288 || 0.0001682 ||align=right| 5,944.5 : 1 |- ! AAAA* | |align=right| 128 || 0.0000748 ||align=right| 13,376.4 : 1 |align=right| 108 || 0.0000631 ||align=right| 15,853.7 : 1 |- ! AAABB | |align=right| 1,344 || 0.0007849 ||align=right| 1,273.0 : 1 |align=right| 1,728 || 0.0010092 ||align=right| 989.9 : 1 |- ! AAAB* | |align=right| 14,336 || 0.0083723 ||align=right| 118.4 : 1 |align=right| 13,824 || 0.0080733 ||align=right| 122.9 : 1 |- ! AABB* | |align=right| 16,128 || 0.0094189 ||align=right| 105.2 : 1 |align=right| 15,552 || 0.0090825 ||align=right| 109.1 : 1 |- ! AAA** | |align=right| 3,840 || 0.0022426 ||align=right| 444.9 : 1 |align=right| 2,376 || 0.0013876 ||align=right| 719.7 : 1 |- ! AAB** | |align=right| 161,280 || 0.0941889 ||align=right| 9.6 : 1 |align=right| 114,048 || 0.0666050 ||align=right| 14.0 : 1 |- ! ??*** | |align=right| 277,760 || 0.1622142 ||align=right| 5.2 : 1 |align=right| 138,600 || 0.0809436 ||align=right| 11.4 : 1 |- ! ?**** | |align=right| 58,240 || 0.0340127 ||align=right| 28.4 : 1 |align=right| 17,820 || 0.0104070 ||align=right| 95.1 : 1 |- ! ***** | |align=right| 4,368 || 0.0025509 ||align=right| 391.0 : 1 |align=right| 792 || 0.0004625 ||align=right| 2,161.0 : 1 |- | colspan=2| Total |align=right| 537,648 || 0.3139910 ||align=right| 2.2 : 1 |align=right| 305,136 || 0.1782020 ||align=right| 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 high cards and low cards left in the deck. The following tables show the derivations when holding one low card and three high cards. {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape x*** (one low card and three high cards) on the flop |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=3| Miss low ! ABC | |align=right| 2,240 || 0.1295097 ||align=right| 6.7 : 1 |align=right| 3,584 || 0.2072155 ||align=right| 3.8 : 1 |- ! xAB | |align=right| 1,008 || 0.0582794 ||align=right| 16.2 : 1 |align=right| 1,344 || 0.0777058 ||align=right| 11.9 : 1 |- | colspan=2| Total |align=right| 3,248 || 0.1877891 ||align=right| 4.3 : 1 |align=right| 4,928 || 0.2849214 ||align=right| 2.5 : 1 |- ! rowspan=9| No low ! AAA | |align=right| 28 || 0.0016189 ||align=right| 616.7 : 1 |align=right| 32 || 0.0018501 ||align=right| 539.5 : 1 |- ! AAB | |align=right| 1,008 || 0.0582794 ||align=right| 16.2 : 1 |align=right| 1,344 || 0.0777058 ||align=right| 11.9 : 1 |- ! xxx | |align=right| 1 || 0.0000578 ||align=right| 17,295.0 : 1 |align=right| 1 || 0.0000578 ||align=right| 17,295.0 : 1 |- ! xxA | |align=right| 84 || 0.0048566 ||align=right| 204.9 : 1 |align=right| 96 || 0.0055504 ||align=right| 179.2 : 1 |- ! xAA | |align=right| 126 || 0.0072849 ||align=right| 136.3 : 1 |align=right| 144 || 0.0083256 ||align=right| 119.1 : 1 |- ! ??* | |align=right| 7,905 || 0.4570421 ||align=right| 1.2 : 1 |align=right| 7,735 || 0.4472132 ||align=right| 1.2 : 1 |- ! ?** | |align=right| 4,216 || 0.2437558 ||align=right| 3.1 : 1 |align=right| 2,730 || 0.1578400 ||align=right| 5.3 : 1 |- ! *** | |align=right| 680 || 0.0393154 ||align=right| 24.4 : 1 |align=right| 286 || 0.0165356 ||align=right| 59.5 : 1 |- | colspan=2| Total |align=right| 14,048 || 0.8122109 ||align=right| 0.2 : 1 |align=right| 12,368 || 0.7150786 ||align=right| 0.4 : 1 |} {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape x*** (one low card and three high cards) on the turn |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=8| Miss low ! ABCD | |align=right| 8,960 || 0.0460479 ||align=right| 20.7 : 1 |align=right| 17,920 || 0.0920958 ||align=right| 9.9 : 1 |- ! AABC | |align=right| 10,080 || 0.0518039 ||align=right| 18.3 : 1 |align=right| 16,128 || 0.0828862 ||align=right| 11.1 : 1 |- ! ABC* | |align=right| 38,080 || 0.1957036 ||align=right| 4.1 : 1 |align=right| 46,592 || 0.2394491 ||align=right| 3.2 : 1 |- ! xxAB | |align=right| 1,008 || 0.0051804 ||align=right| 192.0 : 1 |align=right| 1,344 || 0.0069072 ||align=right| 143.8 : 1 |- ! xABC | |align=right| 6,720 || 0.0345359 ||align=right| 28.0 : 1 |align=right| 10,752 || 0.0552575 ||align=right| 17.1 : 1 |- ! xAAB | |align=right| 3,024 || 0.0155412 ||align=right| 63.3 : 1 |align=right| 4,032 || 0.0207216 ||align=right| 47.3 : 1 |- ! xAB* | |align=right| 17,136 || 0.0880666 ||align=right| 10.4 : 1 |align=right| 17,472 || 0.0897934 ||align=right| 10.1 : 1 |- | colspan=2| Total |align=right| 85,008 || 0.4368794 ||align=right| 1.3 : 1 |align=right| 114,240 || 0.5871107 ||align=right| 0.7 : 1 |- ! rowspan=15| No low ! AAAA | |align=right| 7 || 0.0000360 ||align=right| 27,796.1 : 1 |align=right| 8 || 0.0000411 ||align=right| 24,321.5 : 1 |- ! AAAB | |align=right| 672 || 0.0034536 ||align=right| 288.6 : 1 |align=right| 896 || 0.0046048 ||align=right| 216.2 : 1 |- ! AABB | |align=right| 756 || 0.0038853 ||align=right| 256.4 : 1 |align=right| 1,008 || 0.0051804 ||align=right| 192.0 : 1 |- ! AAA* | |align=right| 476 || 0.0024463 ||align=right| 407.8 : 1 |align=right| 416 || 0.0021379 ||align=right| 466.7 : 1 |- ! AAB* | |align=right| 17,136 || 0.0880666 ||align=right| 10.4 : 1 |align=right| 17,472 || 0.0897934 ||align=right| 10.1 : 1 |- ! xxx* | |align=right| 17 || 0.0000874 ||align=right| 11,444.9 : 1 |align=right| 13 || 0.0000668 ||align=right| 14,966.7 : 1 |- ! xxxA | |align=right| 28 || 0.0001439 ||align=right| 6,948.3 : 1 |align=right| 32 || 0.0001645 ||align=right| 6,079.6 : 1 |- ! xxAA | |align=right| 126 || 0.0006475 ||align=right| 1,543.3 : 1 |align=right| 144 || 0.0007401 ||align=right| 1,350.3 : 1 |- ! xxA* | |align=right| 1,428 || 0.0073389 ||align=right| 135.3 : 1 |align=right| 1,248 || 0.0064138 ||align=right| 154.9 : 1 |- ! xAAA | |align=right| 84 || 0.0004317 ||align=right| 2,315.4 : 1 |align=right| 96 || 0.0004934 ||align=right| 2,025.9 : 1 |- ! xAA* | |align=right| 2,142 || 0.0110083 ||align=right| 89.8 : 1 |align=right| 1,872 || 0.0096207 ||align=right| 102.9 : 1 |- ! ??** | |align=right| 63,240 || 0.3250077 ||align=right| 2.1 : 1 |align=right| 46,410 || 0.2385137 ||align=right| 3.2 : 1 |- ! ?*** | |align=right| 21,080 || 0.1083359 ||align=right| 8.2 : 1 |align=right| 10,010 || 0.0514441 ||align=right| 18.4 : 1 |- ! **** | |align=right| 2,380 || 0.0122315 ||align=right| 80.8 : 1 |align=right| 715 || 0.0036746 ||align=right| 271.1 : 1 |- | colspan=2| Total |align=right| 109,572 || 0.5631206 ||align=right| 0.8 : 1 |align=right| 80,340 || 0.4128893 ||align=right| 1.4 : 1 |} {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape x*** (one low card and three high cards) on the river |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=19| Miss low ! ABCDE | |align=right| 21,504 || 0.0125585 ||align=right| 78.6 : 1 |align=right| 57,344 || 0.0334894 ||align=right| 28.9 : 1 |- ! AABCD | |align=right| 53,760 || 0.0313963 ||align=right| 30.9 : 1 |align=right| 107,520 || 0.0627926 ||align=right| 14.9 : 1 |- ! ABCD* | |align=right| 152,320 || 0.0889562 ||align=right| 10.2 : 1 |align=right| 232,960 || 0.1360506 ||align=right| 6.4 : 1 |- ! AABBC | |align=right| 15,120 || 0.0088302 ||align=right| 112.2 : 1 |align=right| 24,192 || 0.0141283 ||align=right| 69.8 : 1 |- ! AAABC | |align=right| 6,720 || 0.0039245 ||align=right| 253.8 : 1 |align=right| 10,752 || 0.0062793 ||align=right| 158.3 : 1 |- ! AABC* | |align=right| 171,360 || 0.1000757 ||align=right| 9.0 : 1 |align=right| 209,664 || 0.1224455 ||align=right| 7.2 : 1 |- ! ABC** | |align=right| 304,640 || 0.1779123 ||align=right| 4.6 : 1 |align=right| 279,552 || 0.1632607 ||align=right| 5.1 : 1 |- ! xABCD | |align=right| 26,880 || 0.0156981 ||align=right| 62.7 : 1 |align=right| 53,760 || 0.0313963 ||align=right| 30.9 : 1 |- ! xABC* | |align=right| 114,240 || 0.0667171 ||align=right| 14.0 : 1 |align=right| 139,776 || 0.0816304 ||align=right| 11.3 : 1 |- ! xAABC | |align=right| 30,240 || 0.0176604 ||align=right| 55.6 : 1 |align=right| 48,384 || 0.0282567 ||align=right| 34.4 : 1 |- ! xAAB* | |align=right| 51,408 || 0.0300227 ||align=right| 32.3 : 1 |align=right| 52,416 || 0.0306114 ||align=right| 31.7 : 1 |- ! xAAAB | |align=right| 2,016 || 0.0011774 ||align=right| 848.4 : 1 |align=right| 2,688 || 0.0015698 ||align=right| 636.0 : 1 |- ! xAABB | |align=right| 2,268 || 0.0013245 ||align=right| 754.0 : 1 |align=right| 3,024 || 0.0017660 ||align=right| 565.2 : 1 |- ! xAB** | |align=right| 137,088 || 0.0800605 ||align=right| 11.5 : 1 |align=right| 104,832 || 0.0612228 ||align=right| 15.3 : 1 |- ! xxABC | |align=right| 6,720 || 0.0039245 ||align=right| 253.8 : 1 |align=right| 10,752 || 0.0062793 ||align=right| 158.3 : 1 |- ! xxAB* | |align=right| 17,136 || 0.0100076 ||align=right| 98.9 : 1 |align=right| 17,472 || 0.0102038 ||align=right| 97.0 : 1 |- ! xxAAB | |align=right| 3,024 || 0.0017660 ||align=right| 565.2 : 1 |align=right| 4,032 || 0.0023547 ||align=right| 423.7 : 1 |- ! xxxAB | |align=right| 336 || 0.0001962 ||align=right| 5,095.1 : 1 |align=right| 448 || 0.0002616 ||align=right| 3,821.1 : 1 |- | colspan=2| Total |align=right|1,116,780 || 0.6522090 ||align=right| 0.5 : 1 |align=right|1,359,568 || 0.7939992 ||align=right| 0.3 : 1 |- ! rowspan=20| No low ! AAAAB | |align=right| 168 || 0.0000981 ||align=right| 10,191.3 : 1 |align=right| 224 || 0.0001308 ||align=right| 7,643.2 : 1 |- ! AAAA* | |align=right| 119 || 0.0000695 ||align=right| 14,388.1 : 1 |align=right| 104 || 0.0000607 ||align=right| 16,463.5 : 1 |- ! AAABB | |align=right| 1,008 || 0.0005887 ||align=right| 1,697.7 : 1 |align=right| 1,344 || 0.0007849 ||align=right| 1,273.0 : 1 |- ! AAAB* | |align=right| 11,424 || 0.0066717 ||align=right| 148.9 : 1 |align=right| 11,648 || 0.0068025 ||align=right| 146.0 : 1 |- ! AABB* | |align=right| 12,852 || 0.0075057 ||align=right| 132.2 : 1 |align=right| 13,104 || 0.0076528 ||align=right| 129.7 : 1 |- ! AAA** | |align=right| 3,808 || 0.0022239 ||align=right| 448.7 : 1 |align=right| 2,496 || 0.0014577 ||align=right| 685.0 : 1 |- ! AAB** | |align=right| 137,088 || 0.0800605 ||align=right| 11.5 : 1 |align=right| 104,832 || 0.0612228 ||align=right| 15.3 : 1 |- ! xxxAA | |align=right| 42 || 0.0000245 ||align=right| 40,768.1 : 1 |align=right| 48 || 0.0000280 ||align=right| 35,672.0 : 1 |- ! xxxA* | |align=right| 476 || 0.0002780 ||align=right| 3,596.3 : 1 |align=right| 416 || 0.0002429 ||align=right| 4,115.1 : 1 |- ! xxx** | |align=right| 136 || 0.0000794 ||align=right| 12,589.5 : 1 |align=right| 78 || 0.0000456 ||align=right| 21,951.6 : 1 |- ! xxAAA | |align=right| 84 || 0.0000491 ||align=right| 20,383.6 : 1 |align=right| 96 || 0.0000561 ||align=right| 17,835.5 : 1 |- ! xxAA* | |align=right| 2,142 || 0.0012509 ||align=right| 798.4 : 1 |align=right| 1,872 || 0.0010933 ||align=right| 913.7 : 1 |- ! xxA** | |align=right| 11,424 || 0.0066717 ||align=right| 148.9 : 1 |align=right| 7,488 || 0.0043731 ||align=right| 227.7 : 1 |- ! xAAAA | |align=right| 21 || 0.0000123 ||align=right| 81,537.3 : 1 |align=right| 24 || 0.0000140 ||align=right| 71,345.0 : 1 |- ! xAAA* | |align=right| 1,428 || 0.0008340 ||align=right| 1,198.1 : 1 |align=right| 1,248 || 0.0007288 ||align=right| 1,371.0 : 1 |- ! xAA** | |align=right| 17,136 || 0.0100076 ||align=right| 98.9 : 1 |align=right| 11,232 || 0.0065596 ||align=right| 151.4 : 1 |- ! ??*** | |align=right| 316,200 || 0.1846635 ||align=right| 4.4 : 1 |align=right| 170,170 || 0.0993807 ||align=right| 9.1 : 1 |- ! ?**** | |align=right| 73,780 || 0.0430881 ||align=right| 22.2 : 1 |align=right| 25,025 || 0.0146148 ||align=right| 67.4 : 1 |- ! ***** | |align=right| 6,188 || 0.0036138 ||align=right| 275.7 : 1 |align=right| 1,287 || 0.0007516 ||align=right| 1,329.5 : 1 |- | colspan=2| Total |align=right| 595,524 || 0.3477910 ||align=right| 1.9 : 1 |align=right| 352,736 || 0.2060008 ||align=right| 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 high cards and low cards left in the deck. The following tables show the derivations when holding a low pair and two high cards. {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape xx** (a low pair and two high cards) on the flop |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=3| Miss low ! ABC | |align=right| 2,240 || 0.1295097 ||align=right| 6.7 : 1 |align=right| 3,584 || 0.2072155 ||align=right| 3.8 : 1 |- ! xAB | |align=right| 672 || 0.0388529 ||align=right| 24.7 : 1 |align=right| 896 || 0.0518039 ||align=right| 18.3 : 1 |- | colspan=2| Total |align=right| 2,912 || 0.1683626 ||align=right| 4.9 : 1 |align=right| 4,480 || 0.2590194 ||align=right| 2.9 : 1 |- ! rowspan=8| No low ! AAA | |align=right| 28 || 0.0016189 ||align=right| 616.7 : 1 |align=right| 32 || 0.0018501 ||align=right| 539.5 : 1 |- ! AAB | |align=right| 1,008 || 0.0582794 ||align=right| 16.2 : 1 |align=right| 1,344 || 0.0777058 ||align=right| 11.9 : 1 |- ! xxA | |align=right| 28 || 0.0016189 ||align=right| 616.7 : 1 |align=right| 32 || 0.0018501 ||align=right| 539.5 : 1 |- ! xAA | |align=right| 84 || 0.0048566 ||align=right| 204.9 : 1 |align=right| 96 || 0.0055504 ||align=right| 179.2 : 1 |- ! ??* | |align=right| 7,830 || 0.4527058 ||align=right| 1.2 : 1 |align=right| 7,854 || 0.4540934 ||align=right| 1.2 : 1 |- ! ?** | |align=right| 4,590 || 0.2653793 ||align=right| 2.8 : 1 |align=right| 3,094 || 0.1788853 ||align=right| 4.6 : 1 |- ! *** | |align=right| 816 || 0.0471785 ||align=right| 20.2 : 1 |align=right| 364 || 0.0210453 ||align=right| 46.5 : 1 |- | colspan=2| Total |align=right| 14,384 || 0.8316374 ||align=right| 0.2 : 1 |align=right| 12,816 || 0.7409806 ||align=right| 0.3 : 1 |} {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape xx** (a low pair and two high cards) on the turn |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=8| Miss low ! ABCD | |align=right| 8,960 || 0.0460479 ||align=right| 20.7 : 1 |align=right| 17,920 || 0.0920958 ||align=right| 9.9 : 1 |- ! AABC | |align=right| 10,080 || 0.0518039 ||align=right| 18.3 : 1 |align=right| 16,128 || 0.0828862 ||align=right| 11.1 : 1 |- ! ABC* | |align=right| 40,320 || 0.2072155 ||align=right| 3.8 : 1 |align=right| 50,176 || 0.2578682 ||align=right| 2.9 : 1 |- ! xxAB | |align=right| 336 || 0.0017268 ||align=right| 578.1 : 1 |align=right| 448 || 0.0023024 ||align=right| 433.3 : 1 |- ! xABC | |align=right| 4,480 || 0.0230239 ||align=right| 42.4 : 1 |align=right| 7,168 || 0.0368383 ||align=right| 26.1 : 1 |- ! xAAB | |align=right| 2,016 || 0.0103608 ||align=right| 95.5 : 1 |align=right| 2,688 || 0.0138144 ||align=right| 71.4 : 1 |- ! xAB* | |align=right| 12,096 || 0.0621647 ||align=right| 15.1 : 1 |align=right| 12,544 || 0.0644671 ||align=right| 14.5 : 1 |- | colspan=2| Total |align=right| 78,288 || 0.4023435 ||align=right| 1.5 : 1 |align=right| 107,072 || 0.5502724 ||align=right| 0.8 : 1 |- ! rowspan=13| No low ! AAAA | |align=right| 7 || 0.0000360 ||align=right| 27,796.1 : 1 |align=right| 8 || 0.0000411 ||align=right| 24,321.5 : 1 |- ! AAAB | |align=right| 672 || 0.0034536 ||align=right| 288.6 : 1 |align=right| 896 || 0.0046048 ||align=right| 216.2 : 1 |- ! AABB | |align=right| 756 || 0.0038853 ||align=right| 256.4 : 1 |align=right| 1,008 || 0.0051804 ||align=right| 192.0 : 1 |- ! AAA* | |align=right| 504 || 0.0025902 ||align=right| 385.1 : 1 |align=right| 448 || 0.0023024 ||align=right| 433.3 : 1 |- ! AAB* | |align=right| 18,144 || 0.0932470 ||align=right| 9.7 : 1 |align=right| 18,816 || 0.0967006 ||align=right| 9.3 : 1 |- ! xxAA | |align=right| 42 || 0.0002158 ||align=right| 4,631.9 : 1 |align=right| 48 || 0.0002467 ||align=right| 4,052.8 : 1 |- ! xxA* | |align=right| 504 || 0.0025902 ||align=right| 385.1 : 1 |align=right| 448 || 0.0023024 ||align=right| 433.3 : 1 |- ! xAAA | |align=right| 56 || 0.0002878 ||align=right| 3,473.6 : 1 |align=right| 64 || 0.0003289 ||align=right| 3,039.3 : 1 |- ! xAA* | |align=right| 1,512 || 0.0077706 ||align=right| 127.7 : 1 |align=right| 1,344 || 0.0069072 ||align=right| 143.8 : 1 |- ! ??** | |align=right| 66,555 || 0.3420444 ||align=right| 1.9 : 1 |align=right| 51,051 || 0.2623651 ||align=right| 2.8 : 1 |- ! ?*** | |align=right| 24,480 || 0.1258094 ||align=right| 6.9 : 1 |align=right| 12,376 || 0.0636037 ||align=right| 14.7 : 1 |- ! **** | |align=right| 3,060 || 0.0157262 ||align=right| 62.6 : 1 |align=right| 1,001 || 0.0051444 ||align=right| 193.4 : 1 |- | colspan=2| Total |align=right| 116,292 || 0.5976565 ||align=right| 0.7 : 1 |align=right| 87,508 || 0.4497276 ||align=right| 1.2 : 1 |} {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape xx** (a low pair and two high cards) on the river |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=18| Miss low ! ABCDE | |align=right| 21,504 || 0.0125585 ||align=right| 78.6 : 1 |align=right| 57,344 || 0.0334894 ||align=right| 28.9 : 1 |- ! AABCD | |align=right| 53,760 || 0.0313963 ||align=right| 30.9 : 1 |align=right| 107,520 || 0.0627926 ||align=right| 14.9 : 1 |- ! ABCD* | |align=right| 161,280 || 0.0941889 ||align=right| 9.6 : 1 |align=right| 250,880 || 0.1465160 ||align=right| 5.8 : 1 |- ! AABBC | |align=right| 15,120 || 0.0088302 ||align=right| 112.2 : 1 |align=right| 24,192 || 0.0141283 ||align=right| 69.8 : 1 |- ! AAABC | |align=right| 6,720 || 0.0039245 ||align=right| 253.8 : 1 |align=right| 10,752 || 0.0062793 ||align=right| 158.3 : 1 |- ! AABC* | |align=right| 181,440 || 0.1059625 ||align=right| 8.4 : 1 |align=right| 225,792 || 0.1318644 ||align=right| 6.6 : 1 |- ! ABC** | |align=right| 342,720 || 0.2001514 ||align=right| 4.0 : 1 |align=right| 326,144 || 0.1904709 ||align=right| 4.3 : 1 |- ! xABCD | |align=right| 17,920 || 0.0104654 ||align=right| 94.6 : 1 |align=right| 35,840 || 0.0209309 ||align=right| 46.8 : 1 |- ! xABC* | |align=right| 80,640 || 0.0470944 ||align=right| 20.2 : 1 |align=right| 100,352 || 0.0586064 ||align=right| 16.1 : 1 |- ! xAABC | |align=right| 20,160 || 0.0117736 ||align=right| 83.9 : 1 |align=right| 32,256 || 0.0188378 ||align=right| 52.1 : 1 |- ! xAAB* | |align=right| 36,288 || 0.0211925 ||align=right| 46.2 : 1 |align=right| 37,632 || 0.0219774 ||align=right| 44.5 : 1 |- ! xAAAB | |align=right| 1,344 || 0.0007849 ||align=right| 1,273.0 : 1 |align=right| 1,792 || 0.0010465 ||align=right| 954.5 : 1 |- ! xAABB | |align=right| 1,512 || 0.0008830 ||align=right| 1,131.5 : 1 |align=right| 2,016 || 0.0011774 ||align=right| 848.4 : 1 |- ! xAB** | |align=right| 102,816 || 0.0600454 ||align=right| 15.7 : 1 |align=right| 81,536 || 0.0476177 ||align=right| 20.0 : 1 |- ! xxABC | |align=right| 2,240 || 0.0013082 ||align=right| 763.4 : 1 |align=right| 3,584 || 0.0020931 ||align=right| 476.8 : 1 |- ! xxAB* | |align=right| 6,048 || 0.0035321 ||align=right| 282.1 : 1 |align=right| 6,272 || 0.0036629 ||align=right| 272.0 : 1 |- ! xxAAB | |align=right| 1,008 || 0.0005887 ||align=right| 1,697.7 : 1 |align=right| 1,344 || 0.0007849 ||align=right| 1,273.0 : 1 |- | colspan=2| Total |align=right|1,052,520 || 0.6146806 ||align=right| 0.6 : 1 |align=right|1,305,248 || 0.7622759 ||align=right| 0.3 : 1 |- ! rowspan=17| No low ! AAAAB | |align=right| 168 || 0.0000981 ||align=right| 10,191.3 : 1 |align=right| 224 || 0.0001308 ||align=right| 7,643.2 : 1 |- ! AAAA* | |align=right| 126 || 0.0000736 ||align=right| 13,588.7 : 1 |align=right| 112 || 0.0000654 ||align=right| 15,287.4 : 1 |- ! AAABB | |align=right| 1,008 || 0.0005887 ||align=right| 1,697.7 : 1 |align=right| 1,344 || 0.0007849 ||align=right| 1,273.0 : 1 |- ! AAAB* | |align=right| 12,096 || 0.0070642 ||align=right| 140.6 : 1 |align=right| 12,544 || 0.0073258 ||align=right| 135.5 : 1 |- ! AABB* | |align=right| 13,608 || 0.0079472 ||align=right| 124.8 : 1 |align=right| 14,112 || 0.0082415 ||align=right| 120.3 : 1 |- ! AAA** | |align=right| 4,284 || 0.0025019 ||align=right| 398.7 : 1 |align=right| 2,912 || 0.0017006 ||align=right| 587.0 : 1 |- ! AAB** | |align=right| 154,224 || 0.0900681 ||align=right| 10.1 : 1 |align=right| 122,304 || 0.0714266 ||align=right| 13.0 : 1 |- ! xxAAA | |align=right| 28 || 0.0000164 ||align=right| 61,152.7 : 1 |align=right| 32 || 0.0000187 ||align=right| 53,508.5 : 1 |- ! xxAA* | |align=right| 756 || 0.0004415 ||align=right| 2,264.0 : 1 |align=right| 672 || 0.0003925 ||align=right| 2,547.1 : 1 |- ! xxA** | |align=right| 4,284 || 0.0025019 ||align=right| 398.7 : 1 |align=right| 2,912 || 0.0017006 ||align=right| 587.0 : 1 |- ! xAAAA | |align=right| 14 || 0.0000082 ||align=right|122,306.4 : 1 |align=right| 16 || 0.0000093 ||align=right|107,018.0 : 1 |- ! xAAA* | |align=right| 1,008 || 0.0005887 ||align=right| 1,697.7 : 1 |align=right| 896 || 0.0005233 ||align=right| 1,910.1 : 1 |- ! xAA** | |align=right| 12,852 || 0.0075057 ||align=right| 132.2 : 1 |align=right| 8,736 || 0.0051019 ||align=right| 195.0 : 1 |- ! ??*** | |align=right| 354,960 || 0.2072996 ||align=right| 3.8 : 1 |align=right| 204,204 || 0.1192569 ||align=right| 7.4 : 1 |- ! ?**** | |align=right| 91,800 || 0.0536120 ||align=right| 17.7 : 1 |align=right| 34,034 || 0.0198761 ||align=right| 49.3 : 1 |- ! ***** | |align=right| 8,568 || 0.0050038 ||align=right| 198.8 : 1 |align=right| 2,002 || 0.0011692 ||align=right| 854.3 : 1 |- | colspan=2| Total |align=right| 659,784 || 0.3853194 ||align=right| 1.6 : 1 |align=right| 407,056 || 0.2377241 ||align=right| 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 high cards and low cards left in the deck. The following tables show the derivations when holding a low three of a kind and a high card. {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape xxx* (a low three of a kind and a high card) on the flop |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=3| Miss low ! ABC | |align=right| 2,240 || 0.1295097 ||align=right| 6.7 : 1 |align=right| 3,584 || 0.2072155 ||align=right| 3.8 : 1 |- ! xAB | |align=right| 336 || 0.0194265 ||align=right| 50.5 : 1 |align=right| 448 || 0.0259019 ||align=right| 37.6 : 1 |- | colspan=2| Total |align=right| 2,576 || 0.1489362 ||align=right| 5.7 : 1 |align=right| 4,032 || 0.2331175 ||align=right| 3.3 : 1 |- ! rowspan=7| No low ! AAA | |align=right| 28 || 0.0016189 ||align=right| 616.7 : 1 |align=right| 32 || 0.0018501 ||align=right| 539.5 : 1 |- ! AAB | |align=right| 1,008 || 0.0582794 ||align=right| 16.2 : 1 |align=right| 1,344 || 0.0777058 ||align=right| 11.9 : 1 |- ! xAA | |align=right| 42 || 0.0024283 ||align=right| 410.8 : 1 |align=right| 48 || 0.0027752 ||align=right| 359.3 : 1 |- ! ??* | |align=right| 7,714 || 0.4459991 ||align=right| 1.2 : 1 |align=right| 7,920 || 0.4579093 ||align=right| 1.2 : 1 |- ! ?** | |align=right| 4,959 || 0.2867137 ||align=right| 2.5 : 1 |align=right| 3,465 || 0.2003353 ||align=right| 4.0 : 1 |- ! *** | |align=right| 969 || 0.0560245 ||align=right| 16.8 : 1 |align=right| 455 || 0.0263067 ||align=right| 37.0 : 1 |- | colspan=2| Total |align=right| 14,720 || 0.8510638 ||align=right| 0.2 : 1 |align=right| 13,264 || 0.7668825 ||align=right| 0.3 : 1 |} {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape xxx* (a low three of a kind and a high card) on the turn |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=7| Miss low ! ABCD | |align=right| 8,960 || 0.0460479 ||align=right| 20.7 : 1 |align=right| 17,920 || 0.0920958 ||align=right| 9.9 : 1 |- ! AABC | |align=right| 10,080 || 0.0518039 ||align=right| 18.3 : 1 |align=right| 16,128 || 0.0828862 ||align=right| 11.1 : 1 |- ! ABC* | |align=right| 42,560 || 0.2187275 ||align=right| 3.6 : 1 |align=right| 53,760 || 0.2762874 ||align=right| 2.6 : 1 |- ! xABC | |align=right| 2,240 || 0.0115120 ||align=right| 85.9 : 1 |align=right| 3,584 || 0.0184192 ||align=right| 53.3 : 1 |- ! xAAB | |align=right| 1,008 || 0.0051804 ||align=right| 192.0 : 1 |align=right| 1,344 || 0.0069072 ||align=right| 143.8 : 1 |- ! xAB* | |align=right| 6,384 || 0.0328091 ||align=right| 29.5 : 1 |align=right| 6,720 || 0.0345359 ||align=right| 28.0 : 1 |- | colspan=2| Total |align=right| 71,232 || 0.3660808 ||align=right| 1.7 : 1 |align=right| 99,456 || 0.5111317 ||align=right| 1.0 : 1 |- ! rowspan=11| No low ! AAAA | |align=right| 7 || 0.0000360 ||align=right| 27,796.1 : 1 |align=right| 8 || 0.0000411 ||align=right| 24,321.5 : 1 |- ! AAAB | |align=right| 672 || 0.0034536 ||align=right| 288.6 : 1 |align=right| 896 || 0.0046048 ||align=right| 216.2 : 1 |- ! AABB | |align=right| 756 || 0.0038853 ||align=right| 256.4 : 1 |align=right| 1,008 || 0.0051804 ||align=right| 192.0 : 1 |- ! AAA* | |align=right| 532 || 0.0027341 ||align=right| 364.8 : 1 |align=right| 480 || 0.0024669 ||align=right| 404.4 : 1 |- ! AAB* | |align=right| 19,152 || 0.0984274 ||align=right| 9.2 : 1 |align=right| 20,160 || 0.1036078 ||align=right| 8.7 : 1 |- ! xAAA | |align=right| 28 || 0.0001439 ||align=right| 6,948.3 : 1 |align=right| 32 || 0.0001645 ||align=right| 6,079.6 : 1 |- ! xAA* | |align=right| 798 || 0.0041011 ||align=right| 242.8 : 1 |align=right| 720 || 0.0037003 ||align=right| 269.3 : 1 |- ! ??** | |align=right| 69,426 || 0.3567993 ||align=right| 1.8 : 1 |align=right| 55,440 || 0.2849214 ||align=right| 2.5 : 1 |- ! ?*** | |align=right| 28,101 || 0.1444187 ||align=right| 5.9 : 1 |align=right| 15,015 || 0.0771662 ||align=right| 12.0 : 1 |- ! **** | |align=right| 3,876 || 0.0199198 ||align=right| 49.2 : 1 |align=right| 1,365 || 0.0070151 ||align=right| 141.5 : 1 |- | colspan=2| Total |align=right| 123,348 || 0.6339192 ||align=right| 0.6 : 1 |align=right| 95,124 || 0.4888683 ||align=right| 1.0 : 1 |} {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape xxx* (a low three of a kind and a high card) on the river |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=15| Miss low ! ABCDE | |align=right| 21,504 || 0.0125585 ||align=right| 78.6 : 1 |align=right| 57,344 || 0.0334894 ||align=right| 28.9 : 1 |- ! AABCD | |align=right| 53,760 || 0.0313963 ||align=right| 30.9 : 1 |align=right| 107,520 || 0.0627926 ||align=right| 14.9 : 1 |- ! ABCD* | |align=right| 170,240 || 0.0994216 ||align=right| 9.1 : 1 |align=right| 268,800 || 0.1569815 ||align=right| 5.4 : 1 |- ! AABBC | |align=right| 15,120 || 0.0088302 ||align=right| 112.2 : 1 |align=right| 24,192 || 0.0141283 ||align=right| 69.8 : 1 |- ! AAABC | |align=right| 6,720 || 0.0039245 ||align=right| 253.8 : 1 |align=right| 10,752 || 0.0062793 ||align=right| 158.3 : 1 |- ! AABC* | |align=right| 191,520 || 0.1118493 ||align=right| 7.9 : 1 |align=right| 241,920 || 0.1412833 ||align=right| 6.1 : 1 |- ! ABC** | |align=right| 383,040 || 0.2236986 ||align=right| 3.5 : 1 |align=right| 376,320 || 0.2197741 ||align=right| 3.6 : 1 |- ! xABCD | |align=right| 8,960 || 0.0052327 ||align=right| 190.1 : 1 |align=right| 17,920 || 0.0104654 ||align=right| 94.6 : 1 |- ! xABC* | |align=right| 42,560 || 0.0248554 ||align=right| 39.2 : 1 |align=right| 53,760 || 0.0313963 ||align=right| 30.9 : 1 |- ! xAABC | |align=right| 10,080 || 0.0058868 ||align=right| 168.9 : 1 |align=right| 16,128 || 0.0094189 ||align=right| 105.2 : 1 |- ! xAAB* | |align=right| 19,152 || 0.0111849 ||align=right| 88.4 : 1 |align=right| 20,160 || 0.0117736 ||align=right| 83.9 : 1 |- ! xAAAB | |align=right| 672 || 0.0003925 ||align=right| 2,547.1 : 1 |align=right| 896 || 0.0005233 ||align=right| 1,910.1 : 1 |- ! xAABB | |align=right| 756 || 0.0004415 ||align=right| 2,264.0 : 1 |align=right| 1,008 || 0.0005887 ||align=right| 1,697.7 : 1 |- ! xAB** | |align=right| 57,456 || 0.0335548 ||align=right| 28.8 : 1 |align=right| 47,040 || 0.0274718 ||align=right| 35.4 : 1 |- | colspan=2| Total |align=right| 981,540 || 0.5732277 ||align=right| 0.7 : 1 |align=right|1,243,760 || 0.7263663 ||align=right| 0.4 : 1 |- ! rowspan=14| No low ! AAAAB | |align=right| 168 || 0.0000981 ||align=right| 10,191.3 : 1 |align=right| 224 || 0.0001308 ||align=right| 7,643.2 : 1 |- ! AAAA* | |align=right| 133 || 0.0000777 ||align=right| 12,873.5 : 1 |align=right| 120 || 0.0000701 ||align=right| 14,268.2 : 1 |- ! AAABB | |align=right| 1,008 || 0.0005887 ||align=right| 1,697.7 : 1 |align=right| 1,344 || 0.0007849 ||align=right| 1,273.0 : 1 |- ! AAAB* | |align=right| 12,768 || 0.0074566 ||align=right| 133.1 : 1 |align=right| 13,440 || 0.0078491 ||align=right| 126.4 : 1 |- ! AABB* | |align=right| 14,364 || 0.0083887 ||align=right| 118.2 : 1 |align=right| 15,120 || 0.0088302 ||align=right| 112.2 : 1 |- ! AAA** | |align=right| 4,788 || 0.0027962 ||align=right| 356.6 : 1 |align=right| 3,360 || 0.0019623 ||align=right| 508.6 : 1 |- ! AAB** | |align=right| 172,368 || 0.1006644 ||align=right| 8.9 : 1 |align=right| 141,120 || 0.0824153 ||align=right| 11.1 : 1 |- ! xAAAA | |align=right| 7 || 0.0000041 ||align=right|244,613.9 : 1 |align=right| 8 || 0.0000047 ||align=right|214,037.0 : 1 |- ! xAAA* | |align=right| 532 || 0.0003107 ||align=right| 3,217.6 : 1 |align=right| 480 || 0.0002803 ||align=right| 3,566.3 : 1 |- ! xAA** | |align=right| 7,182 || 0.0041943 ||align=right| 237.4 : 1 |align=right| 5,040 || 0.0029434 ||align=right| 338.7 : 1 |- ! ??*** | |align=right| 393,414 || 0.2297571 ||align=right| 3.4 : 1 |align=right| 240,240 || 0.1403022 ||align=right| 6.1 : 1 |- ! ?**** | |align=right| 112,404 || 0.0656449 ||align=right| 14.2 : 1 |align=right| 45,045 || 0.0263067 ||align=right| 37.0 : 1 |- ! ***** | |align=right| 11,628 || 0.0067909 ||align=right| 146.3 : 1 |align=right| 3,003 || 0.0017538 ||align=right| 569.2 : 1 |- | colspan=2| Total |align=right| 730,764 || 0.4267723 ||align=right| 1.3 : 1 |align=right| 468,544 || 0.2736337 ||align=right| 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 high cards and low cards left in the deck. The following tables show the derivations when holding a low four of a kind. {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape xxxx (a low four of a kind) on the flop |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=1| Miss low ! ABC | |align=right| 2,240 || 0.1295097 ||align=right| 6.7 : 1 |align=right| 3,584 || 0.2072155 ||align=right| 3.8 : 1 |- ! rowspan=6| No low ! AAA | |align=right| 28 || 0.0016189 ||align=right| 616.7 : 1 |align=right| 32 || 0.0018501 ||align=right| 539.5 : 1 |- ! AAB | |align=right| 1,008 || 0.0582794 ||align=right| 16.2 : 1 |align=right| 1,344 || 0.0777058 ||align=right| 11.9 : 1 |- ! ??* | |align=right| 7,560 || 0.4370953 ||align=right| 1.3 : 1 |align=right| 7,936 || 0.4588344 ||align=right| 1.2 : 1 |- ! ?** | |align=right| 5,320 || 0.3075856 ||align=right| 2.3 : 1 |align=right| 3,840 || 0.2220167 ||align=right| 3.5 : 1 |- ! *** | |align=right| 1,140 || 0.0659112 ||align=right| 14.2 : 1 |align=right| 560 || 0.0323774 ||align=right| 29.9 : 1 |- | colspan=2| Total |align=right| 15,056 || 0.8704903 ||align=right| 0.1 : 1 |align=right| 13,712 || 0.7927845 ||align=right| 0.3 : 1 |} {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape xxxx (a low four of a kind) on the turn |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=4| Miss low ! ABCD | |align=right| 8,960 || 0.0460479 ||align=right| 20.7 : 1 |align=right| 17,920 || 0.0920958 ||align=right| 9.9 : 1 |- ! AABC | |align=right| 10,080 || 0.0518039 ||align=right| 18.3 : 1 |align=right| 16,128 || 0.0828862 ||align=right| 11.1 : 1 |- ! ABC* | |align=right| 44,800 || 0.2302395 ||align=right| 3.3 : 1 |align=right| 57,344 || 0.2947065 ||align=right| 2.4 : 1 |- | colspan=2| Total |align=right| 63,840 || 0.3280913 ||align=right| 2.0 : 1 |align=right| 91,392 || 0.4696886 ||align=right| 1.1 : 1 |- ! rowspan=9| No low ! AAAA | |align=right| 7 || 0.0000360 ||align=right| 27,796.1 : 1 |align=right| 8 || 0.0000411 ||align=right| 24,321.5 : 1 |- ! AAAB | |align=right| 672 || 0.0034536 ||align=right| 288.6 : 1 |align=right| 896 || 0.0046048 ||align=right| 216.2 : 1 |- ! AABB | |align=right| 756 || 0.0038853 ||align=right| 256.4 : 1 |align=right| 1,008 || 0.0051804 ||align=right| 192.0 : 1 |- ! AAA* | |align=right| 560 || 0.0028780 ||align=right| 346.5 : 1 |align=right| 512 || 0.0026313 ||align=right| 379.0 : 1 |- ! AAB* | |align=right| 20,160 || 0.1036078 ||align=right| 8.7 : 1 |align=right| 21,504 || 0.1105150 ||align=right| 8.0 : 1 |- ! ??** | |align=right| 71,820 || 0.3691027 ||align=right| 1.7 : 1 |align=right| 59,520 || 0.3058896 ||align=right| 2.3 : 1 |- ! ?*** | |align=right| 31,920 || 0.1640456 ||align=right| 5.1 : 1 |align=right| 17,920 || 0.0920958 ||align=right| 9.9 : 1 |- ! **** | |align=right| 4,845 || 0.0248998 ||align=right| 39.2 : 1 |align=right| 1,820 || 0.0093535 ||align=right| 105.9 : 1 |- | colspan=2| Total |align=right| 130,740 || 0.6719087 ||align=right| 0.5 : 1 |align=right| 103,188 || 0.5303114 ||align=right| 0.9 : 1 |} {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape xxxx (a low four of a kind) on the river |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=8| Miss low ! ABCDE | |align=right| 21,504 || 0.0125585 ||align=right| 78.6 : 1 |align=right| 57,344 || 0.0334894 ||align=right| 28.9 : 1 |- ! AABCD | |align=right| 53,760 || 0.0313963 ||align=right| 30.9 : 1 |align=right| 107,520 || 0.0627926 ||align=right| 14.9 : 1 |- ! ABCD* | |align=right| 179,200 || 0.1046543 ||align=right| 8.6 : 1 |align=right| 286,720 || 0.1674469 ||align=right| 5.0 : 1 |- ! AABBC | |align=right| 15,120 || 0.0088302 ||align=right| 112.2 : 1 |align=right| 24,192 || 0.0141283 ||align=right| 69.8 : 1 |- ! AAABC | |align=right| 6,720 || 0.0039245 ||align=right| 253.8 : 1 |align=right| 10,752 || 0.0062793 ||align=right| 158.3 : 1 |- ! AABC* | |align=right| 201,600 || 0.1177361 ||align=right| 7.5 : 1 |align=right| 258,048 || 0.1507022 ||align=right| 5.6 : 1 |- ! ABC** | |align=right| 425,600 || 0.2485540 ||align=right| 3.0 : 1 |align=right| 430,080 || 0.2511704 ||align=right| 3.0 : 1 |- | colspan=2| Total |align=right| 903,504 || 0.5276540 ||align=right| 0.9 : 1 |align=right|1,174,656 || 0.6860090 ||align=right| 0.5 : 1 |- ! rowspan=11| No low ! AAAAB | |align=right| 168 || 0.0000981 ||align=right| 10,191.3 : 1 |align=right| 224 || 0.0001308 ||align=right| 7,643.2 : 1 |- ! AAAA* | |align=right| 140 || 0.0000818 ||align=right| 12,229.7 : 1 |align=right| 128 || 0.0000748 ||align=right| 13,376.4 : 1 |- ! AAABB | |align=right| 1,008 || 0.0005887 ||align=right| 1,697.7 : 1 |align=right| 1,344 || 0.0007849 ||align=right| 1,273.0 : 1 |- ! AAAB* | |align=right| 13,440 || 0.0078491 ||align=right| 126.4 : 1 |align=right| 14,336 || 0.0083723 ||align=right| 118.4 : 1 |- ! AABB* | |align=right| 15,120 || 0.0088302 ||align=right| 112.2 : 1 |align=right| 16,128 || 0.0094189 ||align=right| 105.2 : 1 |- ! AAA** | |align=right| 5,320 || 0.0031069 ||align=right| 320.9 : 1 |align=right| 3,840 || 0.0022426 ||align=right| 444.9 : 1 |- ! AAB** | |align=right| 191,520 || 0.1118493 ||align=right| 7.9 : 1 |align=right| 161,280 || 0.0941889 ||align=right| 9.6 : 1 |- ! ??*** | |align=right| 430,920 || 0.2516609 ||align=right| 3.0 : 1 |align=right| 277,760 || 0.1622142 ||align=right| 5.2 : 1 |- ! ?**** | |align=right| 135,660 || 0.0792266 ||align=right| 11.6 : 1 |align=right| 58,240 || 0.0340127 ||align=right| 28.4 : 1 |- ! ***** | |align=right| 15,504 || 0.0090545 ||align=right| 109.4 : 1 |align=right| 4,368 || 0.0025509 ||align=right| 391.0 : 1 |- | colspan=2| Total |align=right| 808,800 || 0.4723460 ||align=right| 1.1 : 1 |align=right| 537,648 || 0.3139910 ||align=right| 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 high cards and low cards left in the deck. The following tables show the derivations when holding two low ranks and two high cards. {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape xy** (two low ranks and two high cards) on the flop |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=1| Make low ! ABC | |align=right| 1,280 || 0.0740056 ||align=right| 12.5 : 1 |align=right| 2,240 || 0.1295097 ||align=right| 6.7 : 1 |- ! rowspan=3| Miss low ! xAB | |align=right| 1,440 || 0.0832562 ||align=right| 11.0 : 1 |align=right| 2,016 || 0.1165587 ||align=right| 7.6 : 1 |- ! xyA | |align=right| 216 || 0.0124884 ||align=right| 79.1 : 1 |align=right| 252 || 0.0145698 ||align=right| 67.6 : 1 |- | colspan=2| Total |align=right| 1,656 || 0.0957447 ||align=right| 9.4 : 1 |align=right| 2,268 || 0.1311286 ||align=right| 6.6 : 1 |- ! rowspan=10| No low ! AAA | |align=right| 24 || 0.0013876 ||align=right| 719.7 : 1 |align=right| 28 || 0.0016189 ||align=right| 616.7 : 1 |- ! AAB | |align=right| 720 || 0.0416281 ||align=right| 23.0 : 1 |align=right| 1,008 || 0.0582794 ||align=right| 16.2 : 1 |- ! xxx | |align=right| 2 || 0.0001156 ||align=right| 8,647.0 : 1 |align=right| 2 || 0.0001156 ||align=right| 8,647.0 : 1 |- ! xxy | |align=right| 18 || 0.0010407 ||align=right| 959.9 : 1 |align=right| 18 || 0.0010407 ||align=right| 959.9 : 1 |- ! xxA | |align=right| 144 || 0.0083256 ||align=right| 119.1 : 1 |align=right| 168 || 0.0097132 ||align=right| 102.0 : 1 |- ! xAA | |align=right| 216 || 0.0124884 ||align=right| 79.1 : 1 |align=right| 252 || 0.0145698 ||align=right| 67.6 : 1 |- ! ??* | |align=right| 7,830 || 0.4527058 ||align=right| 1.2 : 1 |align=right| 7,854 || 0.4540934 ||align=right| 1.2 : 1 |- ! ?** | |align=right| 4,590 || 0.2653793 ||align=right| 2.8 : 1 |align=right| 3,094 || 0.1788853 ||align=right| 4.6 : 1 |- ! *** | |align=right| 816 || 0.0471785 ||align=right| 20.2 : 1 |align=right| 364 || 0.0210453 ||align=right| 46.5 : 1 |- | colspan=2| Total |align=right| 14,360 || 0.8302498 ||align=right| 0.2 : 1 |align=right| 12,788 || 0.7393617 ||align=right| 0.4 : 1 |} {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape xy** (two low ranks and two high cards) on the turn |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=5| Make low ! ABCD | |align=right| 3,840 || 0.0197348 ||align=right| 49.7 : 1 |align=right| 8,960 || 0.0460479 ||align=right| 20.7 : 1 |- ! AABC | |align=right| 5,760 || 0.0296022 ||align=right| 32.8 : 1 |align=right| 10,080 || 0.0518039 ||align=right| 18.3 : 1 |- ! ABC* | |align=right| 23,040 || 0.1184089 ||align=right| 7.4 : 1 |align=right| 31,360 || 0.1611676 ||align=right| 5.2 : 1 |- ! xABC | |align=right| 7,680 || 0.0394696 ||align=right| 24.3 : 1 |align=right| 13,440 || 0.0690718 ||align=right| 13.5 : 1 |- | colspan=2| Total |align=right| 40,320 || 0.2072155 ||align=right| 3.8 : 1 |align=right| 63,840 || 0.3280913 ||align=right| 2.0 : 1 |- ! rowspan=8| Miss low ! xxyA | |align=right| 432 || 0.0022202 ||align=right| 449.4 : 1 |align=right| 504 || 0.0025902 ||align=right| 385.1 : 1 |- ! xxAB | |align=right| 1,440 || 0.0074006 ||align=right| 134.1 : 1 |align=right| 2,016 || 0.0103608 ||align=right| 95.5 : 1 |- ! xyAA | |align=right| 324 || 0.0016651 ||align=right| 599.6 : 1 |align=right| 378 || 0.0019426 ||align=right| 513.8 : 1 |- ! xAAB | |align=right| 4,320 || 0.0222017 ||align=right| 44.0 : 1 |align=right| 6,048 || 0.0310823 ||align=right| 31.2 : 1 |- ! xyAB | |align=right| 2,160 || 0.0111008 ||align=right| 89.1 : 1 |align=right| 3,024 || 0.0155412 ||align=right| 63.3 : 1 |- ! xyA* | |align=right| 3,888 || 0.0199815 ||align=right| 49.0 : 1 |align=right| 3,528 || 0.0181314 ||align=right| 54.2 : 1 |- ! xAB* | |align=right| 25,920 || 0.1332100 ||align=right| 6.5 : 1 |align=right| 28,224 || 0.1450509 ||align=right| 5.9 : 1 |- | colspan=2| Total |align=right| 38,484 || 0.1977798 ||align=right| 4.1 : 1 |align=right| 43,722 || 0.2246994 ||align=right| 3.5 : 1 |- ! rowspan=18| No low ! AAAA | |align=right| 6 || 0.0000308 ||align=right| 32,429.0 : 1 |align=right| 7 || 0.0000360 ||align=right| 27,796.1 : 1 |- ! AAAB | |align=right| 480 || 0.0024669 ||align=right| 404.4 : 1 |align=right| 672 || 0.0034536 ||align=right| 288.6 : 1 |- ! AABB | |align=right| 540 || 0.0027752 ||align=right| 359.3 : 1 |align=right| 756 || 0.0038853 ||align=right| 256.4 : 1 |- ! AAA* | |align=right| 432 || 0.0022202 ||align=right| 449.4 : 1 |align=right| 392 || 0.0020146 ||align=right| 495.4 : 1 |- ! AAB* | |align=right| 14,112 || 0.0725254 ||align=right| 12.8 : 1 |align=right| 12,960 || 0.0666050 ||align=right| 14.0 : 1 |- ! xxxy | |align=right| 6 || 0.0000308 ||align=right| 32,429.0 : 1 |align=right| 6 || 0.0000308 ||align=right| 32,429.0 : 1 |- ! xxxA | |align=right| 56 || 0.0002878 ||align=right| 3,473.6 : 1 |align=right| 48 || 0.0002467 ||align=right| 4,052.8 : 1 |- ! xxx* | |align=right| 28 || 0.0001439 ||align=right| 6,948.3 : 1 |align=right| 36 || 0.0001850 ||align=right| 5,404.0 : 1 |- ! xxyy | |align=right| 9 || 0.0000463 ||align=right| 21,619.0 : 1 |align=right| 9 || 0.0000463 ||align=right| 21,619.0 : 1 |- ! xxy* | |align=right| 324 || 0.0016651 ||align=right| 599.6 : 1 |align=right| 252 || 0.0012951 ||align=right| 771.1 : 1 |- ! xxAA | |align=right| 216 || 0.0011101 ||align=right| 899.8 : 1 |align=right| 252 || 0.0012951 ||align=right| 771.1 : 1 |- ! xxA* | |align=right| 2,592 || 0.0133210 ||align=right| 74.1 : 1 |align=right| 2,352 || 0.0120876 ||align=right| 81.7 : 1 |- ! xAAA | |align=right| 144 || 0.0007401 ||align=right| 1,350.3 : 1 |align=right| 168 || 0.0008634 ||align=right| 1,157.2 : 1 |- ! xAA* | |align=right| 3,888 || 0.0199815 ||align=right| 49.0 : 1 |align=right| 3,528 || 0.0181314 ||align=right| 54.2 : 1 |- ! ??** | |align=right| 66,555 || 0.3420444 ||align=right| 1.9 : 1 |align=right| 51,051 || 0.2623651 ||align=right| 2.8 : 1 |- ! ?*** | |align=right| 24,480 || 0.1258094 ||align=right| 6.9 : 1 |align=right| 12,376 || 0.0636037 ||align=right| 14.7 : 1 |- ! **** | |align=right| 3,060 || 0.0157262 ||align=right| 62.6 : 1 |align=right| 1,001 || 0.0051444 ||align=right| 193.4 : 1 |- | colspan=2| Total |align=right| 115,776 || 0.5950046 ||align=right| 0.7 : 1 |align=right| 87,018 || 0.4472094 ||align=right| 1.2 : 1 |} {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape xy** (two low ranks and two high cards) on the river |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=13| Make low ! ABCDE | |align=right| 6,144 || 0.0035881 ||align=right| 277.7 : 1 |align=right| 21,504 || 0.0125585 ||align=right| 78.6 : 1 |- ! AABCD | |align=right| 23,040 || 0.0134556 ||align=right| 73.3 : 1 |align=right| 53,760 || 0.0313963 ||align=right| 30.9 : 1 |- ! ABCD* | |align=right| 69,120 || 0.0403667 ||align=right| 23.8 : 1 |align=right| 125,440 || 0.0732580 ||align=right| 12.7 : 1 |- ! AABBC | |align=right| 8,640 || 0.0050458 ||align=right| 197.2 : 1 |align=right| 15,120 || 0.0088302 ||align=right| 112.2 : 1 |- ! AABC* | |align=right| 103,680 || 0.0605500 ||align=right| 15.5 : 1 |align=right| 141,120 || 0.0824153 ||align=right| 11.1 : 1 |- ! AAABC | |align=right| 3,840 || 0.0022426 ||align=right| 444.9 : 1 |align=right| 6,720 || 0.0039245 ||align=right| 253.8 : 1 |- ! ABC** | |align=right| 195,840 || 0.1143722 ||align=right| 7.7 : 1 |align=right| 203,840 || 0.1190443 ||align=right| 7.4 : 1 |- ! xABCD | |align=right| 23,040 || 0.0134556 ||align=right| 73.3 : 1 |align=right| 53,760 || 0.0313963 ||align=right| 30.9 : 1 |- ! xABC* | |align=right| 138,240 || 0.0807333 ||align=right| 11.4 : 1 |align=right| 188,160 || 0.1098870 ||align=right| 8.1 : 1 |- ! xAABC | |align=right| 34,560 || 0.0201833 ||align=right| 48.5 : 1 |align=right| 60,480 || 0.0353208 ||align=right| 27.3 : 1 |- ! xxABC | |align=right| 7,680 || 0.0044852 ||align=right| 222.0 : 1 |align=right| 13,440 || 0.0078491 ||align=right| 126.4 : 1 |- ! xyABC | |align=right| 11,520 || 0.0067278 ||align=right| 147.6 : 1 |align=right| 20,160 || 0.0117736 ||align=right| 83.9 : 1 |- | colspan=2| Total |align=right| 625,344 || 0.3652062 ||align=right| 1.7 : 1 |align=right| 903,504 || 0.5276540 ||align=right| 0.9 : 1 |- ! rowspan=18| Miss low ! xxxyA | |align=right| 144 || 0.0000841 ||align=right| 11,890.0 : 1 |align=right| 168 || 0.0000981 ||align=right| 10,191.3 : 1 |- ! xxxAB | |align=right| 480 || 0.0002803 ||align=right| 3,566.3 : 1 |align=right| 672 || 0.0003925 ||align=right| 2,547.1 : 1 |- ! xxyyA | |align=right| 216 || 0.0001261 ||align=right| 7,926.3 : 1 |align=right| 252 || 0.0001472 ||align=right| 6,793.9 : 1 |- ! xxyAB | |align=right| 4,320 || 0.0025229 ||align=right| 395.4 : 1 |align=right| 6,048 || 0.0035321 ||align=right| 282.1 : 1 |- ! xxyAA | |align=right| 648 || 0.0003784 ||align=right| 2,641.4 : 1 |align=right| 756 || 0.0004415 ||align=right| 2,264.0 : 1 |- ! xxyA* | |align=right| 7,776 || 0.0045412 ||align=right| 219.2 : 1 |align=right| 7,056 || 0.0041208 ||align=right| 241.7 : 1 |- ! xxAAB | |align=right| 4,320 || 0.0025229 ||align=right| 395.4 : 1 |align=right| 6,048 || 0.0035321 ||align=right| 282.1 : 1 |- ! xxAB* | |align=right| 25,920 || 0.0151375 ||align=right| 65.1 : 1 |align=right| 28,224 || 0.0164831 ||align=right| 59.7 : 1 |- ! xyAAA | |align=right| 216 || 0.0001261 ||align=right| 7,926.3 : 1 |align=right| 252 || 0.0001472 ||align=right| 6,793.9 : 1 |- ! xyAAB | |align=right| 6,480 || 0.0037844 ||align=right| 263.2 : 1 |align=right| 9,072 || 0.0052981 ||align=right| 187.7 : 1 |- ! xyAA* | |align=right| 5,832 || 0.0034059 ||align=right| 292.6 : 1 |align=right| 5,292 || 0.0030906 ||align=right| 322.6 : 1 |- ! xyAB* | |align=right| 38,880 || 0.0227062 ||align=right| 43.0 : 1 |align=right| 42,336 || 0.0247246 ||align=right| 39.4 : 1 |- ! xyA** | |align=right| 33,048 || 0.0193003 ||align=right| 50.8 : 1 |align=right| 22,932 || 0.0133925 ||align=right| 73.7 : 1 |- ! xAAAB | |align=right| 2,880 || 0.0016819 ||align=right| 593.6 : 1 |align=right| 4,032 || 0.0023547 ||align=right| 423.7 : 1 |- ! xAABB | |align=right| 3,240 || 0.0018922 ||align=right| 527.5 : 1 |align=right| 4,536 || 0.0026491 ||align=right| 376.5 : 1 |- ! xAAB* | |align=right| 77,760 || 0.0454125 ||align=right| 21.0 : 1 |align=right| 84,672 || 0.0494492 ||align=right| 19.2 : 1 |- ! xAB** | |align=right| 220,320 || 0.1286687 ||align=right| 6.8 : 1 |align=right| 183,456 || 0.1071399 ||align=right| 8.3 : 1 |- | colspan=2| Total |align=right| 432,480 || 0.2525720 ||align=right| 3.0 : 1 |align=right| 405,804 || 0.2369930 ||align=right| 3.2 : 1 |- ! rowspan=24| No low ! AAAAB | |align=right| 120 || 0.0000701 ||align=right| 14,268.2 : 1 |align=right| 168 || 0.0000981 ||align=right| 10,191.3 : 1 |- ! AAAA* | |align=right| 108 || 0.0000631 ||align=right| 15,853.7 : 1 |align=right| 98 || 0.0000572 ||align=right| 17,471.5 : 1 |- ! AAABB | |align=right| 720 || 0.0004205 ||align=right| 2,377.2 : 1 |align=right| 1,008 || 0.0005887 ||align=right| 1,697.7 : 1 |- ! AAAB* | |align=right| 8,640 || 0.0050458 ||align=right| 197.2 : 1 |align=right| 9,408 || 0.0054944 ||align=right| 181.0 : 1 |- ! AABB* | |align=right| 9,720 || 0.0056766 ||align=right| 175.2 : 1 |align=right| 10,584 || 0.0061811 ||align=right| 160.8 : 1 |- ! AAA** | |align=right| 3,672 || 0.0021445 ||align=right| 465.3 : 1 |align=right| 2,548 || 0.0014881 ||align=right| 671.0 : 1 |- ! AAB** | |align=right| 110,160 || 0.0643344 ||align=right| 14.5 : 1 |align=right| 91,728 || 0.0535699 ||align=right| 17.7 : 1 |- ! xxxyy | |align=right| 6 || 0.0000035 ||align=right|285,383.0 : 1 |align=right| 6 || 0.0000035 ||align=right|285,383.0 : 1 |- ! xxxy* | |align=right| 108 || 0.0000631 ||align=right| 15,853.7 : 1 |align=right| 84 || 0.0000491 ||align=right| 20,383.6 : 1 |- ! xxxAA | |align=right| 72 || 0.0000420 ||align=right| 23,781.0 : 1 |align=right| 84 || 0.0000491 ||align=right| 20,383.6 : 1 |- ! xxxA* | |align=right| 864 || 0.0005046 ||align=right| 1,980.8 : 1 |align=right| 784 || 0.0004579 ||align=right| 2,183.1 : 1 |- ! xxx** | |align=right| 306 || 0.0001787 ||align=right| 5,594.8 : 1 |align=right| 182 || 0.0001063 ||align=right| 9,407.3 : 1 |- ! xxyy* | |align=right| 162 || 0.0000946 ||align=right| 10,568.8 : 1 |align=right| 126 || 0.0000736 ||align=right| 13,588.7 : 1 |- ! xxy** | |align=right| 2,754 || 0.0016084 ||align=right| 620.8 : 1 |align=right| 1,638 || 0.0009566 ||align=right| 1,044.4 : 1 |- ! xxAAA | |align=right| 144 || 0.0000841 ||align=right| 11,890.0 : 1 |align=right| 168 || 0.0000981 ||align=right| 10,191.3 : 1 |- ! xxAA* | |align=right| 3,888 || 0.0022706 ||align=right| 439.4 : 1 |align=right| 3,528 || 0.0020604 ||align=right| 484.3 : 1 |- ! xxA** | |align=right| 22,032 || 0.0128669 ||align=right| 76.7 : 1 |align=right| 15,288 || 0.0089283 ||align=right| 111.0 : 1 |- ! xAAAA | |align=right| 36 || 0.0000210 ||align=right| 47,563.0 : 1 |align=right| 42 || 0.0000245 ||align=right| 40,768.1 : 1 |- ! xAAA* | |align=right| 2,592 || 0.0015137 ||align=right| 659.6 : 1 |align=right| 2,352 || 0.0013736 ||align=right| 727.0 : 1 |- ! xAA** | |align=right| 33,048 || 0.0193003 ||align=right| 50.8 : 1 |align=right| 22,932 || 0.0133925 ||align=right| 73.7 : 1 |- ! ??*** | |align=right| 354,960 || 0.2072996 ||align=right| 3.8 : 1 |align=right| 204,204 || 0.1192569 ||align=right| 7.4 : 1 |- ! ?**** | |align=right| 91,800 || 0.0536120 ||align=right| 17.7 : 1 |align=right| 34,034 || 0.0198761 ||align=right| 49.3 : 1 |- ! ***** | |align=right| 8,568 || 0.0050038 ||align=right| 198.8 : 1 |align=right| 2,002 || 0.0011692 ||align=right| 854.3 : 1 |- | colspan=2| Total |align=right| 654,480 || 0.3822218 ||align=right| 1.6 : 1 |align=right| 402,996 || 0.2353531 ||align=right| 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 high cards and low cards left in the deck. The following tables show the derivations when holding a low pair, a low card and a high card. {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape xxy* (a low pair, a low card and a high card) on the flop |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=1| Make low ! ABC | |align=right| 1,280 || 0.0740056 ||align=right| 12.5 : 1 |align=right| 2,240 || 0.1295097 ||align=right| 6.7 : 1 |- ! rowspan=4| Miss low ! xAB | |align=right| 480 || 0.0277521 ||align=right| 35.0 : 1 |align=right| 672 || 0.0388529 ||align=right| 24.7 : 1 |- ! yAB | |align=right| 720 || 0.0416281 ||align=right| 23.0 : 1 |align=right| 1,008 || 0.0582794 ||align=right| 16.2 : 1 |- ! xyA | |align=right| 144 || 0.0083256 ||align=right| 119.1 : 1 |align=right| 168 || 0.0097132 ||align=right| 102.0 : 1 |- | colspan=2| Total |align=right| 1,344 || 0.0777058 ||align=right| 11.9 : 1 |align=right| 1,848 || 0.1068455 ||align=right| 8.4 : 1 |- ! rowspan=13| No low ! AAA | |align=right| 24 || 0.0013876 ||align=right| 719.7 : 1 |align=right| 28 || 0.0016189 ||align=right| 616.7 : 1 |- ! AAB | |align=right| 720 || 0.0416281 ||align=right| 23.0 : 1 |align=right| 1,008 || 0.0582794 ||align=right| 16.2 : 1 |- ! xxy | |align=right| 3 || 0.0001735 ||align=right| 5,764.3 : 1 |align=right| 3 || 0.0001735 ||align=right| 5,764.3 : 1 |- ! xxA | |align=right| 24 || 0.0013876 ||align=right| 719.7 : 1 |align=right| 28 || 0.0016189 ||align=right| 616.7 : 1 |- ! xyy | |align=right| 6 || 0.0003469 ||align=right| 2,881.7 : 1 |align=right| 6 || 0.0003469 ||align=right| 2,881.7 : 1 |- ! xAA | |align=right| 72 || 0.0041628 ||align=right| 239.2 : 1 |align=right| 84 || 0.0048566 ||align=right| 204.9 : 1 |- ! yyy | |align=right| 1 || 0.0000578 ||align=right| 17,295.0 : 1 |align=right| 1 || 0.0000578 ||align=right| 17,295.0 : 1 |- ! yyA | |align=right| 108 || 0.0062442 ||align=right| 159.1 : 1 |align=right| 126 || 0.0072849 ||align=right| 136.3 : 1 |- ! yAA | |align=right| 216 || 0.0124884 ||align=right| 79.1 : 1 |align=right| 252 || 0.0145698 ||align=right| 67.6 : 1 |- ! ??* | |align=right| 7,714 || 0.4459991 ||align=right| 1.2 : 1 |align=right| 7,920 || 0.4579093 ||align=right| 1.2 : 1 |- ! ?** | |align=right| 4,959 || 0.2867137 ||align=right| 2.5 : 1 |align=right| 3,465 || 0.2003353 ||align=right| 4.0 : 1 |- ! *** | |align=right| 969 || 0.0560245 ||align=right| 16.8 : 1 |align=right| 455 || 0.0263067 ||align=right| 37.0 : 1 |- | colspan=2| Total |align=right| 14,672 || 0.8482886 ||align=right| 0.2 : 1 |align=right| 13,208 || 0.7636448 ||align=right| 0.3 : 1 |} {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape xxy* (a low pair, a low card and a high card) on the turn |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=6| Make low ! ABCD | |align=right| 3,840 || 0.0197348 ||align=right| 49.7 : 1 |align=right| 8,960 || 0.0460479 ||align=right| 20.7 : 1 |- ! AABC | |align=right| 5,760 || 0.0296022 ||align=right| 32.8 : 1 |align=right| 10,080 || 0.0518039 ||align=right| 18.3 : 1 |- ! ABC* | |align=right| 24,320 || 0.1249872 ||align=right| 7.0 : 1 |align=right| 33,600 || 0.1726796 ||align=right| 4.8 : 1 |- ! xABC | |align=right| 2,560 || 0.0131565 ||align=right| 75.0 : 1 |align=right| 4,480 || 0.0230239 ||align=right| 42.4 : 1 |- ! yABC | |align=right| 3,840 || 0.0197348 ||align=right| 49.7 : 1 |align=right| 6,720 || 0.0345359 ||align=right| 28.0 : 1 |- | colspan=2| Total |align=right| 40,320 || 0.2072155 ||align=right| 3.8 : 1 |align=right| 63,840 || 0.3280913 ||align=right| 2.0 : 1 |- ! rowspan=12| Miss low ! xxyA | |align=right| 72 || 0.0003700 ||align=right| 2,701.5 : 1 |align=right| 84 || 0.0004317 ||align=right| 2,315.4 : 1 |- ! xxAB | |align=right| 240 || 0.0012334 ||align=right| 809.8 : 1 |align=right| 336 || 0.0017268 ||align=right| 578.1 : 1 |- ! xyyA | |align=right| 144 || 0.0007401 ||align=right| 1,350.3 : 1 |align=right| 168 || 0.0008634 ||align=right| 1,157.2 : 1 |- ! xyAA | |align=right| 216 || 0.0124884 ||align=right| 79.1 : 1 |align=right| 252 || 0.0145698 ||align=right| 67.6 : 1 |- ! xAAB | |align=right| 1,440 || 0.0074006 ||align=right| 134.1 : 1 |align=right| 2,016 || 0.0103608 ||align=right| 95.5 : 1 |- ! xyAB | |align=right| 1,440 || 0.0074006 ||align=right| 134.1 : 1 |align=right| 2,016 || 0.0103608 ||align=right| 95.5 : 1 |- ! xyA* | |align=right| 2,736 || 0.0140611 ||align=right| 70.1 : 1 |align=right| 2,520 || 0.0129510 ||align=right| 76.2 : 1 |- ! xAB* | |align=right| 9,120 || 0.0468702 ||align=right| 20.3 : 1 |align=right| 10,080 || 0.0518039 ||align=right| 18.3 : 1 |- ! yyAB | |align=right| 720 || 0.0037003 ||align=right| 269.3 : 1 |align=right| 1,008 || 0.0051804 ||align=right| 192.0 : 1 |- ! yAAB | |align=right| 2,160 || 0.0111008 ||align=right| 89.1 : 1 |align=right| 3,024 || 0.0155412 ||align=right| 63.3 : 1 |- ! yAB* | |align=right| 13,680 || 0.0703053 ||align=right| 13.2 : 1 |align=right| 15,120 || 0.0777058 ||align=right| 11.9 : 1 |- | colspan=2| Total |align=right| 31,968 || 0.1642923 ||align=right| 5.1 : 1 |align=right| 36,624 || 0.1882208 ||align=right| 4.3 : 1 |- ! rowspan=23| No low ! AAAA | |align=right| 6 || 0.0000308 ||align=right| 32,429.0 : 1 |align=right| 7 || 0.0000360 ||align=right| 27,796.1 : 1 |- ! AAAB | |align=right| 480 || 0.0024669 ||align=right| 404.4 : 1 |align=right| 672 || 0.0034536 ||align=right| 288.6 : 1 |- ! AABB | |align=right| 540 || 0.0027752 ||align=right| 359.3 : 1 |align=right| 756 || 0.0038853 ||align=right| 256.4 : 1 |- ! AAA* | |align=right| 456 || 0.0023435 ||align=right| 425.7 : 1 |align=right| 420 || 0.0021585 ||align=right| 462.3 : 1 |- ! AAB* | |align=right| 13,680 || 0.0703053 ||align=right| 13.2 : 1 |align=right| 15,120 || 0.0777058 ||align=right| 11.9 : 1 |- ! xxyy | |align=right| 3 || 0.0000154 ||align=right| 64,859.0 : 1 |align=right| 3 || 0.0000154 ||align=right| 64,859.0 : 1 |- ! xxy* | |align=right| 57 || 0.0002929 ||align=right| 3,412.7 : 1 |align=right| 45 || 0.0002313 ||align=right| 4,323.0 : 1 |- ! xxAA | |align=right| 36 || 0.0001850 ||align=right| 5,404.0 : 1 |align=right| 42 || 0.0002158 ||align=right| 4,631.9 : 1 |- ! xxA* | |align=right| 456 || 0.0023435 ||align=right| 425.7 : 1 |align=right| 420 || 0.0021585 ||align=right| 462.3 : 1 |- ! xyyy | |align=right| 2 || 0.0000103 ||align=right| 97,289.0 : 1 |align=right| 2 || 0.0000103 ||align=right| 97,289.0 : 1 |- ! xyy* | |align=right| 114 || 0.0005859 ||align=right| 1,705.8 : 1 |align=right| 90 || 0.0004625 ||align=right| 2,161.0 : 1 |- ! xAAA | |align=right| 48 || 0.0002467 ||align=right| 4,052.8 : 1 |align=right| 56 || 0.0002878 ||align=right| 3,473.6 : 1 |- ! xAA* | |align=right| 1,368 || 0.0070305 ||align=right| 141.2 : 1 |align=right| 1,260 || 0.0064755 ||align=right| 153.4 : 1 |- ! yyyA | |align=right| 24 || 0.0001233 ||align=right| 8,106.5 : 1 |align=right| 28 || 0.0001439 ||align=right| 6,948.3 : 1 |- ! yyy* | |align=right| 19 || 0.0000976 ||align=right| 10,240.1 : 1 |align=right| 15 || 0.0000771 ||align=right| 12,971.0 : 1 |- ! yyAA | |align=right| 108 || 0.0005550 ||align=right| 1,800.7 : 1 |align=right| 126 || 0.0006475 ||align=right| 1,543.3 : 1 |- ! yyA* | |align=right| 1,368 || 0.0070305 ||align=right| 141.2 : 1 |align=right| 1,260 || 0.0064755 ||align=right| 153.4 : 1 |- ! yAAA | |align=right| 72 || 0.0003700 ||align=right| 2,701.5 : 1 |align=right| 84 || 0.0004317 ||align=right| 2,315.4 : 1 |- ! yAA* | |align=right| 2,052 || 0.0105458 ||align=right| 93.8 : 1 |align=right| 1,890 || 0.0097132 ||align=right| 102.0 : 1 |- ! ??** | |align=right| 69,426 || 0.3567993 ||align=right| 1.8 : 1 |align=right| 55,440 || 0.2849214 ||align=right| 2.5 : 1 |- ! ?*** | |align=right| 28,101 || 0.1444187 ||align=right| 5.9 : 1 |align=right| 15,015 || 0.0771662 ||align=right| 12.0 : 1 |- ! **** | |align=right| 3,876 || 0.0199198 ||align=right| 49.2 : 1 |align=right| 1,365 || 0.0070151 ||align=right| 141.5 : 1 |- | colspan=2| Total |align=right| 122,292 || 0.6284921 ||align=right| 0.6 : 1 |align=right| 94,116 || 0.4836879 ||align=right| 1.1 : 1 |} {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape xxy* (a low pair, a low card and a high card) on the river |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=17| Make low ! ABCDE | |align=right| 6,144 || 0.0035881 ||align=right| 277.7 : 1 |align=right| 21,504 || 0.0125585 ||align=right| 78.6 : 1 |- ! AABCD | |align=right| 23,040 || 0.0134556 ||align=right| 73.3 : 1 |align=right| 53,760 || 0.0313963 ||align=right| 30.9 : 1 |- ! ABCD* | |align=right| 72,960 || 0.0426093 ||align=right| 22.5 : 1 |align=right| 134,400 || 0.0784907 ||align=right| 11.7 : 1 |- ! AABBC | |align=right| 8,640 || 0.0050458 ||align=right| 197.2 : 1 |align=right| 15,120 || 0.0088302 ||align=right| 112.2 : 1 |- ! AABC* | |align=right| 109,440 || 0.0639139 ||align=right| 14.6 : 1 |align=right| 151,200 || 0.0883021 ||align=right| 10.3 : 1 |- ! AAABC | |align=right| 3,840 || 0.0022426 ||align=right| 444.9 : 1 |align=right| 6,720 || 0.0039245 ||align=right| 253.8 : 1 |- ! ABC** | |align=right| 218,880 || 0.1278278 ||align=right| 6.8 : 1 |align=right| 235,200 || 0.1373588 ||align=right| 6.3 : 1 |- ! xABCD | |align=right| 7,680 || 0.0044852 ||align=right| 222.0 : 1 |align=right| 17,920 || 0.0104654 ||align=right| 94.6 : 1 |- ! xABC* | |align=right| 48,640 || 0.0284062 ||align=right| 34.2 : 1 |align=right| 67,200 || 0.0392454 ||align=right| 24.5 : 1 |- ! xAABC | |align=right| 11,520 || 0.0067278 ||align=right| 147.6 : 1 |align=right| 20,160 || 0.0117736 ||align=right| 83.9 : 1 |- ! xxABC | |align=right| 1,280 || 0.0007475 ||align=right| 1,336.7 : 1 |align=right| 2,240 || 0.0013082 ||align=right| 763.4 : 1 |- ! xyABC | |align=right| 7,680 || 0.0044852 ||align=right| 222.0 : 1 |align=right| 13,440 || 0.0078491 ||align=right| 126.4 : 1 |- ! yyABC | |align=right| 3,840 || 0.0022426 ||align=right| 444.9 : 1 |align=right| 6,720 || 0.0039245 ||align=right| 253.8 : 1 |- ! yABCD | |align=right| 11,520 || 0.0067278 ||align=right| 147.6 : 1 |align=right| 26,880 || 0.0156981 ||align=right| 62.7 : 1 |- ! yABC* | |align=right| 72,960 || 0.0426093 ||align=right| 22.5 : 1 |align=right| 100,800 || 0.0588681 ||align=right| 16.0 : 1 |- ! yAABC | |align=right| 17,280 || 0.0100917 ||align=right| 98.1 : 1 |align=right| 30,240 || 0.0176604 ||align=right| 55.6 : 1 |- | colspan=2| Total |align=right| 625,344 || 0.3652062 ||align=right| 1.7 : 1 |align=right| 903,504 || 0.5276540 ||align=right| 0.9 : 1 |- ! rowspan=27| Miss low ! xxyyA | |align=right| 72 || 0.0000420 ||align=right| 23,781.0 : 1 |align=right| 84 || 0.0000491 ||align=right| 20,383.6 : 1 |- ! xxyAB | |align=right| 720 || 0.0004205 ||align=right| 2,377.2 : 1 |align=right| 1,008 || 0.0005887 ||align=right| 1,697.7 : 1 |- ! xxyAA | |align=right| 108 || 0.0000631 ||align=right| 15,853.7 : 1 |align=right| 126 || 0.0000736 ||align=right| 13,588.7 : 1 |- ! xxyA* | |align=right| 1,368 || 0.0007989 ||align=right| 1,250.7 : 1 |align=right| 1,260 || 0.0007359 ||align=right| 1,358.0 : 1 |- ! xxAAB | |align=right| 720 || 0.0004205 ||align=right| 2,377.2 : 1 |align=right| 1,008 || 0.0005887 ||align=right| 1,697.7 : 1 |- ! xxAB* | |align=right| 4,560 || 0.0026631 ||align=right| 374.5  : 1 |align=right| 5,040 || 0.0029434 ||align=right| 338.7 : 1 |- ! xyyyA | |align=right| 48 || 0.0000280 ||align=right| 35,672.0 : 1 |align=right| 56 || 0.0000327 ||align=right| 30,575.9 : 1 |- ! xyyAA | |align=right| 216 || 0.0001261 ||align=right| 7,926.3 : 1 |align=right| 252 || 0.0001472 ||align=right| 6,793.9 : 1 |- ! xyyAB | |align=right| 1,440 || 0.0008410 ||align=right| 1,188.1 : 1 |align=right| 2,016 || 0.0011774 ||align=right| 848.4 : 1 |- ! xyyA* | |align=right| 2,736 || 0.0015978 ||align=right| 624.8 : 1 |align=right| 2,520 || 0.0014717 ||align=right| 678.5 : 1 |- ! xyAAA | |align=right| 144 || 0.0000841 ||align=right| 11,890.0 : 1 |align=right| 168 || 0.0000981 ||align=right| 10,191.3 : 1 |- ! xyAAB | |align=right| 4,320 || 0.0025229 ||align=right| 395.4 : 1 |align=right| 6,048 || 0.0035321 ||align=right| 282.1 : 1 |- ! xyAA* | |align=right| 4,104 || 0.0023968 ||align=right| 416.2 : 1 |align=right| 3,780 || 0.0022076 ||align=right| 452.0 : 1 |- ! xyAB* | |align=right| 27,360 || 0.0159785 ||align=right| 61.6 : 1 |align=right| 30,240 || 0.0176604 ||align=right| 55.6 : 1 |- ! xyA** | |align=right| 24,624 || 0.0143806 ||align=right| 68.5 : 1 |align=right| 17,640 || 0.0103019 ||align=right| 96.1 : 1 |- ! xAAAB | |align=right| 960 || 0.0005606 ||align=right| 1,782.7 : 1 |align=right| 1,344 || 0.0007849 ||align=right| 1,273.0 : 1 |- ! xAABB | |align=right| 1,080 || 0.0006307 ||align=right| 1,584.5 : 1 |align=right| 1,512 || 0.0008830 ||align=right| 1,131.5 : 1 |- ! xAAB* | |align=right| 27,360 || 0.0159785 ||align=right| 61.6 : 1 |align=right| 30,240 || 0.0176604 ||align=right| 55.6 : 1 |- ! xAB** | |align=right| 82,080 || 0.0479354 ||align=right| 19.9 : 1 |align=right| 70,560 || 0.0412076 ||align=right| 23.3 : 1 |- ! yyyAB | |align=right| 240 || 0.0001402 ||align=right| 7,133.6 : 1 |align=right| 336 || 0.0001962 ||align=right| 5,095.1 : 1 |- ! yyAAB | |align=right| 2,160 || 0.0012615 ||align=right| 791.7 : 1 |align=right| 3,024 || 0.0017660 ||align=right| 565.2 : 1 |- ! yyAB* | |align=right| 13,680 || 0.0079892 ||align=right| 124.2 : 1 |align=right| 15,120 || 0.0088302 ||align=right| 112.2 : 1 |- ! yAAAB | |align=right| 1,440 || 0.0008410 ||align=right| 1,188.1 : 1 |align=right| 2,016 || 0.0011774 ||align=right| 848.4 : 1 |- ! yAABB | |align=right| 1,620 || 0.0009461 ||align=right| 1,056.0 : 1 |align=right| 2,268 || 0.0013245 ||align=right| 754.0 : 1 |- ! yAAB* | |align=right| 41,040 || 0.0239677 ||align=right| 40.7 : 1 |align=right| 45,360 || 0.0264906 ||align=right| 36.7 : 1 |- ! yAB** | |align=right| 123,120 || 0.0719031 ||align=right| 12.9 : 1 |align=right| 105,840 || 0.0618115 ||align=right| 15.2 : 1 |- | colspan=2| Total |align=right| 367,320 || 0.2145180 ||align=right| 3.7 : 1 |align=right| 348,866 || 0.2037407 ||align=right| 3.9 : 1 |- ! rowspan=31| No low ! AAAAB | |align=right| 120 || 0.0000701 ||align=right| 15,019.2 : 1 |align=right| 168 || 0.0000613 ||align=right| 16,306.7 : 1 |- ! AAAA* | |align=right| 114 || 0.0000666 ||align=right| 15,853.7 : 1 |align=right| 105 || 0.0000572 ||align=right| 17,471.5 : 1 |- ! AAABB | |align=right| 720 || 0.0004205 ||align=right| 2,377.2 : 1 |align=right| 1,008 || 0.0005887 ||align=right| 1,697.7 : 1 |- ! AAAB* | |align=right| 10,080 || 0.0058868 ||align=right| 168.9 : 1 |align=right| 9,120 || 0.0053262 ||align=right| 186.8 : 1 |- ! AABB* | |align=right| 11,340 || 0.0066227 ||align=right| 150.0 : 1 |align=right| 10,260 || 0.0059919 ||align=right| 165.9 : 1 |- ! AAA** | |align=right| 2,940 || 0.0017170 ||align=right| 581.4 : 1 |align=right| 4,104 || 0.0023968 ||align=right| 416.2 : 1 |- ! AAB** | |align=right| 105,840 || 0.0618115 ||align=right| 15.2 : 1 |align=right| 123,120 || 0.0719031 ||align=right| 12.9 : 1 |- ! xxyyy | |align=right| 1 || 0.0000006 ||align=right|1,712,303.0 : 1 |align=right| 1 || 0.0000006 ||align=right|1,712,303.0 : 1 |- ! xxyy* | |align=right| 57 || 0.0000333 ||align=right| 30,039.4 : 1 |align=right| 45 || 0.0000263 ||align=right| 38,050.2 : 1 |- ! xxy** | |align=right| 513 || 0.0002996 ||align=right| 3,336.8 : 1 |align=right| 315 || 0.0001840 ||align=right| 5,434.9 : 1 |- ! xxAAA | |align=right| 24 || 0.0000140 ||align=right| 71,345.0 : 1 |align=right| 28 || 0.0000164 ||align=right| 61,152.7 : 1 |- ! xxAA* | |align=right| 684 || 0.0003995 ||align=right| 2,502.4 : 1 |align=right| 630 || 0.0003679 ||align=right| 2,716.9 : 1 |- ! xxA** | |align=right| 4,104 || 0.0023968 ||align=right| 416.2 : 1 |align=right| 2,940 || 0.0017170 ||align=right| 581.4 : 1 |- ! xyyy* | |align=right| 38 || 0.0000222 ||align=right| 45,059.6 : 1 |align=right| 30 || 0.0000175 ||align=right| 57,075.8 : 1 |- ! xyy** | |align=right| 1,026 || 0.0005992 ||align=right| 1,667.9 : 1 |align=right| 630 || 0.0003679 ||align=right| 2,716.9 : 1 |- ! xAAAA | |align=right| 12 || 0.0000070 ||align=right|142,691.0 : 1 |align=right| 14 || 0.0000082 ||align=right|122,306.4 : 1 |- ! xAAA* | |align=right| 912 || 0.0005326 ||align=right| 1,876.5 : 1 |align=right| 840 || 0.0004906 ||align=right| 2,037.5 : 1 |- ! xAA** | |align=right| 12,312 || 0.0071903 ||align=right| 138.1 : 1 |align=right| 8,820 || 0.0051510 ||align=right| 193.1 : 1 |- ! yyyAA | |align=right| 36 || 0.0000210 ||align=right| 47,563.0 : 1 |align=right| 42 || 0.0000245 ||align=right| 40,768.1 : 1 |- ! yyyA* | |align=right| 456 || 0.0002663 ||align=right| 3,754.1 : 1 |align=right| 420 || 0.0002453 ||align=right| 4,075.9 : 1 |- ! yyy** | |align=right| 171 || 0.0000999 ||align=right| 10,012.5 : 1 |align=right| 105 || 0.0000613 ||align=right| 16,306.7 : 1 |- ! yyAAA | |align=right| 72 || 0.0000420 ||align=right| 23,781.0 : 1 |align=right| 84 || 0.0000491 ||align=right| 20,383.6 : 1 |- ! yyAA* | |align=right| 2,052 || 0.0011984 ||align=right| 833.5 : 1 |align=right| 1,890 || 0.0011038 ||align=right| 905.0 : 1 |- ! yyA** | |align=right| 12,312 || 0.0071903 ||align=right| 138.1 : 1 |align=right| 8,820 || 0.0051510 ||align=right| 193.1 : 1 |- ! yAAAA | |align=right| 18 || 0.0000105 ||align=right| 95,127.0 : 1 |align=right| 21 || 0.0000123 ||align=right| 81,537.3 : 1 |- ! yAAA* | |align=right| 1,368 || 0.0007989 ||align=right| 1,250.7 : 1 |align=right| 1,260 || 0.0007359 ||align=right| 1,358.0 : 1 |- ! yAA** | |align=right| 18,468 || 0.0107855 ||align=right| 91.7 : 1 |align=right| 13,230 || 0.0077264 ||align=right| 128.4 : 1 |- ! ??*** | |align=right| 393,414 || 0.2297571 ||align=right| 3.4 : 1 |align=right| 240,240 || 0.1403022 ||align=right| 6.1 : 1 |- ! ?**** | |align=right| 112,404 || 0.0656449 ||align=right| 14.2 : 1 |align=right| 45,045 || 0.0263067 ||align=right| 37.0 : 1 |- ! ***** | |align=right| 11,628 || 0.0067909 ||align=right| 146.3 : 1 |align=right| 3,003 || 0.0017538 ||align=right| 569.2 : 1 |- | colspan=2| Total |align=right| 719,640 || 0.4202758 ||align=right| 1.4 : 1 |align=right| 459,934 || 0.2686053 ||align=right| 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 high cards and low cards left in the deck. The following tables show the derivations when holding a low three of a kind and a low card. {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape xxxy (a low three of a kind and a low card) on the flop |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=1| Make low ! ABC | |align=right| 1,280 || 0.0740056 ||align=right| 12.5 : 1 |align=right| 2,240 || 0.1295097 ||align=right| 6.7 : 1 |- ! rowspan=4| Miss low ! xAB | |align=right| 240 || 0.0138760 ||align=right| 71.1 : 1 |align=right| 336 || 0.0194265 ||align=right| 50.5 : 1 |- ! yAB | |align=right| 720 || 0.0416281 ||align=right| 23.0 : 1 |align=right| 1,008 || 0.0582794 ||align=right| 16.2 : 1 |- ! xyA | |align=right| 72 || 0.0041628 ||align=right| 239.2 : 1 |align=right| 84 || 0.0048566 ||align=right| 204.9 : 1 |- | colspan=2| Total |align=right| 1,032 || 0.0596670 ||align=right| 15.8 : 1 |align=right| 1,428 || 0.0825624 ||align=right| 11.1 : 1 |- ! rowspan=11| No low ! AAA | |align=right| 24 || 0.0013876 ||align=right| 719.7 : 1 |align=right| 28 || 0.0016189 ||align=right| 616.7 : 1 |- ! AAB | |align=right| 720 || 0.0416281 ||align=right| 23.0 : 1 |align=right| 1,008 || 0.0582794 ||align=right| 16.2 : 1 |- ! xyy | |align=right| 3 || 0.0001735 ||align=right| 5,764.3 : 1 |align=right| 3 || 0.0001735 ||align=right| 5,764.3 : 1 |- ! xAA | |align=right| 36 || 0.0020814 ||align=right| 479.4 : 1 |align=right| 42 || 0.0024283 ||align=right| 410.8 : 1 |- ! yyy | |align=right| 1 || 0.0000578 ||align=right| 17,295.0 : 1 |align=right| 1 || 0.0000578 ||align=right| 17,295.0 : 1 |- ! yyA | |align=right| 108 || 0.0062442 ||align=right| 159.1 : 1 |align=right| 126 || 0.0072849 ||align=right| 136.3 : 1 |- ! yAA | |align=right| 216 || 0.0124884 ||align=right| 79.1 : 1 |align=right| 252 || 0.0145698 ||align=right| 67.6 : 1 |- ! ??* | |align=right| 7,560 || 0.4370953 ||align=right| 1.3 : 1 |align=right| 7,936 || 0.4588344 ||align=right| 1.2 : 1 |- ! ?** | |align=right| 5,320 || 0.3075856 ||align=right| 2.3 : 1 |align=right| 3,840 || 0.2220167 ||align=right| 3.5 : 1 |- ! *** | |align=right| 1,140 || 0.0659112 ||align=right| 14.2 : 1 |align=right| 560 || 0.0323774 ||align=right| 29.9 : 1 |- | colspan=2| Total |align=right| 14,984 || 0.8663275 ||align=right| 0.2 : 1 |align=right| 13,628 || 0.7879278 ||align=right| 0.3 : 1 |} {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape xxxy (a low three of a kind and a low card) on the turn |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=6| Make low ! ABCD | |align=right| 8,960 || 0.0460479 ||align=right| 20.7 : 1 |align=right| 17,920 || 0.0920958 ||align=right| 9.9 : 1 |- ! AABC | |align=right| 10,080 || 0.0518039 ||align=right| 18.3 : 1 |align=right| 16,128 || 0.0828862 ||align=right| 11.1 : 1 |- ! ABC* | |align=right| 25,600 || 0.1315654 ||align=right| 6.6 : 1 |align=right| 35,840 || 0.1841916 ||align=right| 4.4 : 1 |- ! xABC | |align=right| 1,280 || 0.0065783 ||align=right| 151.0 : 1 |align=right| 2,240 || 0.0115120 ||align=right| 85.9 : 1 |- ! yABC | |align=right| 3,840 || 0.0197348 ||align=right| 49.7 : 1 |align=right| 6,720 || 0.0345359 ||align=right| 28.0 : 1 |- | colspan=2| Total |align=right| 40,320 || 0.2072155 ||align=right| 3.8 : 1 |align=right| 63,840 || 0.3280913 ||align=right| 2.0 : 1 |- ! rowspan=10| Miss low ! xyyA | |align=right| 72 || 0.0003700 ||align=right| 2,701.5 : 1 |align=right| 84 || 0.0004317 ||align=right| 2,315.4 : 1 |- ! xyAA | |align=right| 108 || 0.0005550 ||align=right| 1,800.7 : 1 |align=right| 126 || 0.0006475 ||align=right| 1,543.3 : 1 |- ! xAAB | |align=right| 720 || 0.0037003 ||align=right| 269.3 : 1 |align=right| 1,008 || 0.0051804 ||align=right| 192.0 : 1 |- ! xyAB | |align=right| 720 || 0.0037003 ||align=right| 269.3 : 1 |align=right| 1,008 || 0.0051804 ||align=right| 192.0 : 1 |- ! xyA* | |align=right| 1,440 || 0.0074006 ||align=right| 134.1 : 1 |align=right| 1,344 || 0.0069072 ||align=right| 143.8 : 1 |- ! xAB* | |align=right| 4,800 || 0.0246685 ||align=right| 39.5 : 1 |align=right| 5,376 || 0.0276287 ||align=right| 35.2 : 1 |- ! yyAB | |align=right| 720 || 0.0037003 ||align=right| 269.3 : 1 |align=right| 1,008 || 0.0051804 ||align=right| 192.0 : 1 |- ! yAAB | |align=right| 2,160 || 0.0111008 ||align=right| 89.1 : 1 |align=right| 3,024 || 0.0155412 ||align=right| 63.3 : 1 |- ! yAB* | |align=right| 14,400 || 0.0740056 ||align=right| 12.5 : 1 |align=right| 16,128 || 0.0828862 ||align=right| 11.1 : 1 |- | colspan=2| Total |align=right| 25,140 || 0.1292014 ||align=right| 6.7 : 1 |align=right| 29,106 || 0.1495837 ||align=right| 5.7 : 1 |- ! rowspan=19| No low ! AAAA | |align=right| 6 || 0.0000308 ||align=right| 32,429.0 : 1 |align=right| 7 || 0.0000360 ||align=right| 27,796.1 : 1 |- ! AAAB | |align=right| 480 || 0.0024669 ||align=right| 404.4 : 1 |align=right| 672 || 0.0034536 ||align=right| 288.6 : 1 |- ! AABB | |align=right| 540 || 0.0027752 ||align=right| 359.3 : 1 |align=right| 756 || 0.0038853 ||align=right| 256.4 : 1 |- ! AAA* | |align=right| 480 || 0.0024669 ||align=right| 404.4 : 1 |align=right| 448 || 0.0023024 ||align=right| 433.3 : 1 |- ! AAB* | |align=right| 14,400 || 0.0740056 ||align=right| 12.5 : 1 |align=right| 16,128 || 0.0828862 ||align=right| 11.1 : 1 |- ! xyyy | |align=right| 1 || 0.0000051 ||align=right|194,579.0 : 1 |align=right| 1 || 0.0000051 ||align=right|194,579.0 : 1 |- ! xyy* | |align=right| 60 || 0.0003084 ||align=right| 3,242.0 : 1 |align=right| 48 || 0.0002467 ||align=right| 4,052.8 : 1 |- ! xAAA | |align=right| 24 || 0.0001233 ||align=right| 8,106.5 : 1 |align=right| 28 || 0.0001439 ||align=right| 6,948.3 : 1 |- ! xAA* | |align=right| 720 || 0.0037003 ||align=right| 269.3 : 1 |align=right| 672 || 0.0034536 ||align=right| 288.6 : 1 |- ! yyyA | |align=right| 24 || 0.0001233 ||align=right| 8,106.5 : 1 |align=right| 28 || 0.0001439 ||align=right| 6,948.3 : 1 |- ! yyy* | |align=right| 20 || 0.0001028 ||align=right| 9,728.0 : 1 |align=right| 16 || 0.0000822 ||align=right| 12,160.3 : 1 |- ! yyAA | |align=right| 108 || 0.0005550 ||align=right| 1,800.7 : 1 |align=right| 126 || 0.0006475 ||align=right| 1,543.3 : 1 |- ! yyA* | |align=right| 1,440 || 0.0074006 ||align=right| 134.1 : 1 |align=right| 1,344 || 0.0069072 ||align=right| 143.8 : 1 |- ! yAAA | |align=right| 72 || 0.0003700 ||align=right| 2,701.5 : 1 |align=right| 84 || 0.0004317 ||align=right| 2,315.4 : 1 |- ! yAA* | |align=right| 2,160 || 0.0111008 ||align=right| 89.1 : 1 |align=right| 2,016 || 0.0103608 ||align=right| 95.5 : 1 |- ! ??** | |align=right| 71,820 || 0.3691027 ||align=right| 1.7 : 1 |align=right| 59,520 || 0.3058896 ||align=right| 2.3 : 1 |- ! ?*** | |align=right| 31,920 || 0.1640456 ||align=right| 5.1 : 1 |align=right| 17,920 || 0.0920958 ||align=right| 9.9 : 1 |- ! **** | |align=right| 4,845 || 0.0248998 ||align=right| 39.2 : 1 |align=right| 1,820 || 0.0093535 ||align=right| 105.9 : 1 |- | colspan=2| Total |align=right| 129,120 || 0.6635831 ||align=right| 0.5 : 1 |align=right| 101,634 || 0.5223250 ||align=right| 0.9 : 1 |} {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape xxxy (a low three of a kind and a low card) on the river |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=16| Make low ! ABCDE | |align=right| 6,144 || 0.0035881 ||align=right| 277.7 : 1 |align=right| 21,504 || 0.0125585 ||align=right| 78.6 : 1 |- ! AABCD | |align=right| 23,040 || 0.0134556 ||align=right| 73.3 : 1 |align=right| 53,760 || 0.0313963 ||align=right| 30.9 : 1 |- ! ABCD* | |align=right| 76,800 || 0.0448518 ||align=right| 21.3 : 1 |align=right| 143,360 || 0.0837235 ||align=right| 10.9 : 1 |- ! AABBC | |align=right| 8,640 || 0.0050458 ||align=right| 197.2 : 1 |align=right| 15,120 || 0.0088302 ||align=right| 112.2 : 1 |- ! AABC* | |align=right| 115,200 || 0.0672778 ||align=right| 13.9 : 1 |align=right| 161,280 || 0.0941889 ||align=right| 9.6 : 1 |- ! AAABC | |align=right| 3,840 || 0.0022426 ||align=right| 444.9 : 1 |align=right| 6,720 || 0.0039245 ||align=right| 253.8 : 1 |- ! ABC** | |align=right| 243,200 || 0.1420309 ||align=right| 6.0 : 1 |align=right| 268,800 || 0.1569815 ||align=right| 5.4 : 1 |- ! xABCD | |align=right| 3,840 || 0.0022426 ||align=right| 444.9 : 1 |align=right| 8,960 || 0.0052327 ||align=right| 190.1 : 1 |- ! xABC* | |align=right| 25,600 || 0.0149506 ||align=right| 65.9 : 1 |align=right| 35,840 || 0.0209309 ||align=right| 46.8 : 1 |- ! xAABC | |align=right| 5,760 || 0.0033639 ||align=right| 296.3 : 1 |align=right| 10,080 || 0.0058868 ||align=right| 168.9 : 1 |- ! xyABC | |align=right| 3,840 || 0.0022426 ||align=right| 444.9 : 1 |align=right| 6,720 || 0.0039245 ||align=right| 253.8 : 1 |- ! yyABC | |align=right| 3,840 || 0.0022426 ||align=right| 444.9 : 1 |align=right| 6,720 || 0.0039245 ||align=right| 253.8 : 1 |- ! yABCD | |align=right| 11,520 || 0.0067278 ||align=right| 147.6 : 1 |align=right| 26,880 || 0.0156981 ||align=right| 62.7 : 1 |- ! yABC* | |align=right| 76,800 || 0.0448518 ||align=right| 21.3 : 1 |align=right| 107,520 || 0.0627926 ||align=right| 14.9 : 1 |- ! yAABC | |align=right| 17,280 || 0.0100917 ||align=right| 98.1 : 1 |align=right| 30,240 || 0.0176604 ||align=right| 55.6 : 1 |- | colspan=2| Total |align=right| 625,344 || 0.3652062 ||align=right| 1.7 : 1 |align=right| 903,504 || 0.5276540 ||align=right| 0.9 : 1 |- ! rowspan=21| Miss low ! xyyyA | |align=right| 24 || 0.0000140 ||align=right| 71,345.0 : 1 |align=right| 28 || 0.0000164 ||align=right| 61,152.7 : 1 |- ! xyyAA | |align=right| 108 || 0.0000631 ||align=right| 15,853.7 : 1 |align=right| 126 || 0.0000736 ||align=right| 13,588.7 : 1 |- ! xyyAB | |align=right| 720 || 0.0004205 ||align=right| 2,377.2 : 1 |align=right| 1,008 || 0.0005887 ||align=right| 1,697.7 : 1 |- ! xyyA* | |align=right| 1,440 || 0.0008410 ||align=right| 1,188.1 : 1 |align=right| 1,344 || 0.0007849 ||align=right| 1,273.0 : 1 |- ! xyAAA | |align=right| 72 || 0.0000420 ||align=right| 23,781.0 : 1 |align=right| 84 || 0.0000491 ||align=right| 20,383.6 : 1 |- ! xyAAB | |align=right| 2,160 || 0.0012615 ||align=right| 791.7 : 1 |align=right| 3,024 || 0.0017660 ||align=right| 565.2 : 1 |- ! xyAA* | |align=right| 2,160 || 0.0012615 ||align=right| 791.7 : 1 |align=right| 2,016 || 0.0011774 ||align=right| 848.4 : 1 |- ! xyAB* | |align=right| 14,400 || 0.0084097 ||align=right| 117.9 : 1 |align=right| 16,128 || 0.0094189 ||align=right| 105.2 : 1 |- ! xyA** | |align=right| 13,680 || 0.0079892 ||align=right| 124.2 : 1 |align=right| 10,080 || 0.0058868 ||align=right| 168.9 : 1 |- ! xAAAB | |align=right| 480 || 0.0002803 ||align=right| 3,566.3 : 1 |align=right| 672 || 0.0003925 ||align=right| 2,547.1 : 1 |- ! xAABB | |align=right| 540 || 0.0003154 ||align=right| 3,169.9 : 1 |align=right| 756 || 0.0004415 ||align=right| 2,264.0 : 1 |- ! xAAB* | |align=right| 14,400 || 0.0084097 ||align=right| 117.9 : 1 |align=right| 16,128 || 0.0094189 ||align=right| 105.2 : 1 |- ! xAB** | |align=right| 45,600 || 0.0266308 ||align=right| 36.6 : 1 |align=right| 40,320 || 0.0235472 ||align=right| 41.5 : 1 |- ! yyyAB | |align=right| 240 || 0.0001402 ||align=right| 7,133.6 : 1 |align=right| 336 || 0.0001962 ||align=right| 5,095.1 : 1 |- ! yyAAB | |align=right| 2,160 || 0.0012615 ||align=right| 791.7 : 1 |align=right| 3,024 || 0.0017660 ||align=right| 565.2 : 1 |- ! yyAB* | |align=right| 14,400 || 0.0084097 ||align=right| 117.9 : 1 |align=right| 16,128 || 0.0094189 ||align=right| 105.2 : 1 |- ! yAAAB | |align=right| 1,440 || 0.0008410 ||align=right| 1,188.1 : 1 |align=right| 2,016 || 0.0011774 ||align=right| 848.4 : 1 |- ! yAABB | |align=right| 1,620 || 0.0009461 ||align=right| 1,056.0 : 1 |align=right| 2,268 || 0.0013245 ||align=right| 754.0 : 1 |- ! yAAB* | |align=right| 43,200 || 0.0252292 ||align=right| 38.6 : 1 |align=right| 48,384 || 0.0282567 ||align=right| 34.4 : 1 |- ! yAB** | |align=right| 136,800 || 0.0798924 ||align=right| 11.5 : 1 |align=right| 120,960 || 0.0706417 ||align=right| 13.2 : 1 |- | colspan=2| Total |align=right| 295,644 || 0.1726586 ||align=right| 4.8 : 1 |align=right| 284,830 || 0.1663431 ||align=right| 5.0 : 1 |- ! rowspan=25| No low ! AAAAB | |align=right| 120 || 0.0000701 ||align=right| 15,019.2 : 1 |align=right| 168 || 0.0000613 ||align=right| 16,306.7 : 1 |- ! AAAA* | |align=right| 120 || 0.0000701 ||align=right| 14,268.2 : 1 |align=right| 112 || 0.0000654 ||align=right| 15,287.4 : 1 |- ! AAABB | |align=right| 720 || 0.0004205 ||align=right| 2,377.2 : 1 |align=right| 1,008 || 0.0005887 ||align=right| 1,697.7 : 1 |- ! AAAB* | |align=right| 9,600 || 0.0056065 ||align=right| 177.4 : 1 |align=right| 10,752 || 0.0062793 ||align=right| 158.3 : 1 |- ! AABB* | |align=right| 10,800 || 0.0063073 ||align=right| 157.5 : 1 |align=right| 12,096 || 0.0070642 ||align=right| 140.6 : 1 |- ! AAA** | |align=right| 4,560 || 0.0026631 ||align=right| 374.5 : 1 |align=right| 3,360 || 0.0019623 ||align=right| 508.6 : 1 |- ! AAB** | |align=right| 136,800 || 0.0798924 ||align=right| 11.5 : 1 |align=right| 120,960 || 0.0706417 ||align=right| 13.2 : 1 |- ! xyyy* | |align=right| 20 || 0.0000117 ||align=right| 85,614.2 : 1 |align=right| 16 || 0.0000093 ||align=right|107,018.0 : 1 |- ! xyy** | |align=right| 570 || 0.0003329 ||align=right| 3,003.0 : 1 |align=right| 360 || 0.0002102 ||align=right| 4,755.4 : 1 |- ! xAAAA | |align=right| 6 || 0.0000035 ||align=right|285,383.0 : 1 |align=right| 7 || 0.0000041 ||align=right|244,613.9 : 1 |- ! xAAA* | |align=right| 480 || 0.0002803 ||align=right| 3,566.3 : 1 |align=right| 448 || 0.0002616 ||align=right| 3,821.1 : 1 |- ! xAA** | |align=right| 6,840 || 0.0039946 ||align=right| 249.3 : 1 |align=right| 5,040 || 0.0029434 ||align=right| 338.7 : 1 |- ! yyyAA | |align=right| 36 || 0.0000210 ||align=right| 47,563.0 : 1 |align=right| 42 || 0.0000245 ||align=right| 40,768.1 : 1 |- ! yyyA* | |align=right| 480 || 0.0002803 ||align=right| 3,566.3 : 1 |align=right| 448 || 0.0002616 ||align=right| 3,821.1 : 1 |- ! yyy** | |align=right| 190 || 0.0001110 ||align=right| 9,011.1 : 1 |align=right| 120 || 0.0000701 ||align=right| 14,268.2 : 1 |- ! yyAAA | |align=right| 72 || 0.0000420 ||align=right| 23,781.0 : 1 |align=right| 84 || 0.0000491 ||align=right| 20,383.6 : 1 |- ! yyAA* | |align=right| 2,160 || 0.0012615 ||align=right| 791.7 : 1 |align=right| 2,016 || 0.0011774 ||align=right| 848.4 : 1 |- ! yyA** | |align=right| 13,680 || 0.0079892 ||align=right| 124.2 : 1 |align=right| 10,080 || 0.0058868 ||align=right| 168.9 : 1 |- ! yAAAA | |align=right| 18 || 0.0000105 ||align=right| 95,127.0 : 1 |align=right| 21 || 0.0000123 ||align=right| 81,537.3 : 1 |- ! yAAA* | |align=right| 1,440 || 0.0008410 ||align=right| 1,188.1 : 1 |align=right| 1,344 || 0.0007849 ||align=right| 1,273.0 : 1 |- ! yAA** | |align=right| 20,520 || 0.0119839 ||align=right| 82.4 : 1 |align=right| 15,120 || 0.0088302 ||align=right| 112.2 : 1 |- ! ??*** | |align=right| 430,920 || 0.2516609 ||align=right| 3.0 : 1 |align=right| 277,760 || 0.1622142 ||align=right| 5.2 : 1 |- ! ?**** | |align=right| 135,660 || 0.0792266 ||align=right| 11.6 : 1 |align=right| 58,240 || 0.0340127 ||align=right| 28.4 : 1 |- ! ***** | |align=right| 15,504 || 0.0090545 ||align=right| 109.4 : 1 |align=right| 4,368 || 0.0025509 ||align=right| 391.0 : 1 |- | colspan=2| Total |align=right| 791,316 || 0.4621352 ||align=right| 1.2 : 1 |align=right| 523,970 || 0.3060029 ||align=right| 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 high cards and low cards left in the deck. The following tables show the derivations when holding two low pair. {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape xxyy (two low pair) on the flop |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=1| Make low ! ABC | |align=right| 1,280 || 0.0740056 ||align=right| 12.5 : 1 |align=right| 2,240 || 0.1295097 ||align=right| 6.7 : 1 |- ! rowspan=3| Miss low ! xAB | |align=right| 960 || 0.0555042 ||align=right| 17.0 : 1 |align=right| 1,344 || 0.0777058 ||align=right| 11.9 : 1 |- ! xyA | |align=right| 96 || 0.0055504 ||align=right| 179.2 : 1 |align=right| 112 || 0.0064755 ||align=right| 153.4 : 1 |- | colspan=2| Total |align=right| 1,056 || 0.0610546 ||align=right| 15.4 : 1 |align=right| 1,456 || 0.0841813 ||align=right| 10.9 : 1 |- ! rowspan=9| No low ! AAA | |align=right| 24 || 0.0013876 ||align=right| 719.7 : 1 |align=right| 28 || 0.0016189 ||align=right| 616.7 : 1 |- ! AAB | |align=right| 720 || 0.0416281 ||align=right| 23.0 : 1 |align=right| 1,008 || 0.0582794 ||align=right| 16.2 : 1 |- ! xxy | |align=right| 4 || 0.0002313 ||align=right| 4,323.0 : 1 |align=right| 4 || 0.0002313 ||align=right| 4,323.0 : 1 |- ! xxA | |align=right| 48 || 0.0027752 ||align=right| 359.3 : 1 |align=right| 56 || 0.0032377 ||align=right| 307.9 : 1 |- ! xAA | |align=right| 144 || 0.0083256 ||align=right| 119.1 : 1 |align=right| 168 || 0.0097132 ||align=right| 102.0 : 1 |- ! ??* | |align=right| 7,560 || 0.4370953 ||align=right| 1.3 : 1 |align=right| 7,936 || 0.4588344 ||align=right| 1.2 : 1 |- ! ?** | |align=right| 5,320 || 0.3075856 ||align=right| 2.3 : 1 |align=right| 3,840 || 0.2220167 ||align=right| 3.5 : 1 |- ! *** | |align=right| 1,140 || 0.0659112 ||align=right| 14.2 : 1 |align=right| 560 || 0.0323774 ||align=right| 29.9 : 1 |- | colspan=2| Total |align=right| 14,960 || 0.8649399 ||align=right| 0.2 : 1 |align=right| 13,600 || 0.7863090 ||align=right| 0.3 : 1 |} {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape xxyy (two low pair) on the turn |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=5| Make low ! ABCD | |align=right| 3,840 || 0.0197348 ||align=right| 49.7 : 1 |align=right| 8,960 || 0.0460479 ||align=right| 20.7 : 1 |- ! AABC | |align=right| 5,760 || 0.0296022 ||align=right| 32.8 : 1 |align=right| 10,080 || 0.0518039 ||align=right| 18.3 : 1 |- ! ABC* | |align=right| 25,600 || 0.1315654 ||align=right| 6.6 : 1 |align=right| 35,840 || 0.1841916 ||align=right| 4.4 : 1 |- ! xABC | |align=right| 5,120 || 0.0263131 ||align=right| 37.0 : 1 |align=right| 8,960 || 0.0460479 ||align=right| 20.7 : 1 |- | colspan=2| Total |align=right| 40,320 || 0.2072155 ||align=right| 3.8 : 1 |align=right| 63,840 || 0.3280913 ||align=right| 2.0 : 1 |- ! rowspan=8| Miss low ! xxyA | |align=right| 96 || 0.0004934 ||align=right| 2,025.9 : 1 |align=right| 112 || 0.0005756 ||align=right| 1,736.3 : 1 |- ! xxAB | |align=right| 480 || 0.0024669 ||align=right| 404.4 : 1 |align=right| 672 || 0.0034536 ||align=right| 288.6 : 1 |- ! xyAA | |align=right| 144 || 0.0007401 ||align=right| 1,350.3 : 1 |align=right| 168 || 0.0008634 ||align=right| 1,157.2 : 1 |- ! xAAB | |align=right| 2,880 || 0.0148011 ||align=right| 66.6 : 1 |align=right| 4,032 || 0.0207216 ||align=right| 47.3 : 1 |- ! xyAB | |align=right| 960 || 0.0049337 ||align=right| 201.7 : 1 |align=right| 1,344 || 0.0069072 ||align=right| 143.8 : 1 |- ! xyA* | |align=right| 1,920 || 0.0098674 ||align=right| 100.3 : 1 |align=right| 1,792 || 0.0092096 ||align=right| 107.6 : 1 |- ! xAB* | |align=right| 19,200 || 0.0986741 ||align=right| 9.1 : 1 |align=right| 21,504 || 0.1105150 ||align=right| 8.0 : 1 |- | colspan=2| Total |align=right| 25,680 || 0.1319766 ||align=right| 6.6 : 1 |align=right| 29,624 || 0.1522459 ||align=right| 5.6 : 1 |- ! rowspan=15| No low ! AAAA | |align=right| 6 || 0.0000308 ||align=right| 32,429.0 : 1 |align=right| 7 || 0.0000360 ||align=right| 27,796.1 : 1 |- ! AAAB | |align=right| 480 || 0.0024669 ||align=right| 404.4 : 1 |align=right| 672 || 0.0034536 ||align=right| 288.6 : 1 |- ! AABB | |align=right| 540 || 0.0027752 ||align=right| 359.3 : 1 |align=right| 756 || 0.0038853 ||align=right| 256.4 : 1 |- ! AAA* | |align=right| 480 || 0.0024669 ||align=right| 404.4 : 1 |align=right| 448 || 0.0023024 ||align=right| 433.3 : 1 |- ! AAB* | |align=right| 14,400 || 0.0740056 ||align=right| 12.5 : 1 |align=right| 16,128 || 0.0828862 ||align=right| 11.1 : 1 |- ! xxyy | |align=right| 1 || 0.0000051 ||align=right|194,579.0 : 1 |align=right| 1 || 0.0000051 ||align=right|194,579.0 : 1 |- ! xxy* | |align=right| 80 || 0.0004111 ||align=right| 2,431.3 : 1 |align=right| 64 || 0.0003289 ||align=right| 3,039.3 : 1 |- ! xxAA | |align=right| 72 || 0.0003700 ||align=right| 2,701.5 : 1 |align=right| 84 || 0.0004317 ||align=right| 2,315.4 : 1 |- ! xxA* | |align=right| 960 || 0.0049337 ||align=right| 201.7 : 1 |align=right| 896 || 0.0046048 ||align=right| 216.2 : 1 |- ! xAAA | |align=right| 96 || 0.0004934 ||align=right| 2,025.9 : 1 |align=right| 112 || 0.0005756 ||align=right| 1,736.3 : 1 |- ! xAA* | |align=right| 2,880 || 0.0148011 ||align=right| 66.6 : 1 |align=right| 2,688 || 0.0138144 ||align=right| 71.4 : 1 |- ! ??** | |align=right| 71,820 || 0.3691027 ||align=right| 1.7 : 1 |align=right| 59,520 || 0.3058896 ||align=right| 2.3 : 1 |- ! ?*** | |align=right| 31,920 || 0.1640456 ||align=right| 5.1 : 1 |align=right| 17,920 || 0.0920958 ||align=right| 9.9 : 1 |- ! **** | |align=right| 4,845 || 0.0248998 ||align=right| 39.2 : 1 |align=right| 1,820 || 0.0093535 ||align=right| 105.9 : 1 |- | colspan=2| Total |align=right| 128,580 || 0.6608079 ||align=right| 0.5 : 1 |align=right| 101,116 || 0.5196629 ||align=right| 0.9 : 1 |} {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape xxyy (two low pair) on the river |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=13| Make low ! ABCDE | |align=right| 6,144 || 0.0035881 ||align=right| 277.7 : 1 |align=right| 21,504 || 0.0125585 ||align=right| 78.6 : 1 |- ! AABCD | |align=right| 23,040 || 0.0134556 ||align=right| 73.3 : 1 |align=right| 53,760 || 0.0313963 ||align=right| 30.9 : 1 |- ! ABCD* | |align=right| 76,800 || 0.0448518 ||align=right| 21.3 : 1 |align=right| 143,360 || 0.0837235 ||align=right| 10.9 : 1 |- ! AABBC | |align=right| 8,640 || 0.0050458 ||align=right| 197.2 : 1 |align=right| 15,120 || 0.0088302 ||align=right| 112.2 : 1 |- ! AABC* | |align=right| 115,200 || 0.0672778 ||align=right| 13.9 : 1 |align=right| 161,280 || 0.0941889 ||align=right| 9.6 : 1 |- ! AAABC | |align=right| 3,840 || 0.0022426 ||align=right| 444.9 : 1 |align=right| 6,720 || 0.0039245 ||align=right| 253.8 : 1 |- ! ABC** | |align=right| 243,200 || 0.1420309 ||align=right| 6.0 : 1 |align=right| 268,800 || 0.1569815 ||align=right| 5.4 : 1 |- ! xABCD | |align=right| 15,360 || 0.0089704 ||align=right| 110.5 : 1 |align=right| 35,840 || 0.0209309 ||align=right| 46.8 : 1 |- ! xABC* | |align=right| 102,400 || 0.0598025 ||align=right| 15.7 : 1 |align=right| 143,360 || 0.0837235 ||align=right| 10.9 : 1 |- ! xAABC | |align=right| 23,040 || 0.0134556 ||align=right| 73.3 : 1 |align=right| 40,320 || 0.0235472 ||align=right| 41.5 : 1 |- ! xxABC | |align=right| 2,560 || 0.0014951 ||align=right| 667.9 : 1 |align=right| 4,480 || 0.0026164 ||align=right| 381.2 : 1 |- ! xyABC | |align=right| 5,120 || 0.0029901 ||align=right| 333.4 : 1 |align=right| 8,960 || 0.0052327 ||align=right| 190.1 : 1 |- | colspan=2| Total |align=right| 625,344 || 0.3652062 ||align=right| 1.7 : 1 |align=right| 903,504 || 0.5276540 ||align=right| 0.9 : 1 |- ! rowspan=16| Miss low ! xxyyA | |align=right| 24 || 0.0000140 ||align=right| 71,345.0 : 1 |align=right| 28 || 0.0000164 ||align=right| 61,152.7 : 1 |- ! xxyAB | |align=right| 960 || 0.0005606 ||align=right| 1,782.7 : 1 |align=right| 1,344 || 0.0007849 ||align=right| 1,273.0 : 1 |- ! xxyAA | |align=right| 144 || 0.0000841 ||align=right| 11,890.0 : 1 |align=right| 168 || 0.0000981 ||align=right| 10,191.3 : 1 |- ! xxyA* | |align=right| 1,920 || 0.0011213 ||align=right| 890.8 : 1 |align=right| 1,792 || 0.0010465 ||align=right| 954.5 : 1 |- ! xxAAB | |align=right| 1,440 || 0.0008410 ||align=right| 1,188.1 : 1 |align=right| 2,016 || 0.0011774 ||align=right| 848.4 : 1 |- ! xxAB* | |align=right| 9,600 || 0.0056065 ||align=right| 177.4 : 1 |align=right| 10,752 || 0.0062793 ||align=right| 158.3 : 1 |- ! xyAAA | |align=right| 96 || 0.0000561 ||align=right| 17,835.5 : 1 |align=right| 112 || 0.0000654 ||align=right| 15,287.4 : 1 |- ! xyAAB | |align=right| 2,880 || 0.0016819 ||align=right| 593.6 : 1 |align=right| 4,032 || 0.0023547 ||align=right| 423.7 : 1 |- ! xyAA* | |align=right| 2,880 || 0.0016819 ||align=right| 593.6 : 1 |align=right| 2,688 || 0.0015698 ||align=right| 636.0 : 1 |- ! xyAB* | |align=right| 19,200 || 0.0112130 ||align=right| 88.2 : 1 |align=right| 21,504 || 0.0125585 ||align=right| 78.6 : 1 |- ! xyA** | |align=right| 18,240 || 0.0106523 ||align=right| 92.9 : 1 |align=right| 13,440 || 0.0078491 ||align=right| 126.4 : 1 |- ! xAAAB | |align=right| 1,920 || 0.0011213 ||align=right| 890.8 : 1 |align=right| 2,688 || 0.0015698 ||align=right| 636.0 : 1 |- ! xAABB | |align=right| 2,160 || 0.0012615 ||align=right| 791.7 : 1 |align=right| 3,024 || 0.0017660 ||align=right| 565.2 : 1 |- ! xAAB* | |align=right| 57,600 || 0.0336389 ||align=right| 28.7 : 1 |align=right| 64,512 || 0.0376756 ||align=right| 25.5 : 1 |- ! xAB** | |align=right| 182,400 || 0.1065231 ||align=right| 8.4 : 1 |align=right| 161,280 || 0.0941889 ||align=right| 9.6 : 1 |- | colspan=2| Total |align=right| 301,464 || 0.1760575 ||align=right| 4.7 : 1 |align=right| 289,380 || 0.1690004 ||align=right| 4.9 : 1 |- ! rowspan=19| No low ! AAAAB | |align=right| 120 || 0.0000701 ||align=right| 14,268.2 : 1 |align=right| 168 || 0.0000981 ||align=right| 10,191.3 : 1 |- ! AAAA* | |align=right| 120 || 0.0000701 ||align=right| 14,268.2 : 1 |align=right| 112 || 0.0000654 ||align=right| 15,287.4 : 1 |- ! AAABB | |align=right| 720 || 0.0004205 ||align=right| 2,377.2 : 1 |align=right| 1,008 || 0.0005887 ||align=right| 1,697.7 : 1 |- ! AAAB* | |align=right| 9,600 || 0.0056065 ||align=right| 177.4 : 1 |align=right| 10,752 || 0.0062793 ||align=right| 158.3 : 1 |- ! AABB* | |align=right| 10,800 || 0.0063073 ||align=right| 157.5 : 1 |align=right| 12,096 || 0.0070642 ||align=right| 140.6 : 1 |- ! AAA** | |align=right| 4,560 || 0.0026631 ||align=right| 374.5 : 1 |align=right| 3,360 || 0.0019623 ||align=right| 508.6 : 1 |- ! AAB** | |align=right| 136,800 || 0.0798924 ||align=right| 11.5 : 1 |align=right| 120,960 || 0.0706417 ||align=right| 13.2 : 1 |- ! xxyy* | |align=right| 20 || 0.0000117 ||align=right| 85,614.2 : 1 |align=right| 16 || 0.0000093 ||align=right|107,018.0 : 1 |- ! xxy** | |align=right| 760 || 0.0004438 ||align=right| 2,252.0 : 1 |align=right| 480 || 0.0002803 ||align=right| 3,566.3 : 1 |- ! xxAAA | |align=right| 48 || 0.0000280 ||align=right| 35,672.0 : 1 |align=right| 56 || 0.0000327 ||align=right| 30,575.9 : 1 |- ! xxAA* | |align=right| 1,440 || 0.0008410 ||align=right| 1,188.1 : 1 |align=right| 1,344 || 0.0007849 ||align=right| 1,273.0 : 1 |- ! xxA** | |align=right| 9,120 || 0.0053262 ||align=right| 186.8 : 1 |align=right| 6,720 || 0.0039245 ||align=right| 253.8 : 1 |- ! xAAAA | |align=right| 24 || 0.0000140 ||align=right| 71,345.0 : 1 |align=right| 28 || 0.0000164 ||align=right| 61,152.7 : 1 |- ! xAAA* | |align=right| 1,920 || 0.0011213 ||align=right| 890.8 : 1 |align=right| 1,792 || 0.0010465 ||align=right| 954.5 : 1 |- ! xAA** | |align=right| 27,360 || 0.0159785 ||align=right| 61.6 : 1 |align=right| 20,160 || 0.0117736 ||align=right| 83.9 : 1 |- ! ??*** | |align=right| 430,920 || 0.2516609 ||align=right| 3.0 : 1 |align=right| 277,760 || 0.1622142 ||align=right| 5.2 : 1 |- ! ?**** | |align=right| 135,660 || 0.0792266 ||align=right| 11.6 : 1 |align=right| 58,240 || 0.0340127 ||align=right| 28.4 : 1 |- ! ***** | |align=right| 15,504 || 0.0090545 ||align=right| 109.4 : 1 |align=right| 4,368 || 0.0025509 ||align=right| 391.0 : 1 |- | colspan=2| Total |align=right| 785,496 || 0.4587363 ||align=right| 1.2 : 1 |align=right| 519,420 || 0.3033457 ||align=right| 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 high cards and low cards left in the deck. The following tables show the derivations when holding three low ranks and a high card. {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape xyz* (three low ranks and a high card) on the flop |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=3| Make low ! ABC | |align=right| 640 || 0.0370028 ||align=right| 26.0 : 1 |align=right| 1,280 || 0.0740056 ||align=right| 12.5 : 1 |- ! xAB | |align=right| 1,440 || 0.0832562 ||align=right| 11.0 : 1 |align=right| 2,160 || 0.1248844 ||align=right| 7.0 : 1 |- | colspan=2| Total |align=right| 2,080 || 0.1202590 ||align=right| 7.3 : 1 |align=right| 3,440 || 0.1988899 ||align=right| 4.0 : 1 |- ! rowspan=3| Miss low ! xyA | |align=right| 540 || 0.0312211 ||align=right| 31.0 : 1 |align=right| 648 || 0.0374653 ||align=right| 25.7 : 1 |- ! xyz | |align=right| 27 || 0.0015611 ||align=right| 639.6 : 1 |align=right| 27 || 0.0015611 ||align=right| 639.6 : 1 |- | colspan=2| Total |align=right| 567 || 0.0327821 ||align=right| 29.5 : 1 |align=right| 675 || 0.0390264 ||align=right| 24.6 : 1 |- ! rowspan=10| No low ! AAA | |align=right| 20 || 0.0011563 ||align=right| 863.8 : 1 |align=right| 24 || 0.0013876 ||align=right| 719.7 : 1 |- ! AAB | |align=right| 480 || 0.0277521 ||align=right| 35.0 : 1 |align=right| 720 || 0.0416281 ||align=right| 23.0 : 1 |- ! xxx | |align=right| 3 || 0.0001735 ||align=right| 5,764.3 : 1 |align=right| 3 || 0.0001735 ||align=right| 5,764.3 : 1 |- ! xxy | |align=right| 54 || 0.0031221 ||align=right| 319.3 : 1 |align=right| 54 || 0.0031221 ||align=right| 319.3 : 1 |- ! xxA | |align=right| 180 || 0.0104070 ||align=right| 95.1 : 1 |align=right| 216 || 0.0124884 ||align=right| 79.1 : 1 |- ! xAA | |align=right| 270 || 0.0156105 ||align=right| 63.1 : 1 |align=right| 324 || 0.0187327 ||align=right| 52.4 : 1 |- ! ??* | |align=right| 7,714 || 0.4459991 ||align=right| 1.2 : 1 |align=right| 7,920 || 0.4579093 ||align=right| 1.2 : 1 |- ! ?** | |align=right| 4,959 || 0.2867137 ||align=right| 2.5 : 1 |align=right| 3,465 || 0.2003353 ||align=right| 4.0 : 1 |- ! *** | |align=right| 969 || 0.0560245 ||align=right| 16.8 : 1 |align=right| 455 || 0.0263067 ||align=right| 37.0 : 1 |- | colspan=2| Total |align=right| 14,649 || 0.8469588 ||align=right| 0.2 : 1 |align=right| 13,181 || 0.7620837 ||align=right| 0.3 : 1 |} {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape xyz* (three low ranks and a high card) on the turn |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=9| Make low ! ABCD | |align=right| 1,280 || 0.0065783 ||align=right| 151.0 : 1 |align=right| 3,840 || 0.0197348 ||align=right| 49.7 : 1 |- ! AABC | |align=right| 2,880 || 0.0148011 ||align=right| 66.6 : 1 |align=right| 5,760 || 0.0296022 ||align=right| 32.8 : 1 |- ! ABC* | |align=right| 12,160 || 0.0624936 ||align=right| 15.0 : 1 |align=right| 19,200 || 0.0986741 ||align=right| 9.1 : 1 |- ! xABC | |align=right| 5,760 || 0.0296022 ||align=right| 32.8 : 1 |align=right| 11,520 || 0.0592044 ||align=right| 15.9 : 1 |- ! xxAB | |align=right| 1,440 || 0.0074006 ||align=right| 134.1 : 1 |align=right| 2,160 || 0.0111008 ||align=right| 89.1 : 1 |- ! xAAB | |align=right| 4,320 || 0.0222017 ||align=right| 44.0 : 1 |align=right| 6,480 || 0.0333025 ||align=right| 29.0 : 1 |- ! xyAB | |align=right| 4,320 || 0.0222017 ||align=right| 44.0 : 1 |align=right| 6,480 || 0.0333025 ||align=right| 29.0 : 1 |- ! xAB* | |align=right| 27,360 || 0.1406105 ||align=right| 6.1 : 1 |align=right| 32,400 || 0.1665125 ||align=right| 5.0 : 1 |- | colspan=2| Total |align=right| 59,520 || 0.3058896 ||align=right| 2.3 : 1 |align=right| 87,840 || 0.4514339 ||align=right| 1.2 : 1 |- ! rowspan=7| Miss low ! xxyA | |align=right| 1,080 || 0.0055504 ||align=right| 179.2 : 1 |align=right| 1,296 || 0.0066605 ||align=right| 149.1 : 1 |- ! xyAA | |align=right| 810 || 0.0041628 ||align=right| 239.2 : 1 |align=right| 972 || 0.0049954 ||align=right| 199.2 : 1 |- ! xyA* | |align=right| 10,260 || 0.0527290 ||align=right| 18.0 : 1 |align=right| 9,720 || 0.0499537 ||align=right| 19.0 : 1 |- ! xxyz | |align=right| 81 || 0.0004163 ||align=right| 2,401.2 : 1 |align=right| 81 || 0.0004163 ||align=right| 2,401.2 : 1 |- ! xyzA | |align=right| 540 || 0.0027752 ||align=right| 359.3 : 1 |align=right| 648 || 0.0033302 ||align=right| 299.3 : 1 |- ! xyz* | |align=right| 513 || 0.0026364 ||align=right| 378.3 : 1 |align=right| 405 || 0.0020814 ||align=right| 479.4 : 1 |- | colspan=2| Total |align=right| 13,284 || 0.0682701 ||align=right| 13.6 : 1 |align=right| 13,122 || 0.0674376 ||align=right| 13.8 : 1 |- ! rowspan=18| No low ! AAAA | |align=right| 5 || 0.0000257 ||align=right| 38,915.0 : 1 |align=right| 6 || 0.0000308 ||align=right| 32,429.0 : 1 |- ! AAAB | |align=right| 320 || 0.0016446 ||align=right| 607.1 : 1 |align=right| 480 || 0.0024669 ||align=right| 404.4 : 1 |- ! AABB | |align=right| 360 || 0.0018501 ||align=right| 539.5 : 1 |align=right| 540 || 0.0027752 ||align=right| 359.3 : 1 |- ! AAA* | |align=right| 380 || 0.0019529 ||align=right| 511.1 : 1 |align=right| 360 || 0.0018501 ||align=right| 539.5 : 1 |- ! AAB* | |align=right| 9,120 || 0.0468702 ||align=right| 20.3 : 1 |align=right| 10,800 || 0.0555042 ||align=right| 17.0 : 1 |- ! xxxy | |align=right| 18 || 0.0000925 ||align=right| 10,809.0 : 1 |align=right| 18 || 0.0000925 ||align=right| 10,809.0 : 1 |- ! xxxA | |align=right| 60 || 0.0003084 ||align=right| 3,242.0 : 1 |align=right| 72 || 0.0003700 ||align=right| 2,701.5 : 1 |- ! xxx* | |align=right| 57 || 0.0002929 ||align=right| 3,412.7 : 1 |align=right| 45 || 0.0002313 ||align=right| 4,323.0 : 1 |- ! xxyy | |align=right| 27 || 0.0001388 ||align=right| 7,205.7 : 1 |align=right| 27 || 0.0001388 ||align=right| 7,205.7 : 1 |- ! xxy* | |align=right| 1,026 || 0.0052729 ||align=right| 188.6 : 1 |align=right| 810 || 0.0041628 ||align=right| 239.2 : 1 |- ! xxAA | |align=right| 270 || 0.0013876 ||align=right| 719.7 : 1 |align=right| 324 || 0.0016651 ||align=right| 599.6 : 1 |- ! xxA* | |align=right| 3,420 || 0.0175763 ||align=right| 55.9 : 1 |align=right| 3,240 || 0.0166512 ||align=right| 59.1 : 1 |- ! xAAA | |align=right| 180 || 0.0009251 ||align=right| 1,080.0 : 1 |align=right| 216 || 0.0011101 ||align=right| 899.8 : 1 |- ! xAA* | |align=right| 5,130 || 0.0263645 ||align=right| 36.9 : 1 |align=right| 4,860 || 0.0249769 ||align=right| 39.0 : 1 |- ! ??** | |align=right| 69,426 || 0.3567993 ||align=right| 1.8 : 1 |align=right| 55,440 || 0.2849214 ||align=right| 2.5 : 1 |- ! ?*** | |align=right| 28,101 || 0.1444187 ||align=right| 5.9 : 1 |align=right| 15,015 || 0.0771662 ||align=right| 12.0 : 1 |- ! **** | |align=right| 3,876 || 0.0199198 ||align=right| 49.2 : 1 |align=right| 1,365 || 0.0070151 ||align=right| 141.5 : 1 |- | colspan=2| Total |align=right| 121,776 || 0.6258403 ||align=right| 0.6 : 1 |align=right| 93,618 || 0.4811286 ||align=right| 1.1 : 1 |} {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape xyz* (three low ranks and a high card) on the river |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=24| Make low ! ABCDE | |align=right| 1,024 || 0.0005980 ||align=right| 1,671.2 : 1 |align=right| 6,144 || 0.0035881 ||align=right| 277.7 : 1 |- ! AABCD | |align=right| 7,680 || 0.0044852 ||align=right| 222.0 : 1 |align=right| 23,040 || 0.0134556 ||align=right| 73.3 : 1 |- ! ABCD* | |align=right| 24,320 || 0.0142031 ||align=right| 69.4 : 1 |align=right| 57,600 || 0.0336389 ||align=right| 28.7 : 1 |- ! AABBC | |align=right| 4,320 || 0.0025229 ||align=right| 395.4 : 1 |align=right| 8,640 || 0.0050458 ||align=right| 197.2 : 1 |- ! AABC* | |align=right| 54,720 || 0.0319569 ||align=right| 30.3 : 1 |align=right| 86,400 || 0.0504583 ||align=right| 18.8 : 1 |- ! AAABC | |align=right| 1,920 || 0.0011213 ||align=right| 890.8 : 1 |align=right| 3,840 || 0.0022426 ||align=right| 444.9 : 1 |- ! ABC** | |align=right| 109,440 || 0.0639139 ||align=right| 14.6 : 1 |align=right| 134,400 || 0.0784907 ||align=right| 11.7 : 1 |- ! xABCD | |align=right| 11,520 || 0.0067278 ||align=right| 147.6 : 1 |align=right| 34,560 || 0.0201833 ||align=right| 48.5 : 1 |- ! xABC* | |align=right| 109,440 || 0.0639139 ||align=right| 14.6 : 1 |align=right| 172,800 || 0.1009167 ||align=right| 8.9 : 1 |- ! xAABC | |align=right| 25,920 || 0.0151375 ||align=right| 65.1 : 1 |align=right| 51,840 || 0.0302750 ||align=right| 32.0 : 1 |- ! xxABC | |align=right| 5,760 || 0.0033639 ||align=right| 296.3 : 1 |align=right| 11,520 || 0.0067278 ||align=right| 147.6 : 1 |- ! xyABC | |align=right| 17,280 || 0.0100917 ||align=right| 98.1 : 1 |align=right| 34,560 || 0.0201833 ||align=right| 48.5 : 1 |- ! xxxAB | |align=right| 480 || 0.0002803 ||align=right| 3,566.3 : 1 |align=right| 720 || 0.0004205 ||align=right| 2,377.2 : 1 |- ! xxyAB | |align=right| 8,640 || 0.0050458 ||align=right| 197.2 : 1 |align=right| 12,960 || 0.0075687 ||align=right| 131.1 : 1 |- ! xxAAB | |align=right| 4,320 || 0.0025229 ||align=right| 395.4 : 1 |align=right| 6,480 || 0.0037844 ||align=right| 263.2 : 1 |- ! xxAB* | |align=right| 27,360 || 0.0159785 ||align=right| 61.6 : 1 |align=right| 32,400 || 0.0189219 ||align=right| 51.8 : 1 |- ! xyAAB | |align=right| 12,960 || 0.0075687 ||align=right| 131.1 : 1 |align=right| 19,440 || 0.0113531 ||align=right| 87.1 : 1 |- ! xyAB* | |align=right| 82,080 || 0.0479354 ||align=right| 19.9 : 1 |align=right| 97,200 || 0.0567656 ||align=right| 16.6 : 1 |- ! xAAAB | |align=right| 2,880 || 0.0016819 ||align=right| 593.6 : 1 |align=right| 4,320 || 0.0025229 ||align=right| 395.4 : 1 |- ! xAABB | |align=right| 3,240 || 0.0018922 ||align=right| 527.5 : 1 |align=right| 4,860 || 0.0028383 ||align=right| 351.3 : 1 |- ! xAAB* | |align=right| 82,080 || 0.0479354 ||align=right| 19.9 : 1 |align=right| 97,200 || 0.0567656 ||align=right| 16.6 : 1 |- ! xAB** | |align=right| 246,240 || 0.1438062 ||align=right| 6.0 : 1 |align=right| 226,800 || 0.1324531 ||align=right| 6.5 : 1 |- ! xyzAB | |align=right| 4,320 || 0.0025229 ||align=right| 395.4 : 1 |align=right| 6,480 || 0.0037844 ||align=right| 263.2 : 1 |- | colspan=2| Total |align=right| 847,944 || 0.4952065 ||align=right| 1.0 : 1 |align=right|1,134,204 || 0.6623847 ||align=right| 0.5 : 1 |- ! rowspan=15| Miss low ! xxxyA | |align=right| 360 || 0.0002102 ||align=right| 4,755.4 : 1 |align=right| 432 || 0.0002523 ||align=right| 3,962.7 : 1 |- ! xxyyA | |align=right| 540 || 0.0003154 ||align=right| 3,169.9 : 1 |align=right| 648 || 0.0003784 ||align=right| 2,641.4 : 1 |- ! xxyAA | |align=right| 1,620 || 0.0009461 ||align=right| 1,056.0 : 1 |align=right| 1,944 || 0.0011353 ||align=right| 879.8 : 1 |- ! xxyA* | |align=right| 20,520 || 0.0119839 ||align=right| 82.4 : 1 |align=right| 19,440 || 0.0113531 ||align=right| 87.1 : 1 |- ! xyAAA | |align=right| 540 || 0.0003154 ||align=right| 3,169.9 : 1 |align=right| 648 || 0.0003784 ||align=right| 2,641.4 : 1 |- ! xyAA* | |align=right| 15,390 || 0.0089879 ||align=right| 110.3 : 1 |align=right| 14,580 || 0.0085148 ||align=right| 116.4 : 1 |- ! xyA** | |align=right| 92,340 || 0.0539273 ||align=right| 17.5 : 1 |align=right| 68,040 || 0.0397359 ||align=right| 24.2 : 1 |- ! xxxyz | |align=right| 27 || 0.0000158 ||align=right| 63,417.7 : 1 |align=right| 27 || 0.0000158 ||align=right| 63,417.7 : 1 |- ! xxyyz | |align=right| 81 || 0.0000473 ||align=right| 21,138.6 : 1 |align=right| 81 || 0.0000473 ||align=right| 21,138.6 : 1 |- ! xxyzA | |align=right| 1,620 || 0.0009461 ||align=right| 1,056.0 : 1 |align=right| 1,944 || 0.0011353 ||align=right| 879.8 : 1 |- ! xxyz* | |align=right| 1,539 || 0.0008988 ||align=right| 1,111.6 : 1 |align=right| 1,215 || 0.0007096 ||align=right| 1,408.3 : 1 |- ! xyzAA | |align=right| 810 || 0.0004730 ||align=right| 2,113.0 : 1 |align=right| 972 || 0.0005677 ||align=right| 1,760.6 : 1 |- ! xyzA* | |align=right| 10,260 || 0.0059919 ||align=right| 165.9 : 1 |align=right| 9,720 || 0.0056766 ||align=right| 175.2 : 1 |- ! xyz** | |align=right| 4,617 || 0.0026964 ||align=right| 369.9 : 1 |align=right| 2,835 || 0.0016557 ||align=right| 603.0 : 1 |- | colspan=2| Total |align=right| 150,264 || 0.0877554 ||align=right| 10.4 : 1 |align=right| 122,526 || 0.0715562 ||align=right| 13.0 : 1 |- ! rowspan=24| No low ! AAAAB | |align=right| 80 || 0.0000467 ||align=right| 21,402.8 : 1 |align=right| 120 || 0.0000701 ||align=right| 14,268.2 : 1 |- ! AAAA* | |align=right| 95 || 0.0000555 ||align=right| 18,023.3 : 1 |align=right| 90 || 0.0000526 ||align=right| 19,024.6 : 1 |- ! AAABB | |align=right| 480 || 0.0002803 ||align=right| 3,566.3 : 1 |align=right| 720 || 0.0004205 ||align=right| 2,377.2 : 1 |- ! AAAB* | |align=right| 6,080 || 0.0035508 ||align=right| 280.6 : 1 |align=right| 7,200 || 0.0042049 ||align=right| 236.8 : 1 |- ! AABB* | |align=right| 6,840 || 0.0039946 ||align=right| 249.3 : 1 |align=right| 8,100 || 0.0047305 ||align=right| 210.4 : 1 |- ! AAA** | |align=right| 3,420 || 0.0019973 ||align=right| 499.7 : 1 |align=right| 2,520 || 0.0014717 ||align=right| 678.5 : 1 |- ! AAB** | |align=right| 82,080 || 0.0479354 ||align=right| 19.9 : 1 |align=right| 75,600 || 0.0441510 ||align=right| 21.6 : 1 |- ! xxxyy | |align=right| 18 || 0.0000105 ||align=right| 95,127.0 : 1 |align=right| 18 || 0.0000105 ||align=right| 95,127.0 : 1 |- ! xxxy* | |align=right| 342 || 0.0001997 ||align=right| 5,005.7 : 1 |align=right| 270 || 0.0001577 ||align=right| 6,340.9 : 1 |- ! xxxAA | |align=right| 90 || 0.0000526 ||align=right| 19,024.6 : 1 |align=right| 108 || 0.0000631 ||align=right| 15,853.7 : 1 |- ! xxxA* | |align=right| 1,140 || 0.0006658 ||align=right| 1,501.0 : 1 |align=right| 1,080 || 0.0006307 ||align=right| 1,584.5 : 1 |- ! xxx** | |align=right| 513 || 0.0002996 ||align=right| 3,336.8 : 1 |align=right| 315 || 0.0001840 ||align=right| 5,434.9 : 1 |- ! xxyy* | |align=right| 513 || 0.0002996 ||align=right| 3,336.8 : 1 |align=right| 405 || 0.0002365 ||align=right| 4,226.9 : 1 |- ! xxy** | |align=right| 9,234 || 0.0053927 ||align=right| 184.4 : 1 |align=right| 5,670 || 0.0033113 ||align=right| 301.0 : 1 |- ! xxAAA | |align=right| 180 || 0.0001051 ||align=right| 9,511.8 : 1 |align=right| 216 || 0.0001261 ||align=right| 7,926.3 : 1 |- ! xxAA* | |align=right| 5,130 || 0.0029960 ||align=right| 332.8 : 1 |align=right| 4,860 || 0.0028383 ||align=right| 351.3 : 1 |- ! xxA** | |align=right| 30,780 || 0.0179758 ||align=right| 54.6 : 1 |align=right| 22,680 || 0.0132453 ||align=right| 74.5 : 1 |- ! xAAAA | |align=right| 45 || 0.0000263 ||align=right| 38,050.2 : 1 |align=right| 54 || 0.0000315 ||align=right| 31,708.3 : 1 |- ! xAAA* | |align=right| 3,420 || 0.0019973 ||align=right| 499.7 : 1 |align=right| 3,240 || 0.0018922 ||align=right| 527.5 : 1 |- ! xAA** | |align=right| 46,170 || 0.0269637 ||align=right| 36.1 : 1 |align=right| 34,020 || 0.0198680 ||align=right| 49.3 : 1 |- ! ??*** | |align=right| 393,414 || 0.2297571 ||align=right| 3.4 : 1 |align=right| 240,240 || 0.1403022 ||align=right| 6.1 : 1 |- ! ?**** | |align=right| 112,404 || 0.0656449 ||align=right| 14.2 : 1 |align=right| 45,045 || 0.0263067 ||align=right| 37.0 : 1 |- ! ***** | |align=right| 11,628 || 0.0067909 ||align=right| 146.3 : 1 |align=right| 3,003 || 0.0017538 ||align=right| 569.2 : 1 |- | colspan=2| Total |align=right| 714,096 || 0.4170381 ||align=right| 1.4 : 1 |align=right| 455,574 || 0.2660591 ||align=right| 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 high cards and low cards left in the deck. The following tables show the derivations when holding a low pair and two low ranks. {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape xxyz (a low pair and two low ranks) on the flop |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=4| Make low ! ABC | |align=right| 640 || 0.0370028 ||align=right| 26.0 : 1 |align=right| 1,280 || 0.0740056 ||align=right| 12.5 : 1 |- ! xAB | |align=right| 320 || 0.0185014 ||align=right| 53.1 : 1 |align=right| 480 || 0.0277521 ||align=right| 35.0 : 1 |- ! yAB | |align=right| 960 || 0.0555042 ||align=right| 17.0 : 1 |align=right| 1,440 || 0.0832562 ||align=right| 11.0 : 1 |- | colspan=2| Total |align=right| 1,920 || 0.1110083 ||align=right| 8.0 : 1 |align=right| 3,200 || 0.1850139 ||align=right| 4.4 : 1 |- ! rowspan=4| Miss low ! xyA | |align=right| 240 || 0.0138760 ||align=right| 71.1 : 1 |align=right| 288 || 0.0166512 ||align=right| 95.1 : 1 |- ! yzA | |align=right| 180 || 0.0104070 ||align=right| 59.1 : 1 |align=right| 216 || 0.0124884 ||align=right| 79.1 : 1 |- ! xyz | |align=right| 18 || 0.0010407 ||align=right| 959.9 : 1 |align=right| 18 || 0.0010407 ||align=right| 959.9 : 1 |- | colspan=2| Total |align=right| 438 || 0.0253238 ||align=right| 38.5 : 1 |align=right| 522 || 0.0301804 ||align=right| 32.1 : 1 |- ! rowspan=14| No low ! AAA | |align=right| 20 || 0.0011563 ||align=right| 863.8 : 1 |align=right| 24 || 0.0013876 ||align=right| 719.7 : 1 |- ! AAB | |align=right| 480 || 0.0277521 ||align=right| 35.0 : 1 |align=right| 720 || 0.0416281 ||align=right| 23.0 : 1 |- ! xxy | |align=right| 6 || 0.0003469 ||align=right| 2,881.7 : 1 |align=right| 6 || 0.0003469 ||align=right| 2,881.7 : 1 |- ! xxA | |align=right| 20 || 0.0011563 ||align=right| 863.8 : 1 |align=right| 24 || 0.0013876 ||align=right| 719.7 : 1 |- ! xyy | |align=right| 12 || 0.0006938 ||align=right| 1,440.3 : 1 |align=right| 12 || 0.0006938 ||align=right| 1,440.3 : 1 |- ! xAA | |align=right| 60 || 0.0034690 ||align=right| 287.3 : 1 |align=right| 72 || 0.0041628 ||align=right| 239.2 : 1 |- ! yyy | |align=right| 2 || 0.0001156 ||align=right| 8,647.0 : 1 |align=right| 2 || 0.0001156 ||align=right| 8,647.0 : 1 |- ! yyA | |align=right| 120 || 0.0069380 ||align=right| 143.1 : 1 |align=right| 144 || 0.0083256 ||align=right| 119.1 : 1 |- ! yAA | |align=right| 180 || 0.0104070 ||align=right| 59.1 : 1 |align=right| 216 || 0.0124884 ||align=right| 79.1 : 1 |- ! yyz | |align=right| 18 || 0.0010407 ||align=right| 959.9 : 1 |align=right| 18 || 0.0010407 ||align=right| 959.9 : 1 |- ! ??* | |align=right| 7,560 || 0.4370953 ||align=right| 1.3 : 1 |align=right| 7,936 || 0.4588344 ||align=right| 1.2 : 1 |- ! ?** | |align=right| 5,320 || 0.3075856 ||align=right| 2.3 : 1 |align=right| 3,840 || 0.2220167 ||align=right| 3.5 : 1 |- ! *** | |align=right| 1,140 || 0.0659112 ||align=right| 14.2 : 1 |align=right| 560 || 0.0323774 ||align=right| 29.9 : 1 |- | colspan=2| Total |align=right| 14,938 || 0.8636679 ||align=right| 0.2 : 1 |align=right| 13,574 || 0.7848057 ||align=right| 0.3 : 1 |} {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape xxyz (a low pair and two low ranks) on the turn |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=14| Make low ! ABCD | |align=right| 1,280 || 0.0065783 ||align=right| 151.0 : 1 |align=right| 3,840 || 0.0197348 ||align=right| 49.7 : 1 |- ! AABC | |align=right| 2,880 || 0.0148011 ||align=right| 66.6 : 1 |align=right| 5,760 || 0.0296022 ||align=right| 32.8 : 1 |- ! ABC* | |align=right| 12,800 || 0.0657827 ||align=right| 14.2 : 1 |align=right| 20,480 || 0.1052523 ||align=right| 8.5 : 1 |- ! xABC | |align=right| 1,280 || 0.0065783 ||align=right| 151.0 : 1 |align=right| 2,560 || 0.0131565 ||align=right| 75.0 : 1 |- ! yABC | |align=right| 6,720 || 0.0345359 ||align=right| 28.0 : 1 |align=right| 7,680 || 0.0394696 ||align=right| 24.3 : 1 |- ! xxAB | |align=right| 160 || 0.0008223 ||align=right| 1,215.1 : 1 |align=right| 240 || 0.0012334 ||align=right| 809.8 : 1 |- ! xAAB | |align=right| 960 || 0.0049337 ||align=right| 201.7 : 1 |align=right| 1,440 || 0.0074006 ||align=right| 134.1 : 1 |- ! xyAB | |align=right| 1,920 || 0.0098674 ||align=right| 100.3 : 1 |align=right| 2,880 || 0.0148011 ||align=right| 66.6 : 1 |- ! xAB* | |align=right| 6,400 || 0.0328914 ||align=right| 29.4 : 1 |align=right| 7,680 || 0.0394696 ||align=right| 24.3 : 1 |- ! yyAB | |align=right| 960 || 0.0049337 ||align=right| 201.7 : 1 |align=right| 1,440 || 0.0074006 ||align=right| 134.1 : 1 |- ! yAAB | |align=right| 2,880 || 0.0148011 ||align=right| 66.6 : 1 |align=right| 4,320 || 0.0222017 ||align=right| 44.0 : 1 |- ! yAB* | |align=right| 19,200 || 0.0986741 ||align=right| 9.1 : 1 |align=right| 23,040 || 0.1184089 ||align=right| 7.4 : 1 |- ! yzAB | |align=right| 1,440 || 0.0074006 ||align=right| 134.1 : 1 |align=right| 2,160 || 0.0111008 ||align=right| 89.1 : 1 |- | colspan=2| Total |align=right| 56,000 || 0.2877994 ||align=right| 2.5 : 1 |align=right| 83,520 || 0.4292322 ||align=right| 1.3 : 1 |- ! rowspan=12| Miss low ! xxyA | |align=right| 120 || 0.0006167 ||align=right| 1,620.5 : 1 |align=right| 144 || 0.0007401 ||align=right| 1,350.3 : 1 |- ! xyyA | |align=right| 240 || 0.0012334 ||align=right| 809.8 : 1 |align=right| 288 || 0.0014801 ||align=right| 674.6 : 1 |- ! xyAA | |align=right| 360 || 0.0018501 ||align=right| 539.5 : 1 |align=right| 432 || 0.0022202 ||align=right| 449.4 : 1 |- ! xyA* | |align=right| 4,800 || 0.0246685 ||align=right| 39.5 : 1 |align=right| 4,608 || 0.0236818 ||align=right| 41.2 : 1 |- ! yyzA | |align=right| 360 || 0.0018501 ||align=right| 539.5 : 1 |align=right| 432 || 0.0022202 ||align=right| 449.4 : 1 |- ! yzAA | |align=right| 270 || 0.0013876 ||align=right| 719.7 : 1 |align=right| 324 || 0.0016651 ||align=right| 599.6 : 1 |- ! yzA* | |align=right| 3,600 || 0.0185014 ||align=right| 53.1 : 1 |align=right| 3,456 || 0.0177613 ||align=right| 55.3 : 1 |- ! xxyz | |align=right| 9 || 0.0000463 ||align=right| 21,619.0 : 1 |align=right| 9 || 0.0000463 ||align=right| 21,619.0 : 1 |- ! xyyz | |align=right| 36 || 0.0001850 ||align=right| 5,404.0 : 1 |align=right| 36 || 0.0001850 ||align=right| 5,404.0 : 1 |- ! xyzA | |align=right| 360 || 0.0018501 ||align=right| 539.5 : 1 |align=right| 432 || 0.0022202 ||align=right| 449.4 : 1 |- ! xyz* | |align=right| 360 || 0.0018501 ||align=right| 539.5 : 1 |align=right| 288 || 0.0014801 ||align=right| 674.6 : 1 |- | colspan=2| Total |align=right| 10,515 || 0.0540395 ||align=right| 17.5 : 1 |align=right| 10,449 || 0.0537003 ||align=right| 17.6 : 1 |- ! rowspan=26| No low ! AAAA | |align=right| 5 || 0.0000257 ||align=right| 38,915.0 : 1 |align=right| 6 || 0.0000308 ||align=right| 32,429.0 : 1 |- ! AAAB | |align=right| 320 || 0.0016446 ||align=right| 607.1 : 1 |align=right| 480 || 0.0024669 ||align=right| 404.4 : 1 |- ! AABB | |align=right| 360 || 0.0018501 ||align=right| 539.5 : 1 |align=right| 540 || 0.0027752 ||align=right| 359.3 : 1 |- ! AAA* | |align=right| 400 || 0.0020557 ||align=right| 485.5 : 1 |align=right| 384 || 0.0019735 ||align=right| 505.7 : 1 |- ! AAB* | |align=right| 9,600 || 0.0493370 ||align=right| 19.3 : 1 |align=right| 11,520 || 0.0592044 ||align=right| 15.9 : 1 |- ! xxyy | |align=right| 6 || 0.0000308 ||align=right| 32,429.0 : 1 |align=right| 6 || 0.0000308 ||align=right| 32,429.0 : 1 |- ! xxy* | |align=right| 120 || 0.0006167 ||align=right| 1,620.5 : 1 |align=right| 96 || 0.0004934 ||align=right| 2,025.9 : 1 |- ! xxAA | |align=right| 30 || 0.0001542 ||align=right| 6,485.0 : 1 |align=right| 36 || 0.0001850 ||align=right| 5,404.0 : 1 |- ! xxA* | |align=right| 400 || 0.0020557 ||align=right| 485.5 : 1 |align=right| 384 || 0.0019735 ||align=right| 505.7 : 1 |- ! xyyy | |align=right| 4 || 0.0000206 ||align=right| 48,644.0 : 1 |align=right| 4 || 0.0000206 ||align=right| 48,644.0 : 1 |- ! xyy* | |align=right| 240 || 0.0012334 ||align=right| 809.8 : 1 |align=right| 192 || 0.0009867 ||align=right| 1,012.4 : 1 |- ! xAAA | |align=right| 40 || 0.0002056 ||align=right| 4,863.5 : 1 |align=right| 48 || 0.0002467 ||align=right| 4,052.8 : 1 |- ! xAA* | |align=right| 1,200 || 0.0061671 ||align=right| 161.2 : 1 |align=right| 1,152 || 0.0059204 ||align=right| 167.9 : 1 |- ! yyyA | |align=right| 40 || 0.0002056 ||align=right| 4,863.5 : 1 |align=right| 48 || 0.0002467 ||align=right| 4,052.8 : 1 |- ! yyy* | |align=right| 40 || 0.0002056 ||align=right| 4,863.5 : 1 |align=right| 32 || 0.0001645 ||align=right| 6,079.6 : 1 |- ! yyAA | |align=right| 180 || 0.0009251 ||align=right| 1,080.0 : 1 |align=right| 216 || 0.0011101 ||align=right| 899.8 : 1 |- ! yyA* | |align=right| 2,400 || 0.0123343 ||align=right| 80.1 : 1 |align=right| 2,304 || 0.0118409 ||align=right| 83.5 : 1 |- ! yAAA | |align=right| 120 || 0.0006167 ||align=right| 1,620.5 : 1 |align=right| 144 || 0.0007401 ||align=right| 1,350.3 : 1 |- ! yAA* | |align=right| 3,600 || 0.0185014 ||align=right| 53.1 : 1 |align=right| 3,456 || 0.0177613 ||align=right| 55.3 : 1 |- ! yyyz | |align=right| 6 || 0.0000308 ||align=right| 32,429.0 : 1 |align=right| 6 || 0.0000308 ||align=right| 32,429.0 : 1 |- ! yyzz | |align=right| 9 || 0.0000463 ||align=right| 21,619.0 : 1 |align=right| 9 || 0.0000463 ||align=right| 21,619.0 : 1 |- ! yyz* | |align=right| 360 || 0.0018501 ||align=right| 539.5 : 1 |align=right| 288 || 0.0014801 ||align=right| 674.6 : 1 |- ! ??** | |align=right| 71,820 || 0.3691027 ||align=right| 1.7 : 1 |align=right| 59,520 || 0.3058896 ||align=right| 2.3 : 1 |- ! ?*** | |align=right| 31,920 || 0.1640456 ||align=right| 5.1 : 1 |align=right| 17,920 || 0.0920958 ||align=right| 9.9 : 1 |- ! **** | |align=right| 4,845 || 0.0248998 ||align=right| 39.2 : 1 |align=right| 1,820 || 0.0093535 ||align=right| 105.9 : 1 |- | colspan=2| Total |align=right| 128,065 || 0.6581612 ||align=right| 0.5 : 1 |align=right| 100,611 || 0.5170675 ||align=right| 0.9 : 1 |} {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape xxyz (a low pair and two low ranks) on the river |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=39| Make low ! ABCDE | |align=right| 1,024 || 0.0005980 ||align=right| 1,671.2 : 1 |align=right| 6,144 || 0.0035881 ||align=right| 277.7 : 1 |- ! AABCD | |align=right| 7,680 || 0.0044852 ||align=right| 30.9 : 1 |align=right| 23,040 || 0.0134556 ||align=right| 73.3 : 1 |- ! ABCD* | |align=right| 25,600 || 0.0149506 ||align=right| 222.0 : 1 |align=right| 61,440 || 0.0358815 ||align=right| 26.9 : 1 |- ! AABBC | |align=right| 4,320 || 0.0025229 ||align=right| 395.4 : 1 |align=right| 8,640 || 0.0050458 ||align=right| 197.2 : 1 |- ! AABC* | |align=right| 57,600 || 0.0336389 ||align=right| 28.7 : 1 |align=right| 92,160 || 0.0538222 ||align=right| 17.6 : 1 |- ! AAABC | |align=right| 1,920 || 0.0011213 ||align=right| 890.8 : 1 |align=right| 3,840 || 0.0022426 ||align=right| 444.9 : 1 |- ! ABC** | |align=right| 121,600 || 0.0710154 ||align=right| 13.1 : 1 |align=right| 153,600 || 0.0897037 ||align=right| 10.1 : 1 |- ! xABCD | |align=right| 2,560 || 0.0014951 ||align=right| 667.9 : 1 |align=right| 7,680 || 0.0044852 ||align=right| 222.0 : 1 |- ! xABC* | |align=right| 25,600 || 0.0149506 ||align=right| 65.9 : 1 |align=right| 40,960 || 0.0239210 ||align=right| 40.8 : 1 |- ! xAABC | |align=right| 5,760 || 0.0033639 ||align=right| 296.3 : 1 |align=right| 11,520 || 0.0067278 ||align=right| 147.6 : 1 |- ! xxABC | |align=right| 640 || 0.0003738 ||align=right| 2,674.5 : 1 |align=right| 1,280 || 0.0007475 ||align=right| 1,336.7 : 1 |- ! xyABC | |align=right| 7,680 || 0.0044852 ||align=right| 222.0 : 1 |align=right| 15,360 || 0.0089704 ||align=right| 110.5 : 1 |- ! yyABC | |align=right| 3,840 || 0.0022426 ||align=right| 444.9 : 1 |align=right| 7,680 || 0.0044852 ||align=right| 222.0 : 1 |- ! yABCD | |align=right| 7,680 || 0.0044852 ||align=right| 222.0 : 1 |align=right| 23,040 || 0.0134556 ||align=right| 73.3 : 1 |- ! yABC* | |align=right| 76,800 || 0.0448518 ||align=right| 21.3 : 1 |align=right| 122,880 || 0.0717630 ||align=right| 12.9 : 1 |- ! yAABC | |align=right| 17,280 || 0.0100917 ||align=right| 98.1 : 1 |align=right| 34,560 || 0.0201833 ||align=right| 48.5 : 1 |- ! xxyAB | |align=right| 960 || 0.0005606 ||align=right| 1,782.7 : 1 |align=right| 1,440 || 0.0008410 ||align=right| 1,188.1 : 1 |- ! xxAAB | |align=right| 480 || 0.0002803 ||align=right| 3,566.3 : 1 |align=right| 720 || 0.0004205 ||align=right| 2,377.2 : 1 |- ! xxAB* | |align=right| 3,200 || 0.0018688 ||align=right| 534.1 : 1 |align=right| 3,840 || 0.0022426 ||align=right| 444.9 : 1 |- ! xyyAB | |align=right| 1,920 || 0.0011213 ||align=right| 890.8 : 1 |align=right| 2,880 || 0.0016819 ||align=right| 593.6 : 1 |- ! xyAAB | |align=right| 5,760 || 0.0033639 ||align=right| 296.3 : 1 |align=right| 8,640 || 0.0050458 ||align=right| 197.2 : 1 |- ! xyAB* | |align=right| 38,400 || 0.0224259 ||align=right| 43.6 : 1 |align=right| 46,080 || 0.0269111 ||align=right| 36.2 : 1 |- ! xAAAB | |align=right| 640 || 0.0003738 ||align=right| 2,674.5 : 1 |align=right| 960 || 0.0005606 ||align=right| 1,782.7 : 1 |- ! xAABB | |align=right| 720 || 0.0004205 ||align=right| 2,377.2 : 1 |align=right| 1,080 || 0.0006307 ||align=right| 1,584.5 : 1 |- ! xAAB* | |align=right| 19,200 || 0.0112130 ||align=right| 88.2 : 1 |align=right| 23,040 || 0.0134556 ||align=right| 73.3 : 1 |- ! xAB** | |align=right| 60,800 || 0.0355077 ||align=right| 27.2 : 1 |align=right| 57,600 || 0.0336389 ||align=right| 28.7 : 1 |- ! yyyAB | |align=right| 320 || 0.0001869 ||align=right| 5,350.0 : 1 |align=right| 480 || 0.0002803 ||align=right| 3,566.3 : 1 |- ! yyAAB | |align=right| 2,880 || 0.0016819 ||align=right| 593.6 : 1 |align=right| 4,320 || 0.0025229 ||align=right| 395.4 : 1 |- ! yyAB* | |align=right| 19,200 || 0.0112130 ||align=right| 88.2 : 1 |align=right| 23,040 || 0.0134556 ||align=right| 73.3 : 1 |- ! yAAAB | |align=right| 1,920 || 0.0011213 ||align=right| 890.8 : 1 |align=right| 2,880 || 0.0016819 ||align=right| 593.6 : 1 |- ! yAABB | |align=right| 1,620 || 0.0009461 ||align=right| 1,056.0 : 1 |align=right| 3,240 || 0.0018922 ||align=right| 527.5 : 1 |- ! yAAB* | |align=right| 57,600 || 0.0336389 ||align=right| 28.7 : 1 |align=right| 69,120 || 0.0403667 ||align=right| 23.8 : 1 |- ! yAB** | |align=right| 182,400 || 0.1065231 ||align=right| 8.4 : 1 |align=right| 172,800 || 0.1009167 ||align=right| 8.9 : 1 |- ! yyzAB | |align=right| 2,880 || 0.0016819 ||align=right| 593.6 : 1 |align=right| 4,320 || 0.0025229 ||align=right| 395.4 : 1 |- ! yzAAB | |align=right| 4,320 || 0.0025229 ||align=right| 395.4 : 1 |align=right| 6,480 || 0.0037844 ||align=right| 263.2 : 1 |- ! yzABC | |align=right| 5,760 || 0.0033639 ||align=right| 296.3 : 1 |align=right| 11,520 || 0.0067278 ||align=right| 147.6 : 1 |- ! yzAB* | |align=right| 28,800 || 0.0168194 ||align=right| 58.5 : 1 |align=right| 34,560 || 0.0201833 ||align=right| 48.5 : 1 |- ! xyzAB | |align=right| 2,880 || 0.0016819 ||align=right| 593.6 : 1 |align=right| 4,320 || 0.0025229 ||align=right| 395.4 : 1 |- | colspan=2| Total |align=right| 810,784 || 0.4735047 ||align=right| 1.1 : 1 |align=right|1,097,184 || 0.6407647 ||align=right| 0.6 : 1 |- ! rowspan=27| Miss low ! xxyyA | |align=right| 120 || 0.0000701 ||align=right| 14,268.2 : 1 |align=right| 144 || 0.0000841 ||align=right| 11,890.0 : 1 |- ! xxyAA | |align=right| 180 || 0.0001051 ||align=right| 9,511.8 : 1 |align=right| 216 || 0.0001261 ||align=right| 7,926.3 : 1 |- ! xxyA* | |align=right| 2,400 || 0.0014016 ||align=right| 712.5 : 1 |align=right| 2,304 || 0.0013456 ||align=right| 742.2 : 1 |- ! xyyyA | |align=right| 80 || 0.0000467 ||align=right| 21,402.8 : 1 |align=right| 96 || 0.0000561 ||align=right| 17,835.5 : 1 |- ! xyyAA | |align=right| 360 || 0.0002102 ||align=right| 4,755.4 : 1 |align=right| 432 || 0.0002523 ||align=right| 3,962.7 : 1 |- ! xyyA* | |align=right| 4,800 || 0.0028032 ||align=right| 355.7 : 1 |align=right| 4,608 || 0.0026911 ||align=right| 370.6 : 1 |- ! xyAAA | |align=right| 240 || 0.0001402 ||align=right| 7,133.6 : 1 |align=right| 288 || 0.0001682 ||align=right| 5,944.5 : 1 |- ! xyAA* | |align=right| 7,200 || 0.0042049 ||align=right| 236.8 : 1 |align=right| 6,912 || 0.0040367 ||align=right| 246.7 : 1 |- ! xyA** | |align=right| 45,600 || 0.0266308 ||align=right| 36.6 : 1 |align=right| 34,560 || 0.0201833 ||align=right| 48.5 : 1 |- ! yyyzA | |align=right| 120 || 0.0000701 ||align=right| 14,268.2 : 1 |align=right| 144 || 0.0000841 ||align=right| 11,890.0 : 1 |- ! yyzzA | |align=right| 180 || 0.0001051 ||align=right| 9,511.8 : 1 |align=right| 216 || 0.0001261 ||align=right| 7,926.3 : 1 |- ! yyzAA | |align=right| 540 || 0.0003154 ||align=right| 3,169.9 : 1 |align=right| 648 || 0.0003784 ||align=right| 2,641.4 : 1 |- ! yyzA* | |align=right| 7,200 || 0.0042049 ||align=right| 236.8 : 1 |align=right| 6,912 || 0.0040367 ||align=right| 246.7 : 1 |- ! yzAAA | |align=right| 180 || 0.0001051 ||align=right| 9,511.8 : 1 |align=right| 216 || 0.0001261 ||align=right| 7,926.3 : 1 |- ! yzAA* | |align=right| 5,400 || 0.0031536 ||align=right| 316.1 : 1 |align=right| 5,184 || 0.0030275 ||align=right| 329.3 : 1 |- ! yzA** | |align=right| 34,200 || 0.0199731 ||align=right| 49.1 : 1 |align=right| 25,920 || 0.0151375 ||align=right| 65.1 : 1 |- ! xxyyz | |align=right| 18 || 0.0000105 ||align=right| 95,127.0 : 1 |align=right| 18 || 0.0000105 ||align=right| 95,127.0 : 1 |- ! xxyzA | |align=right| 180 || 0.0001051 ||align=right| 9,511.8 : 1 |align=right| 216 || 0.0001261 ||align=right| 7,926.3 : 1 |- ! xxyz* | |align=right| 180 || 0.0001051 ||align=right| 9,511.8 : 1 |align=right| 144 || 0.0000841 ||align=right| 11,890.0 : 1 |- ! xyyyz | |align=right| 12 || 0.0000070 ||align=right| 14,2691 : 1 |align=right| 12 || 0.0000070 ||align=right| 14,2691 : 1 |- ! xyyzz | |align=right| 18 || 0.0000105 ||align=right| 95,127.0 : 1 |align=right| 18 || 0.0000105 ||align=right| 95,127.0 : 1 |- ! xyyzA | |align=right| 720 || 0.0004205 ||align=right| 2,377.2 : 1 |align=right| 864 || 0.0005046 ||align=right| 1,980.8 : 1 |- ! xyyz* | |align=right| 720 || 0.0004205 ||align=right| 2,377.2 : 1 |align=right| 576 || 0.0003364 ||align=right| 2,971.8 : 1 |- ! xyzAA | |align=right| 540 || 0.0003154 ||align=right| 3,169.9 : 1 |align=right| 648 || 0.0003784 ||align=right| 2,641.4 : 1 |- ! xyzA* | |align=right| 7,200 || 0.0042049 ||align=right| 236.8 : 1 |align=right| 6,912 || 0.0040367 ||align=right| 246.7 : 1 |- ! xyz** | |align=right| 3,420 || 0.0019973 ||align=right| 499.7 : 1 |align=right| 2,160 || 0.0012615 ||align=right| 791.7 : 1 |- | colspan=2| Total |align=right| 121,808 || 0.0711369 ||align=right| 13.1 : 1 |align=right| 100,368 || 0.0586158 ||align=right| 16.1 : 1 |- ! rowspan=35| No low ! AAAAB | |align=right| 80 || 0.0000467 ||align=right| 21,402.8 : 1 |align=right| 120 || 0.0000701 ||align=right| 15,019.2 : 1 |- ! AAAA* | |align=right| 100 || 0.0000584 ||align=right| 17,122.0 : 1 |align=right| 96 || 0.0000561 ||align=right| 17,835.5 : 1 |- ! AAABB | |align=right| 480 || 0.0002803 ||align=right| 3,566.3 : 1 |align=right| 720 || 0.0004205 ||align=right| 2,377.2 : 1 |- ! AAAB* | |align=right| 6,400 || 0.0037377 ||align=right| 266.5 : 1 |align=right| 7,680 || 0.0044852 ||align=right| 30.9 : 1 |- ! AABB* | |align=right| 7,200 || 0.0042049 ||align=right| 236.8 : 1 |align=right| 8,640 || 0.0050458 ||align=right| 197.2 : 1 |- ! AAA** | |align=right| 3,800 || 0.0022192 ||align=right| 449.6 : 1 |align=right| 2,880 || 0.0016819 ||align=right| 593.6 : 1 |- ! AAB** | |align=right| 91,200 || 0.0532616 ||align=right| 17.8 : 1 |align=right| 86,400 || 0.0504583 ||align=right| 18.8 : 1 |- ! xxyyy | |align=right| 2 || 0.0000012 ||align=right| 856151.0 : 1 |align=right| 2 || 0.0000012 ||align=right| 856151.0 : 1 |- ! xxyy* | |align=right| 120 || 0.0000701 ||align=right| 14,268.2 : 1 |align=right| 96 || 0.0000561 ||align=right| 17,835.5 : 1 |- ! xxy** | |align=right| 1,140 || 0.0006658 ||align=right| 1,501.0 : 1 |align=right| 720 || 0.0004205 ||align=right| 2,377.2 : 1 |- ! xxAAA | |align=right| 20 || 0.0000117 ||align=right| 85,614.2 : 1 |align=right| 24 || 0.0000140 ||align=right| 71,345.0 : 1 |- ! xxAA* | |align=right| 600 || 0.0003504 ||align=right| 2,852.8 : 1 |align=right| 576 || 0.0003364 ||align=right| 2,971.8 : 1 |- ! xxA** | |align=right| 3,800 || 0.0022192 ||align=right| 449.6 : 1 |align=right| 2,880 || 0.0016819 ||align=right| 593.6 : 1 |- ! xyyy* | |align=right| 80 || 0.0000467 ||align=right| 21,402.8 : 1 |align=right| 64 || 0.0000374 ||align=right| 26,753.8 : 1 |- ! xyy** | |align=right| 2,280 || 0.0013315 ||align=right| 750.0 : 1 |align=right| 1,440 || 0.0008410 ||align=right| 1,188.1 : 1 |- ! xAAAA | |align=right| 10 || 0.0000058 ||align=right|171,229.4 : 1 |align=right| 12 || 0.0000070 ||align=right|142,691.0 : 1 |- ! xAAA* | |align=right| 800 || 0.0004672 ||align=right| 2,139.4 : 1 |align=right| 768 || 0.0004485 ||align=right| 2,228.6 : 1 |- ! xAA** | |align=right| 11,400 || 0.0066577 ||align=right| 149.2 : 1 |align=right| 8,640 || 0.0050458 ||align=right| 197.2 : 1 |- ! yyyAA | |align=right| 60 || 0.0000350 ||align=right| 28,537.4 : 1 |align=right| 72 || 0.0000420 ||align=right| 23,781.0 : 1 |- ! yyyA* | |align=right| 800 || 0.0004672 ||align=right| 2,139.4 : 1 |align=right| 768 || 0.0004485 ||align=right| 2,228.6 : 1 |- ! yyy** | |align=right| 380 || 0.0002219 ||align=right| 4,505.1 : 1 |align=right| 240 || 0.0001402 ||align=right| 7,133.6 : 1 |- ! yyAAA | |align=right| 120 || 0.0000701 ||align=right| 14,268.2 : 1 |align=right| 144 || 0.0000841 ||align=right| 11,890.0 : 1 |- ! yyAA* | |align=right| 3,600 || 0.0021024 ||align=right| 474.6 : 1 |align=right| 3,456 || 0.0020183 ||align=right| 494.5 : 1 |- ! yyA** | |align=right| 22,800 || 0.0133154 ||align=right| 74.1 : 1 |align=right| 17,280 || 0.0100917 ||align=right| 98.1 : 1 |- ! yAAAA | |align=right| 30 || 0.0000175 ||align=right| 57,075.8 : 1 |align=right| 36 || 0.0000210 ||align=right| 47,563.0 : 1 |- ! yAAA* | |align=right| 2,400 || 0.0014016 ||align=right| 712.5 : 1 |align=right| 2,304 || 0.0013456 ||align=right| 742.2 : 1 |- ! yAA** | |align=right| 34,200 || 0.0199731 ||align=right| 49.1 : 1 |align=right| 25,920 || 0.0151375 ||align=right| 65.1 : 1 |- ! yyyzz | |align=right| 6 || 0.0000035 ||align=right|285,383.0 : 1 |align=right| 6 || 0.0000035 ||align=right|285,383.0 : 1 |- ! yyyz* | |align=right| 120 || 0.0000701 ||align=right| 14,268.2 : 1 |align=right| 96 || 0.0000561 ||align=right| 17,835.5 : 1 |- ! yyzz* | |align=right| 180 || 0.0001051 ||align=right| 9,511.8 : 1 |align=right| 144 || 0.0000841 ||align=right| 11,890.0 : 1 |- ! yyz** | |align=right| 3,420 || 0.0019973 ||align=right| 499.7 : 1 |align=right| 2,160 || 0.0012615 ||align=right| 791.7 : 1 |- ! ??*** | |align=right| 430,920 || 0.2516609 ||align=right| 3.0 : 1 |align=right| 277,760 || 0.1622142 ||align=right| 5.2 : 1 |- ! ?**** | |align=right| 135,660 || 0.0792266 ||align=right| 11.6 : 1 |align=right| 58,240 || 0.0340127 ||align=right| 28.4 : 1 |- ! ***** | |align=right| 15,504 || 0.0090545 ||align=right| 109.4 : 1 |align=right| 4,368 || 0.0025509 ||align=right| 391.0 : 1 |- | colspan=2| Total |align=right| 779,712 || 0.4553584 ||align=right| 1.2 : 1 |align=right| 514,752 || 0.3006195 ||align=right| 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 high cards and low cards left in the deck. The following tables show the derivations when holding four low ranks. {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape xyzr (four low ranks) on the flop |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=4| Make low ! ABC | |align=right| 256 || 0.0148011 ||align=right| 66.6 : 1 |align=right| 640 || 0.0370028 ||align=right| 26.0 : 1 |- ! xAB | |align=right| 1,152 || 0.0666050 ||align=right| 14.0 : 1 |align=right| 1,920 || 0.1110083 ||align=right| 8.0 : 1 |- ! xyA | |align=right| 864 || 0.0499537 ||align=right| 19.0 : 1 |align=right| 1,080 || 0.0624422 ||align=right| 15.0 : 1 |- | colspan=2| Total |align=right| 2,272 || 0.1313599 ||align=right| 6.6 : 1 |align=right| 3,640 || 0.2104533 ||align=right| 3.8 : 1 |- ! rowspan=1| Miss low ! xyz | |align=right| 108 || 0.0062442 ||align=right| 159.1 : 1 |align=right| 108 || 0.0062442 ||align=right| 159.1 : 1 |- ! rowspan=10| No low ! AAA | |align=right| 16 || 0.0009251 ||align=right| 1,080.0 : 1 |align=right| 20 || 0.0011563 ||align=right| 863.8 : 1 |- ! AAB | |align=right| 288 || 0.0166512 ||align=right| 59.1 : 1 |align=right| 480 || 0.0277521 ||align=right| 35.0 : 1 |- ! xxx | |align=right| 4 || 0.0002313 ||align=right| 4,323.0 : 1 |align=right| 4 || 0.0002313 ||align=right| 4,323.0 : 1 |- ! xxy | |align=right| 108 || 0.0062442 ||align=right| 159.1 : 1 |align=right| 108 || 0.0062442 ||align=right| 159.1 : 1 |- ! xxA | |align=right| 192 || 0.0111008 ||align=right| 89.1 : 1 |align=right| 240 || 0.0138760 ||align=right| 71.1 : 1 |- ! xAA | |align=right| 288 || 0.0166512 ||align=right| 59.1 : 1 |align=right| 360 || 0.0208141 ||align=right| 47.0 : 1 |- ! ??* | |align=right| 7,560 || 0.4370953 ||align=right| 1.3 : 1 |align=right| 7,936 || 0.4588344 ||align=right| 1.2 : 1 |- ! ?** | |align=right| 5,320 || 0.3075856 ||align=right| 2.3 : 1 |align=right| 3,840 || 0.2220167 ||align=right| 3.5 : 1 |- ! *** | |align=right| 1,140 || 0.0659112 ||align=right| 14.2 : 1 |align=right| 560 || 0.0323774 ||align=right| 29.9 : 1 |- | colspan=2| Total |align=right| 14,916 || 0.8623959 ||align=right| 0.2 : 1 |align=right| 13,548 || 0.7833025 ||align=right| 0.3 : 1 |} {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape xyzr (four low ranks) on the turn |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=13| Make low ! ABCD | |align=right| 256 || 0.0013157 ||align=right| 759.1 : 1 |align=right| 1,280 || 0.0065783 ||align=right| 151.0 : 1 |- ! AABC | |align=right| 1,152 || 0.0059204 ||align=right| 167.9 : 1 |align=right| 2,880 || 0.0148011 ||align=right| 66.6 : 1 |- ! ABC* | |align=right| 5,120 || 0.0263131 ||align=right| 37.0 : 1 |align=right| 10,240 || 0.0526262 ||align=right| 18.0 : 1 |- ! xABC | |align=right| 3,072 || 0.0157879 ||align=right| 62.3 : 1 |align=right| 7,680 || 0.0394696 ||align=right| 24.3 : 1 |- ! xxAB | |align=right| 1,152 || 0.0059204 ||align=right| 167.9 : 1 |align=right| 1,920 || 0.0098674 ||align=right| 100.3 : 1 |- ! xAAB | |align=right| 3,456 || 0.0177613 ||align=right| 55.3 : 1 |align=right| 5,760 || 0.0296022 ||align=right| 32.8 : 1 |- ! xyAB | |align=right| 5,184 || 0.0266420 ||align=right| 36.5 : 1 |align=right| 8,640 || 0.0444033 ||align=right| 21.5 : 1 |- ! xAB* | |align=right| 23,040 || 0.1184089 ||align=right| 7.4 : 1 |align=right| 30,720 || 0.1578785 ||align=right| 5.3 : 1 |- ! xxyA | |align=right| 1,728 || 0.0088807 ||align=right| 111.6 : 1 |align=right| 2,160 || 0.0111008 ||align=right| 89.1 : 1 |- ! xyAA | |align=right| 1,296 || 0.0066605 ||align=right| 149.1 : 1 |align=right| 1,620 || 0.0083256 ||align=right| 119.1 : 1 |- ! xyA* | |align=right| 17,280 || 0.0888067 ||align=right| 10.3 : 1 |align=right| 17,280 || 0.0888067 ||align=right| 10.3 : 1 |- ! xyzA | |align=right| 1,728 || 0.0088807 ||align=right| 111.6 : 1 |align=right| 2,160 || 0.0111008 ||align=right| 89.1 : 1 |- | colspan=2| Total |align=right| 64,464 || 0.3312982 ||align=right| 2.0 : 1 |align=right| 92,340 || 0.4745606 ||align=right| 1.1 : 1 |- ! rowspan=4| Miss low ! xxyz | |align=right| 324 || 0.0016651 ||align=right| 599.6 : 1 |align=right| 324 || 0.0016651 ||align=right| 599.6 : 1 |- ! xyzr | |align=right| 81 || 0.0004163 ||align=right| 2,401.2 : 1 |align=right| 81 || 0.0004163 ||align=right| 2,401.2 : 1 |- ! xyz* | |align=right| 2,160 || 0.0111008 ||align=right| 89.1 : 1 |align=right| 1,728 || 0.0088807 ||align=right| 111.6 : 1 |- | colspan=2| Total |align=right| 2,565 || 0.131822 ||align=right| 74.9 : 1 |align=right| 2,133 || 0.109621 ||align=right| 90.2 : 1 |- ! rowspan=18| No low ! AAAA | |align=right| 4 || 0.0000206 ||align=right| 48,644.0 : 1 |align=right| 5 || 0.0000257 ||align=right| 38,915.0 : 1 |- ! AAAB | |align=right| 192 || 0.0009867 ||align=right| 1,012.4 : 1 |align=right| 320 || 0.0016446 ||align=right| 607.1 : 1 |- ! AABB | |align=right| 216 || 0.0011101 ||align=right| 899.8 : 1 |align=right| 360 || 0.0018501 ||align=right| 539.5 : 1 |- ! AAA* | |align=right| 320 || 0.0016446 ||align=right| 607.1 : 1 |align=right| 320 || 0.0016446 ||align=right| 607.1 : 1 |- ! AAB* | |align=right| 5,760 || 0.0296022 ||align=right| 32.8 : 1 |align=right| 7,680 || 0.0394696 ||align=right| 24.3 : 1 |- ! xxxy | |align=right| 36 || 0.0001850 ||align=right| 5,404.0 : 1 |align=right| 36 || 0.0001850 ||align=right| 5,404.0 : 1 |- ! xxxA | |align=right| 64 || 0.0003289 ||align=right| 3,039.3 : 1 |align=right| 80 || 0.0004111 ||align=right| 2,431.3 : 1 |- ! xxx* | |align=right| 80 || 0.0004111 ||align=right| 2,431.3 : 1 |align=right| 64 || 0.0003289 ||align=right| 3,039.3 : 1 |- ! xxyy | |align=right| 54 || 0.0002775 ||align=right| 3,602.3 : 1 |align=right| 54 || 0.0002775 ||align=right| 3,602.3 : 1 |- ! xxy* | |align=right| 2,160 || 0.0111008 ||align=right| 89.1 : 1 |align=right| 1,728 || 0.0088807 ||align=right| 111.6 : 1 |- ! xxAA | |align=right| 270 || 0.0013876 ||align=right| 719.7 : 1 |align=right| 288 || 0.0014801 ||align=right| 674.6 : 1 |- ! xxA* | |align=right| 3,840 || 0.0197348 ||align=right| 49.7 : 1 |align=right| 3,840 || 0.0197348 ||align=right| 49.7 : 1 |- ! xAAA | |align=right| 192 || 0.0009867 ||align=right| 1,012.4 : 1 |align=right| 240 || 0.0012334 ||align=right| 809.8 : 1 |- ! xAA* | |align=right| 5,760 || 0.0296022 ||align=right| 32.8 : 1 |align=right| 5,760 || 0.0296022 ||align=right| 32.8 : 1 |- ! ??** | |align=right| 71,820 || 0.3691027 ||align=right| 1.7 : 1 |align=right| 59,520 || 0.3058896 ||align=right| 2.3 : 1 |- ! ?*** | |align=right| 31,920 || 0.1640456 ||align=right| 5.1 : 1 |align=right| 17,920 || 0.0920958 ||align=right| 9.9 : 1 |- ! **** | |align=right| 4,845 || 0.0248998 ||align=right| 39.2 : 1 |align=right| 1,820 || 0.0093535 ||align=right| 105.9 : 1 |- | colspan=2| Total |align=right| 127,551 || 0.6555196 ||align=right| 0.5 : 1 |align=right| 100,107 || 0.5144773 ||align=right| 0.9 : 1 |} {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;" |+Derivations for low hand shape xyzr (four low ranks) on the river |- style="Background: #eee;;" ! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) |- style="Background: #eee;;" ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds |- ! rowspan=35| Make low ! ABCDE | (see #1 below) |align=right| 0 || 0.0000000 ||align=right| ∞ : 1 |align=right| 1,024 || 0.0005980 ||align=right| 1,671.2 : 1 |- ! AABCD | |align=right| 1,536 || 0.0008970 ||align=right| 1,113.8 : 1 |align=right| 7,680 || 0.0044852 ||align=right| 222.0 : 1 |- ! ABCD* | |align=right| 5,120 || 0.0029901 ||align=right| 333.4 : 1 |align=right| 20,480 || 0.0119605 ||align=right| 82.6 : 1 |- ! AABBC | |align=right| 1,728 || 0.0010092 ||align=right| 989.9 : 1 |align=right| 4,320 || 0.0025229 ||align=right| 395.4 : 1 |- ! AABC* | |align=right| 23,040 || 0.0134556 ||align=right| 73.3 : 1 |align=right| 46,080 || 0.0269111 ||align=right| 36.2 : 1 |- ! AAABC | |align=right| 768 || 0.0004485 ||align=right| 2,228.6 : 1 |align=right| 1,920 || 0.0011213 ||align=right| 890.8 : 1 |- ! ABC** | |align=right| 48,640 || 0.0284062 ||align=right| 34.2 : 1 |align=right| 76,800 || 0.0448518 ||align=right| 21.3 : 1 |- ! xABCD | |align=right| 3,072 || 0.0017941 ||align=right| 556.4 : 1 |align=right| 23,040 || 0.0134556 ||align=right| 73.3 : 1 |- ! xABC* | |align=right| 61,440 || 0.0358815 ||align=right| 26.9 : 1 |align=right| 122,880 || 0.0717630 ||align=right| 12.9 : 1 |- ! xAABC | |align=right| 13,824 || 0.0080733 ||align=right| 122.9 : 1 |align=right| 34,560 || 0.0201833 ||align=right| 48.5 : 1 |- ! xxABC | |align=right| 3,072 || 0.0017941 ||align=right| 556.4 : 1 |align=right| 7,680 || 0.0044852 ||align=right| 222.0 : 1 |- ! xyABC | |align=right| 13,824 || 0.0080733 ||align=right| 122.9 : 1 |align=right| 34,560 || 0.0201833 ||align=right| 48.5 : 1 |- ! xxxAB | |align=right| 384 || 0.0002243 ||align=right| 4,458.1 : 1 |align=right| 640 || 0.0003738 ||align=right| 2,674.5 : 1 |- ! xxyAB | |align=right| 10,368 || 0.0060550 ||align=right| 164.2 : 1 |align=right| 17,280 || 0.0100917 ||align=right| 98.1 : 1 |- ! xxAAB | |align=right| 3,456 || 0.0020183 ||align=right| 494.5 : 1 |align=right| 5,760 || 0.0033639 ||align=right| 296.3 : 1 |- ! xxAB* | |align=right| 23,040 || 0.0134556 ||align=right| 73.3 : 1 |align=right| 30,720 || 0.0179407 ||align=right| 54.7 : 1 |- ! xyAAB | |align=right| 15,552 || 0.0090825 ||align=right| 109.1 : 1 |align=right| 25,920 || 0.0151375 ||align=right| 65.1 : 1 |- ! xyAB* | |align=right| 103,680 || 0.0605500 ||align=right| 15.5 : 1 |align=right| 138,240 || 0.0807333 ||align=right| 11.4 : 1 |- ! xAAAB | |align=right| 2,304 || 0.0013456 ||align=right| 742.2 : 1 |align=right| 3,840 || 0.0022426 ||align=right| 444.9 : 1 |- ! xAABB | |align=right| 2,592 || 0.0015137 ||align=right| 659.6 : 1 |align=right| 4,320 || 0.0025229 ||align=right| 395.4 : 1 |- ! xAAB* | |align=right| 69,120 || 0.0403667 ||align=right| 23.8 : 1 |align=right| 92,160 || 0.0538222 ||align=right| 17.6 : 1 |- ! xAB** | |align=right| 218,880 || 0.1278278 ||align=right| 6.8 : 1 |align=right| 230,400 || 0.1345555 ||align=right| 6.4 : 1 |- ! xyzAB | |align=right| 10,368 || 0.0060550 ||align=right| 164.2 : 1 |align=right| 17,280 || 0.0100917 ||align=right| 98.1 : 1 |- ! xxxyA | |align=right| 576 || 0.0003364 ||align=right| 2,971.8 : 1 |align=right| 720 || 0.0004205 ||align=right| 2,377.2 : 1 |- ! xxyyA | |align=right| 864 || 0.0005046 ||align=right| 1,980.8 : 1 |align=right| 1,080 || 0.0006307 ||align=right| 1,584.5 : 1 |- ! xxyAA | |align=right| 1,620 || 0.0009461 ||align=right| 1,056.0 : 1 |align=right| 2,592 || 0.0015137 ||align=right| 659.6 : 1 |- ! xxyA* | |align=right| 34,560 || 0.0201833 ||align=right| 48.5 : 1 |align=right| 34,560 || 0.0201833 ||align=right| 48.5 : 1 |- ! xyAAA | |align=right| 864 || 0.0005046 ||align=right| 1,980.8 : 1 |align=right| 1,080 || 0.0006307 ||align=right| 1,584.5 : 1 |- ! xyAA* | |align=right| 25,920 || 0.0151375 ||align=right| 65.1 : 1 |align=right| 25,920 || 0.0151375 ||align=right| 65.1 : 1 |- ! xyA** | |align=right| 164,160 || 0.0958708 ||align=right| 9.4 : 1 |align=right| 129,600 || 0.0756875 ||align=right| 12.2 : 1 |- ! xxyzA | |align=right| 5,184 || 0.0030275 ||align=right| 329.3 : 1 |align=right| 6,480 || 0.0037844 ||align=right| 263.2 : 1 |- ! xyzAA | |align=right| 2,592 || 0.0015137 ||align=right| 659.6 : 1 |align=right| 3,240 || 0.0018922 ||align=right| 527.5 : 1 |- ! xyzA* | |align=right| 34,560 || 0.0201833 ||align=right| 48.5 : 1 |align=right| 34,560 || 0.0201833 ||align=right| 48.5 : 1 |- ! xyzrA | |align=right| 1,296 || 0.0007569 ||align=right| 1,320.2 : 1 |align=right| 1,620 || 0.0009461 ||align=right| 1,056.0 : 1 |- | colspan=2| Total |align=right| 908,976 || 0.5308497 ||align=right| 0.9 : 1 |align=right|1,182,004 || 0.6903003 ||align=right| 0.4 : 1 |- ! rowspan=7| Miss low ! xxxyz | |align=right| 108 || 0.0000631 ||align=right| 15,853.7 : 1 |align=right| 108 || 0.0000631 ||align=right| 15,853.7 : 1 |- ! xxyyz | |align=right| 324 || 0.0001892 ||align=right| 5,283.9 : 1 |align=right| 324 || 0.0001892 ||align=right| 5,283.9 : 1 |- ! xxyz* | |align=right| 6,480 || 0.0037844 ||align=right| 263.2 : 1 |align=right| 5,184 || 0.0030275 ||align=right| 329.3 : 1 |- ! xyz** | |align=right| 20,520 || 0.0119839 ||align=right| 82.4 : 1 |align=right| 12,960 || 0.0075687 ||align=right| 131.1 : 1 |- ! xxyzr | |align=right| 324 || 0.0001892 ||align=right| 5,283.9 : 1 |align=right| 324 || 0.0001892 ||align=right| 5,283.9 : 1 |- ! xyzr* | |align=right| 1,620 || 0.0009461 ||align=right| 1,056.0 : 1 |align=right| 1,296 || 0.0007569 ||align=right| 1,320.2 : 1 |- | colspan=2| Total |align=right| 29,376 || 0.0171558 ||align=right| 57.3 : 1 |align=right| 20,196 || 0.0117946 ||align=right| 83.8 : 1 |- ! rowspan=24| No low ! AAAAB | |align=right| 48 || 0.0000280 ||align=right| 35,672.0 : 1 |align=right| 80 || 0.0000467 ||align=right| 21,402.8 : 1 |- ! AAAA* | |align=right| 80 || 0.0000467 ||align=right| 21,402.8 : 1 |align=right| 80 || 0.0000467 ||align=right| 21,402.8 : 1 |- ! AAABB | |align=right| 288 || 0.0001682 ||align=right| 5,944.5 : 1 |align=right| 480 || 0.0002803 ||align=right| 3,566.3 : 1 |- ! AAAB* | |align=right| 3,840 || 0.0022426 ||align=right| 444.9 : 1 |align=right| 5,120 || 0.0029901 ||align=right| 333.4 : 1 |- ! AABB* | |align=right| 4,320 || 0.0025229 ||align=right| 395.4 : 1 |align=right| 5,760 || 0.0033639 ||align=right| 296.3 : 1 |- ! AAA** | |align=right| 3,040 || 0.0017754 ||align=right| 562.3 : 1 |align=right| 2,400 || 0.0014016 ||align=right| 712.5 : 1 |- ! AAB** | |align=right| 54,720 || 0.0319569 ||align=right| 30.3 : 1 |align=right| 57,600 || 0.0336389 ||align=right| 28.7 : 1 |- ! xxxyy | |align=right| 36 || 0.0000210 ||align=right| 47,563.0 : 1 |align=right| 36 || 0.0000210 ||align=right| 47,563.0 : 1 |- ! xxxy* | |align=right| 720 || 0.0004205 ||align=right| 2,377.2 : 1 |align=right| 576 || 0.0003364 ||align=right| 2,971.8 : 1 |- ! xxxAA | |align=right| 96 || 0.0000561 ||align=right| 17,835.5 : 1 |align=right| 120 || 0.0000701 ||align=right| 14,268.2 : 1 |- ! xxxA* | |align=right| 1,280 || 0.0007475 ||align=right| 1,336.7 : 1 |align=right| 1,280 || 0.0007475 ||align=right| 1,336.7 : 1 |- ! xxx** | |align=right| 760 || 0.0004438 ||align=right| 2,252.0 : 1 |align=right| 480 || 0.0002803 ||align=right| 3,566.3 : 1 |- ! xxyy* | |align=right| 1,080 || 0.0006307 ||align=right| 1,584.5 : 1 |align=right| 864 || 0.0005046 ||align=right| 1,980.8 : 1 |- ! xxy** | |align=right| 20,520 || 0.0119839 ||align=right| 82.4 : 1 |align=right| 12,960 || 0.0075687 ||align=right| 131.1 : 1 |- ! xxAAA | |align=right| 192 || 0.0001121 ||align=right| 8,917.3 : 1 |align=right| 240 || 0.0001402 ||align=right| 7,133.6 : 1 |- ! xxAA* | |align=right| 5,760 || 0.0033639 ||align=right| 296.3 : 1 |align=right| 5,760 || 0.0033639 ||align=right| 296.3 : 1 |- ! xxA** | |align=right| 36,480 || 0.0213046 ||align=right| 45.9 : 1 |align=right| 28,800 || 0.0168194 ||align=right| 58.5 : 1 |- ! xAAAA | |align=right| 48 || 0.0000280 ||align=right| 35,672.0 : 1 |align=right| 60 || 0.0000350 ||align=right| 28,537.4 : 1 |- ! xAAA* | |align=right| 3,840 || 0.0022426 ||align=right| 444.9 : 1 |align=right| 3,840 || 0.0022426 ||align=right| 444.9 : 1 |- ! xAA** | |align=right| 54,720 || 0.0319569 ||align=right| 30.3 : 1 |align=right| 43,200 || 0.0252292 ||align=right| 38.6 : 1 |- ! ??*** | |align=right| 430,920 || 0.2516609 ||align=right| 3.0 : 1 |align=right| 277,760 || 0.1622142 ||align=right| 5.2 : 1 |- ! ?**** | |align=right| 135,660 || 0.0792266 ||align=right| 11.6 : 1 |align=right| 58,240 || 0.0340127 ||align=right| 28.4 : 1 |- ! ***** | |align=right| 15,504 || 0.0090545 ||align=right| 109.4 : 1 |align=right| 4,368 || 0.0025509 ||align=right| 391.0 : 1 |- | colspan=2| Total |align=right| 773,952 || 0.4519945 ||align=right| 1.2 : 1 |align=right| 510,104 || 0.2979050 ||align=right| 2.4 : 1 |} The board ABCDE is not possible with an 8-high qualifier since there are only 8 − 4 = 4 low ranks not represented in the hand xyzr. In this case, by definition the binomial coefficient