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

The probability derivations for starting hands making a qualifying low hand in Omaha hold 'em Hi-Lo are separate for each of the starting hand low shapes. The derivations require identifying the individual cases that complete a low hand; make a low hand possible on the board, but don't make a low hand; and that create a board with fewer than three low ranks, making no low hand possible.

The ranks on the board are indicated using lower case letters for matches with the starting hand and upper case letters to indicate qualifying low ranks that don't match the starting hand. So the low hand shape xxyz is any hand with a low pair of x with two additional low ranks y and z and the board xyA represents a flop that contains one x, one of the non-paired ranks y and one other low rank A. Note that since y and z have an identical relationship to the starting hand—each represents an unpaired rank—xyA and xzA represent the same set of boards and are interchangeable, so derivations for this hand choose one of the two choices represented by y. In addition to the upper and lower case letters, * is used to represent any rank higher than the qualifying low. So for the low hand shape xxy*, the board xx* represents a flop that contains two xs and a high rank (either the same high rank as xxy* or a different high rank).

One common calculation regardless of the hand is that no low is possible when there are two or fewer low cards on the board. If $h$ is the number of high cards in the hand ($h$ = 0, 1, 2, 3, 4) and $r$ is the low qualifier ($r$ = 8, 9), then $4r-4+h$ is the number of low cards remaining in the deck and $52-4r-h$ is the number of high cards remaining in the deck. Using ? to represent any low card, the number of boards with 0, 1 or 2 low cards on the flop, turn and river are calculated as follows:

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

Each table in the following subsections shows all of the boards that can make each hand and the derivation for the combinations for that board. Probabilities are determined by dividing the number of combinations for each hand by the ${\begin{matrix}{48 \choose 3}=17,296\end{matrix}}$ boards on the flop, ${\begin{matrix}{48 \choose 4}=194,580\end{matrix}}$ boards on the turn, and ${\begin{matrix}{48 \choose 5}=1,712,304\end{matrix}}$ boards at the river. The probabilities for the boards in each table total 1.0.

Derivations for making low hands with four high cards

Starting hands with four high cards (****) cannot make a low hand, so the only possibilities are having a low hand possible on the board or not. With four high cards in the hand, there are $52-4r-4=48-4r$ high cards and $4r-4+4=4r$ low cards left in the deck. The following tables show the derivations when holding four high cards.

Derivations for low hand shape **** (four high cards) on the flop

Hand to make	Board	Derivation	8-high qualifier (r = 8)			9-high qualifier (r = 9)
			Combos	Probability	Odds	Combos	Probability	Odds
Miss low	ABC	${\begin{matrix}{r \choose 3}{4 \choose 1}^{3}\end{matrix}}$	3,584	0.2072155	3.8 : 1	5,376	0.3108233	2.2 : 1
No low	AAA	${\begin{matrix}{r \choose 1}{4 \choose 3}\end{matrix}}$	32	0.0018501	539.5 : 1	36	0.0020814	479.4 : 1
	AAB	${\begin{matrix}{r \choose 1}{4 \choose 2}{r-1 \choose 1}{4 \choose 1}\end{matrix}}$	1,344	0.0777058	11.9 : 1	1,728	0.0999075	9.0 : 1
	??*	${\begin{matrix}{4r \choose 2}{48-4r \choose 1}\end{matrix}}$	7,168	0.4144311	1.4 : 1	6,912	0.3996300	1.5 : 1
	?**	${\begin{matrix}{4r \choose 1}{48-4r \choose 2}\end{matrix}}$	7,936	0.4588344	1.2 : 1	7,560	0.4370953	1.3 : 1
	***	${\begin{matrix}{48-4r \choose 3}\end{matrix}}$	560	0.0323774	29.9 : 1	220	0.0127197	77.6 : 1
	Total		13,712	0.7927845	0.3 : 1	11,920	0.6891767	0.5 : 1

Derivations for low hand shape **** (four high cards) on the turn

Hand to make	Board	Derivation	8-high qualifier (r = 8)			9-high qualifier (r = 9)
			Combos	Probability	Odds	Combos	Probability	Odds
Miss low	ABCD	${\begin{matrix}{r \choose 4}{4 \choose 1}^{4}\end{matrix}}$	17,920	0.0920958	9.9 : 1	32,256	0.1657724	5.0 : 1
	AABC	${\begin{matrix}{r \choose 1}{4 \choose 2}{r-1 \choose 2}{4 \choose 1}^{2}\end{matrix}}$	16,128	0.0828862	11.1 : 1	24,192	0.1243293	7.0 : 1
	ABC*	${\begin{matrix}{r \choose 3}{4 \choose 1}^{3}{48-4r \choose 1}\end{matrix}}$	57,344	0.2947065	2.4 : 1	64,512	0.3315449	2.0 : 1
	Total		91,392	0.7927845	1.1 : 1	120,960	0.6216466	0.6 : 1
No low	AAAA	${\begin{matrix}{r \choose 1}{4 \choose 4}\end{matrix}}$	8	0.0000411	24,321.5 : 1	9	0.0000463	21,619.0 : 1
	AAAB	${\begin{matrix}{r \choose 1}{4 \choose 3}{r-1 \choose 1}{4 \choose 1}\end{matrix}}$	896	0.0046048	216.2 : 1	1,152	0.0059204	167.9 : 1
	AABB	${\begin{matrix}{r \choose 2}{4 \choose 2}^{2}\end{matrix}}$	1,008	0.0051804	192.0 : 1	1,296	0.0066605	149.1 : 1
	AAA*	${\begin{matrix}{r \choose 1}{4 \choose 3}{48-4r \choose 1}\end{matrix}}$	512	0.0026313	379.0 : 1	432	0.0022202	449.4 : 1
	AAB*	${\begin{matrix}{r \choose 1}{4 \choose 2}{r-1 \choose 1}{4 \choose 1}{48-4r \choose 1}\end{matrix}}$	21,504	0.1105150	8.0 : 1	20,736	0.1065680	8.4 : 1
	??**	${\begin{matrix}{4r \choose 2}{48-4r \choose 2}\end{matrix}}$	59,520	0.3058896	2.3 : 1	41,580	0.2136910	3.7 : 1
	?***	${\begin{matrix}{4r \choose 1}{48-4r \choose 3}\end{matrix}}$	17,920	0.0920958	9.9 : 1	7,920	0.0407031	23.6 : 1
	****	${\begin{matrix}{48-4r \choose 4}\end{matrix}}$	1,820	0.0093535	105.9 : 1	495	0.0025439	392.1 : 1
	Total		103,188	0.5303114	0.9 : 1	73,620	0.3783534	1.6 : 1

Derivations for low hand shape **** (four high cards) on the river

Hand to make	Board	Derivation	8-high qualifier (r = 8)			9-high qualifier (r = 9)
			Combos	Probability	Odds	Combos	Probability	Odds
Miss low	ABCDE	${\begin{matrix}{r \choose 5}{4 \choose 1}^{5}\end{matrix}}$	57,344	0.0334894	28.9 : 1	129,024	0.0753511	12.3 : 1
	AABCD	${\begin{matrix}{r \choose 1}{4 \choose 2}{r-1 \choose 3}{4 \choose 1}^{3}\end{matrix}}$	107,520	0.0627926	14.9 : 1	193,536	0.1130267	7.8 : 1
	ABCD*	${\begin{matrix}{r \choose 4}{4 \choose 1}^{4}{48-4r \choose 1}\end{matrix}}$	286,720	0.1674469	5.0 : 1	387,072	0.2260533	3.4 : 1
	AABBC	${\begin{matrix}{r \choose 2}{4 \choose 2}^{2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$	24,192	0.0141283	69.8 : 1	36,288	0.0211925	46.2 : 1
	AAABC	${\begin{matrix}{r \choose 1}{4 \choose 3}{r-1 \choose 2}{4 \choose 1}^{2}\end{matrix}}$	10,752	0.0062793	158.3 : 1	16,128	0.0094189	105.2 : 1
	AABC*	${\begin{matrix}{r \choose 1}{4 \choose 2}{r-1 \choose 2}{4 \choose 1}^{2}{48-4r \choose 1}\end{matrix}}$	258,048	0.1507022	5.6 : 1	290,304	0.1695400	4.9 : 1
	ABC**	${\begin{matrix}{r \choose 3}{4 \choose 1}^{3}{48-4r \choose 2}\end{matrix}}$	430,080	0.2511704	3.0 : 1	354,816	0.2072155	3.8 : 1
	Total		1,174,656	0.6860090	0.5 : 1	1,407,168	0.8217980	0.2 : 1
No low	AAAAB	${\begin{matrix}{r \choose 1}{4 \choose 4}{r-1 \choose 1}{4 \choose 1}\end{matrix}}$	224	0.0001308	7,643.2 : 1	288	0.0001682	5,944.5 : 1
	AAAA*	${\begin{matrix}{r \choose 1}{4 \choose 4}{48-4r \choose 1}\end{matrix}}$	128	0.0000748	13,376.4 : 1	108	0.0000631	15,853.7 : 1
	AAABB	${\begin{matrix}{r \choose 1}{4 \choose 3}{r-1 \choose 1}{4 \choose 2}\end{matrix}}$	1,344	0.0007849	1,273.0 : 1	1,728	0.0010092	989.9 : 1
	AAAB*	${\begin{matrix}{r \choose 1}{4 \choose 3}{r-1 \choose 1}{4 \choose 1}{48-4r \choose 1}\end{matrix}}$	14,336	0.0083723	118.4 : 1	13,824	0.0080733	122.9 : 1
	AABB*	${\begin{matrix}{r \choose 2}{4 \choose 2}^{2}{48-4r \choose 1}\end{matrix}}$	16,128	0.0094189	105.2 : 1	15,552	0.0090825	109.1 : 1
	AAA**	${\begin{matrix}{r \choose 1}{4 \choose 3}{48-4r \choose 2}\end{matrix}}$	3,840	0.0022426	444.9 : 1	2,376	0.0013876	719.7 : 1
	AAB**	${\begin{matrix}{r \choose 1}{4 \choose 2}{r-1 \choose 1}{4 \choose 1}{48-4r \choose 2}\end{matrix}}$	161,280	0.0941889	9.6 : 1	114,048	0.0666050	14.0 : 1
	??***	${\begin{matrix}{4r \choose 2}{48-4r \choose 3}\end{matrix}}$	277,760	0.1622142	5.2 : 1	138,600	0.0809436	11.4 : 1
	?****	${\begin{matrix}{4r \choose 1}{48-4r \choose 4}\end{matrix}}$	58,240	0.0340127	28.4 : 1	17,820	0.0104070	95.1 : 1
	*****	${\begin{matrix}{48-4r \choose 5}\end{matrix}}$	4,368	0.0025509	391.0 : 1	792	0.0004625	2,161.0 : 1
	Total		537,648	0.3139910	2.2 : 1	305,136	0.1782020	4.6 : 1

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

Starting hands with one low card and three high cards (x***) cannot make a low hand, so the only possibilities are having a low hand possible on the board or not. With three high cards in the hand, there are $52-4r-3=49-4r$ high cards and $4r-4+3=4r-1$ low cards left in the deck. The following tables show the derivations when holding one low card and three high cards.

Derivations for low hand shape x*** (one low card and three high cards) on the flop

Hand to make	Board	Derivation	8-high qualifier (r = 8)			9-high qualifier (r = 9)
			Combos	Probability	Odds	Combos	Probability	Odds
Miss low	ABC	${\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}\end{matrix}}$	2,240	0.1295097	6.7 : 1	3,584	0.2072155	3.8 : 1
	xAB	${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}\end{matrix}}$	1,008	0.0582794	16.2 : 1	1,344	0.0777058	11.9 : 1
	Total		3,248	0.1877891	4.3 : 1	4,928	0.2849214	2.5 : 1
No low	AAA	${\begin{matrix}{r-1 \choose 1}{4 \choose 3}\end{matrix}}$	28	0.0016189	616.7 : 1	32	0.0018501	539.5 : 1
	AAB	${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$	1,008	0.0582794	16.2 : 1	1,344	0.0777058	11.9 : 1
	xxx	${\begin{matrix}{1 \choose 1}{3 \choose 3}\end{matrix}}$	1	0.0000578	17,295.0 : 1	1	0.0000578	17,295.0 : 1
	xxA	${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 1}\end{matrix}}$	84	0.0048566	204.9 : 1	96	0.0055504	179.2 : 1
	xAA	${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}\end{matrix}}$	126	0.0072849	136.3 : 1	144	0.0083256	119.1 : 1
	??*	${\begin{matrix}{4r-1 \choose 2}{49-4r \choose 1}\end{matrix}}$	7,905	0.4570421	1.2 : 1	7,735	0.4472132	1.2 : 1
	?**	${\begin{matrix}{4r-1 \choose 1}{49-4r \choose 2}\end{matrix}}$	4,216	0.2437558	3.1 : 1	2,730	0.1578400	5.3 : 1
	***	${\begin{matrix}{49-4r \choose 3}\end{matrix}}$	680	0.0393154	24.4 : 1	286	0.0165356	59.5 : 1
	Total		14,048	0.8122109	0.2 : 1	12,368	0.7150786	0.4 : 1

Derivations for low hand shape x*** (one low card and three high cards) on the turn

Hand to make	Board	Derivation	8-high qualifier (r = 8)			9-high qualifier (r = 9)
			Combos	Probability	Odds	Combos	Probability	Odds
Miss low	ABCD	${\begin{matrix}{r-1 \choose 4}{4 \choose 1}^{4}\end{matrix}}$	8,960	0.0460479	20.7 : 1	17,920	0.0920958	9.9 : 1
	AABC	${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$	10,080	0.0518039	18.3 : 1	16,128	0.0828862	11.1 : 1
	ABC*	${\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}{49-4r \choose 1}\end{matrix}}$	38,080	0.1957036	4.1 : 1	46,592	0.2394491	3.2 : 1
	xxAB	${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-1 \choose 2}{4 \choose 1}^{2}\end{matrix}}$	1,008	0.0051804	192.0 : 1	1,344	0.0069072	143.8 : 1
	xABC	${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 3}{4 \choose 1}^{3}\end{matrix}}$	6,720	0.0345359	28.0 : 1	10,752	0.0552575	17.1 : 1
	xAAB	${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$	3,024	0.0155412	63.3 : 1	4,032	0.0207216	47.3 : 1
	xAB*	${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}{49-4r \choose 1}\end{matrix}}$	17,136	0.0880666	10.4 : 1	17,472	0.0897934	10.1 : 1
	Total		85,008	0.4368794	1.3 : 1	114,240	0.5871107	0.7 : 1
No low	AAAA	${\begin{matrix}{r-1 \choose 1}{4 \choose 4}\end{matrix}}$	7	0.0000360	27,796.1 : 1	8	0.0000411	24,321.5 : 1
	AAAB	${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$	672	0.0034536	288.6 : 1	896	0.0046048	216.2 : 1
	AABB	${\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}\end{matrix}}$	756	0.0038853	256.4 : 1	1,008	0.0051804	192.0 : 1
	AAA*	${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{49-4r \choose 1}\end{matrix}}$	476	0.0024463	407.8 : 1	416	0.0021379	466.7 : 1
	AAB*	${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{49-4r \choose 1}\end{matrix}}$	17,136	0.0880666	10.4 : 1	17,472	0.0897934	10.1 : 1
	xxx*	${\begin{matrix}{1 \choose 1}{3 \choose 3}{49-4r \choose 1}\end{matrix}}$	17	0.0000874	11,444.9 : 1	13	0.0000668	14,966.7 : 1
	xxxA	${\begin{matrix}{1 \choose 1}{3 \choose 3}{r-1 \choose 1}{4 \choose 1}\end{matrix}}$	28	0.0001439	6,948.3 : 1	32	0.0001645	6,079.6 : 1
	xxAA	${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 2}\end{matrix}}$	126	0.0006475	1,543.3 : 1	144	0.0007401	1,350.3 : 1
	xxA*	${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 1}{49-4r \choose 1}\end{matrix}}$	1,428	0.0073389	135.3 : 1	1,248	0.0064138	154.9 : 1
	xAAA	${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 3}\end{matrix}}$	84	0.0004317	2,315.4 : 1	96	0.0004934	2,025.9 : 1
	xAA*	${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{49-4r \choose 1}\end{matrix}}$	2,142	0.0110083	89.8 : 1	1,872	0.0096207	102.9 : 1
	??**	${\begin{matrix}{4r-1 \choose 2}{49-4r \choose 2}\end{matrix}}$	63,240	0.3250077	2.1 : 1	46,410	0.2385137	3.2 : 1
	?***	${\begin{matrix}{4r-1 \choose 1}{49-4r \choose 3}\end{matrix}}$	21,080	0.1083359	8.2 : 1	10,010	0.0514441	18.4 : 1
	****	${\begin{matrix}{49-4r \choose 4}\end{matrix}}$	2,380	0.0122315	80.8 : 1	715	0.0036746	271.1 : 1
	Total		109,572	0.5631206	0.8 : 1	80,340	0.4128893	1.4 : 1

Derivations for low hand shape x*** (one low card and three high cards) on the river

Hand to make	Board	Derivation	8-high qualifier (r = 8)			9-high qualifier (r = 9)
			Combos	Probability	Odds	Combos	Probability	Odds
Miss low	ABCDE	${\begin{matrix}{r-1 \choose 5}{4 \choose 1}^{5}\end{matrix}}$	21,504	0.0125585	78.6 : 1	57,344	0.0334894	28.9 : 1
	AABCD	${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}$	53,760	0.0313963	30.9 : 1	107,520	0.0627926	14.9 : 1
	ABCD*	${\begin{matrix}{r-1 \choose 4}{4 \choose 1}^{4}{49-4r \choose 1}\end{matrix}}$	152,320	0.0889562	10.2 : 1	232,960	0.1360506	6.4 : 1
	AABBC	${\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$	15,120	0.0088302	112.2 : 1	24,192	0.0141283	69.8 : 1
	AAABC	${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$	6,720	0.0039245	253.8 : 1	10,752	0.0062793	158.3 : 1
	AABC*	${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}{49-4r \choose 1}\end{matrix}}$	171,360	0.1000757	9.0 : 1	209,664	0.1224455	7.2 : 1
	ABC**	${\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}{49-4r \choose 2}\end{matrix}}$	304,640	0.1779123	4.6 : 1	279,552	0.1632607	5.1 : 1
	xABCD	${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 4}{4 \choose 1}^{4}\end{matrix}}$	26,880	0.0156981	62.7 : 1	53,760	0.0313963	30.9 : 1
	xABC*	${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 3}{4 \choose 1}^{3}{49-4r \choose 1}\end{matrix}}$	114,240	0.0667171	14.0 : 1	139,776	0.0816304	11.3 : 1
	xAABC	${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$	30,240	0.0176604	55.6 : 1	48,384	0.0282567	34.4 : 1
	xAAB*	${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{49-4r \choose 1}\end{matrix}}$	51,408	0.0300227	32.3 : 1	52,416	0.0306114	31.7 : 1
	xAAAB	${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$	2,016	0.0011774	848.4 : 1	2,688	0.0015698	636.0 : 1
	xAABB	${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 2}^{2}\end{matrix}}$	2,268	0.0013245	754.0 : 1	3,024	0.0017660	565.2 : 1
	xAB**	${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}{49-4r \choose 2}\end{matrix}}$	137,088	0.0800605	11.5 : 1	104,832	0.0612228	15.3 : 1
	xxABC	${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-1 \choose 3}{4 \choose 1}^{3}\end{matrix}}$	6,720	0.0039245	253.8 : 1	10,752	0.0062793	158.3 : 1
	xxAB*	${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-1 \choose 2}{4 \choose 1}^{2}{49-4r \choose 1}\end{matrix}}$	17,136	0.0100076	98.9 : 1	17,472	0.0102038	97.0 : 1
	xxAAB	${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$	3,024	0.0017660	565.2 : 1	4,032	0.0023547	423.7 : 1
	xxxAB	${\begin{matrix}{1 \choose 1}{3 \choose 3}{r-1 \choose 2}{4 \choose 1}^{2}\end{matrix}}$	336	0.0001962	5,095.1 : 1	448	0.0002616	3,821.1 : 1
	Total		1,116,780	0.6522090	0.5 : 1	1,359,568	0.7939992	0.3 : 1
No low	AAAAB	${\begin{matrix}{r-1 \choose 1}{4 \choose 4}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$	168	0.0000981	10,191.3 : 1	224	0.0001308	7,643.2 : 1
	AAAA*	${\begin{matrix}{r-1 \choose 1}{4 \choose 4}{49-4r \choose 1}\end{matrix}}$	119	0.0000695	14,388.1 : 1	104	0.0000607	16,463.5 : 1
	AAABB	${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 2}\end{matrix}}$	1,008	0.0005887	1,697.7 : 1	1,344	0.0007849	1,273.0 : 1
	AAAB*	${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}{49-4r \choose 1}\end{matrix}}$	11,424	0.0066717	148.9 : 1	11,648	0.0068025	146.0 : 1
	AABB*	${\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}{49-4r \choose 1}\end{matrix}}$	12,852	0.0075057	132.2 : 1	13,104	0.0076528	129.7 : 1
	AAA**	${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{49-4r \choose 2}\end{matrix}}$	3,808	0.0022239	448.7 : 1	2,496	0.0014577	685.0 : 1
	AAB**	${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{49-4r \choose 2}\end{matrix}}$	137,088	0.0800605	11.5 : 1	104,832	0.0612228	15.3 : 1
	xxxAA	${\begin{matrix}{1 \choose 1}{3 \choose 3}{r-1 \choose 1}{4 \choose 2}\end{matrix}}$	42	0.0000245	40,768.1 : 1	48	0.0000280	35,672.0 : 1
	xxxA*	${\begin{matrix}{1 \choose 1}{3 \choose 3}{r-1 \choose 1}{4 \choose 1}{49-4r \choose 1}\end{matrix}}$	476	0.0002780	3,596.3 : 1	416	0.0002429	4,115.1 : 1
	xxx**	${\begin{matrix}{1 \choose 1}{3 \choose 3}{49-4r \choose 2}\end{matrix}}$	136	0.0000794	12,589.5 : 1	78	0.0000456	21,951.6 : 1
	xxAAA	${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 3}\end{matrix}}$	84	0.0000491	20,383.6 : 1	96	0.0000561	17,835.5 : 1
	xxAA*	${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 2}{49-4r \choose 1}\end{matrix}}$	2,142	0.0012509	798.4 : 1	1,872	0.0010933	913.7 : 1
	xxA**	${\begin{matrix}{1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 1}{49-4r \choose 2}\end{matrix}}$	11,424	0.0066717	148.9 : 1	7,488	0.0043731	227.7 : 1
	xAAAA	${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 4}\end{matrix}}$	21	0.0000123	81,537.3 : 1	24	0.0000140	71,345.0 : 1
	xAAA*	${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 3}{49-4r \choose 1}\end{matrix}}$	1,428	0.0008340	1,198.1 : 1	1,248	0.0007288	1,371.0 : 1
	xAA**	${\begin{matrix}{1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{49-4r \choose 2}\end{matrix}}$	17,136	0.0100076	98.9 : 1	11,232	0.0065596	151.4 : 1
	??***	${\begin{matrix}{4r-1 \choose 2}{49-4r \choose 3}\end{matrix}}$	316,200	0.1846635	4.4 : 1	170,170	0.0993807	9.1 : 1
	?****	${\begin{matrix}{4r-1 \choose 1}{49-4r \choose 4}\end{matrix}}$	73,780	0.0430881	22.2 : 1	25,025	0.0146148	67.4 : 1
	*****	${\begin{matrix}{49-4r \choose 5}\end{matrix}}$	6,188	0.0036138	275.7 : 1	1,287	0.0007516	1,329.5 : 1
	Total		595,524	0.3477910	1.9 : 1	352,736	0.2060008	3.9 : 1

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

Starting hands with a low pair and two high cards (xx**) cannot make a low hand, so the only possibilities are having a low hand possible on the board or not. With two high cards in the hand, there are $52-4r-2=50-4r$ high cards and $4r-4+2=4r-2$ low cards left in the deck. The following tables show the derivations when holding a low pair and two high cards.

Derivations for low hand shape xx** (a low pair and two high cards) on the flop

Hand to make	Board	Derivation	8-high qualifier (r = 8)			9-high qualifier (r = 9)
			Combos	Probability	Odds	Combos	Probability	Odds
Miss low	ABC	${\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}\end{matrix}}$	2,240	0.1295097	6.7 : 1	3,584	0.2072155	3.8 : 1
	xAB	${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}\end{matrix}}$	672	0.0388529	24.7 : 1	896	0.0518039	18.3 : 1
	Total		2,912	0.1683626	4.9 : 1	4,480	0.2590194	2.9 : 1
No low	AAA	${\begin{matrix}{r-1 \choose 1}{4 \choose 3}\end{matrix}}$	28	0.0016189	616.7 : 1	32	0.0018501	539.5 : 1
	AAB	${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$	1,008	0.0582794	16.2 : 1	1,344	0.0777058	11.9 : 1
	xxA	${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 1}\end{matrix}}$	28	0.0016189	616.7 : 1	32	0.0018501	539.5 : 1
	xAA	${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}\end{matrix}}$	84	0.0048566	204.9 : 1	96	0.0055504	179.2 : 1
	??*	${\begin{matrix}{4r-2 \choose 2}{50-4r \choose 1}\end{matrix}}$	7,830	0.4527058	1.2 : 1	7,854	0.4540934	1.2 : 1
	?**	${\begin{matrix}{4r-2 \choose 1}{50-4r \choose 2}\end{matrix}}$	4,590	0.2653793	2.8 : 1	3,094	0.1788853	4.6 : 1
	***	${\begin{matrix}{50-4r \choose 3}\end{matrix}}$	816	0.0471785	20.2 : 1	364	0.0210453	46.5 : 1
	Total		14,384	0.8316374	0.2 : 1	12,816	0.7409806	0.3 : 1

Derivations for low hand shape xx** (a low pair and two high cards) on the turn

Hand to make	Board	Derivation	8-high qualifier (r = 8)			9-high qualifier (r = 9)
			Combos	Probability	Odds	Combos	Probability	Odds
Miss low	ABCD	${\begin{matrix}{r-1 \choose 4}{4 \choose 1}^{4}\end{matrix}}$	8,960	0.0460479	20.7 : 1	17,920	0.0920958	9.9 : 1
	AABC	${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$	10,080	0.0518039	18.3 : 1	16,128	0.0828862	11.1 : 1
	ABC*	${\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}{50-4r \choose 1}\end{matrix}}$	40,320	0.2072155	3.8 : 1	50,176	0.2578682	2.9 : 1
	xxAB	${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-1 \choose 2}{4 \choose 1}^{2}\end{matrix}}$	336	0.0017268	578.1 : 1	448	0.0023024	433.3 : 1
	xABC	${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 3}{4 \choose 1}^{3}\end{matrix}}$	4,480	0.0230239	42.4 : 1	7,168	0.0368383	26.1 : 1
	xAAB	${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$	2,016	0.0103608	95.5 : 1	2,688	0.0138144	71.4 : 1
	xAB*	${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}{50-4r \choose 1}\end{matrix}}$	12,096	0.0621647	15.1 : 1	12,544	0.0644671	14.5 : 1
	Total		78,288	0.4023435	1.5 : 1	107,072	0.5502724	0.8 : 1
No low	AAAA	${\begin{matrix}{r-1 \choose 1}{4 \choose 4}\end{matrix}}$	7	0.0000360	27,796.1 : 1	8	0.0000411	24,321.5 : 1
	AAAB	${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$	672	0.0034536	288.6 : 1	896	0.0046048	216.2 : 1
	AABB	${\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}\end{matrix}}$	756	0.0038853	256.4 : 1	1,008	0.0051804	192.0 : 1
	AAA*	${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{50-4r \choose 1}\end{matrix}}$	504	0.0025902	385.1 : 1	448	0.0023024	433.3 : 1
	AAB*	${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1}\end{matrix}}$	18,144	0.0932470	9.7 : 1	18,816	0.0967006	9.3 : 1
	xxAA	${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 2}\end{matrix}}$	42	0.0002158	4,631.9 : 1	48	0.0002467	4,052.8 : 1
	xxA*	${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 1}{50-4r \choose 1}\end{matrix}}$	504	0.0025902	385.1 : 1	448	0.0023024	433.3 : 1
	xAAA	${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 3}\end{matrix}}$	56	0.0002878	3,473.6 : 1	64	0.0003289	3,039.3 : 1
	xAA*	${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}{50-4r \choose 1}\end{matrix}}$	1,512	0.0077706	127.7 : 1	1,344	0.0069072	143.8 : 1
	??**	${\begin{matrix}{4r-2 \choose 2}{50-4r \choose 2}\end{matrix}}$	66,555	0.3420444	1.9 : 1	51,051	0.2623651	2.8 : 1
	?***	${\begin{matrix}{4r-2 \choose 1}{50-4r \choose 3}\end{matrix}}$	24,480	0.1258094	6.9 : 1	12,376	0.0636037	14.7 : 1
	****	${\begin{matrix}{50-4r \choose 4}\end{matrix}}$	3,060	0.0157262	62.6 : 1	1,001	0.0051444	193.4 : 1
	Total		116,292	0.5976565	0.7 : 1	87,508	0.4497276	1.2 : 1

Derivations for low hand shape xx** (a low pair and two high cards) on the river

Hand to make	Board	Derivation	8-high qualifier (r = 8)			9-high qualifier (r = 9)
			Combos	Probability	Odds	Combos	Probability	Odds
Miss low	ABCDE	${\begin{matrix}{r-1 \choose 5}{4 \choose 1}^{5}\end{matrix}}$	21,504	0.0125585	78.6 : 1	57,344	0.0334894	28.9 : 1
	AABCD	${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}$	53,760	0.0313963	30.9 : 1	107,520	0.0627926	14.9 : 1
	ABCD*	${\begin{matrix}{r-1 \choose 4}{4 \choose 1}^{4}{50-4r \choose 1}\end{matrix}}$	161,280	0.0941889	9.6 : 1	250,880	0.1465160	5.8 : 1
	AABBC	${\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$	15,120	0.0088302	112.2 : 1	24,192	0.0141283	69.8 : 1
	AAABC	${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$	6,720	0.0039245	253.8 : 1	10,752	0.0062793	158.3 : 1
	AABC*	${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}{50-4r \choose 1}\end{matrix}}$	181,440	0.1059625	8.4 : 1	225,792	0.1318644	6.6 : 1
	ABC**	${\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}{50-4r \choose 2}\end{matrix}}$	342,720	0.2001514	4.0 : 1	326,144	0.1904709	4.3 : 1
	xABCD	${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 4}{4 \choose 1}^{4}\end{matrix}}$	17,920	0.0104654	94.6 : 1	35,840	0.0209309	46.8 : 1
	xABC*	${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 3}{4 \choose 1}^{3}{50-4r \choose 1}\end{matrix}}$	80,640	0.0470944	20.2 : 1	100,352	0.0586064	16.1 : 1
	xAABC	${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$	20,160	0.0117736	83.9 : 1	32,256	0.0188378	52.1 : 1
	xAAB*	${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1}\end{matrix}}$	36,288	0.0211925	46.2 : 1	37,632	0.0219774	44.5 : 1
	xAAAB	${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$	1,344	0.0007849	1,273.0 : 1	1,792	0.0010465	954.5 : 1
	xAABB	${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 2}^{2}\end{matrix}}$	1,512	0.0008830	1,131.5 : 1	2,016	0.0011774	848.4 : 1
	xAB**	${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}{50-4r \choose 2}\end{matrix}}$	102,816	0.0600454	15.7 : 1	81,536	0.0476177	20.0 : 1
	xxABC	${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-1 \choose 3}{4 \choose 1}^{3}\end{matrix}}$	2,240	0.0013082	763.4 : 1	3,584	0.0020931	476.8 : 1
	xxAB*	${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-1 \choose 2}{4 \choose 1}^{2}{50-4r \choose 1}\end{matrix}}$	6,048	0.0035321	282.1 : 1	6,272	0.0036629	272.0 : 1
	xxAAB	${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$	1,008	0.0005887	1,697.7 : 1	1,344	0.0007849	1,273.0 : 1
	Total		1,052,520	0.6146806	0.6 : 1	1,305,248	0.7622759	0.3 : 1
No low	AAAAB	${\begin{matrix}{r-1 \choose 1}{4 \choose 4}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$	168	0.0000981	10,191.3 : 1	224	0.0001308	7,643.2 : 1
	AAAA*	${\begin{matrix}{r-1 \choose 1}{4 \choose 4}{50-4r \choose 1}\end{matrix}}$	126	0.0000736	13,588.7 : 1	112	0.0000654	15,287.4 : 1
	AAABB	${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 2}\end{matrix}}$	1,008	0.0005887	1,697.7 : 1	1,344	0.0007849	1,273.0 : 1
	AAAB*	${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1}\end{matrix}}$	12,096	0.0070642	140.6 : 1	12,544	0.0073258	135.5 : 1
	AABB*	${\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}{50-4r \choose 1}\end{matrix}}$	13,608	0.0079472	124.8 : 1	14,112	0.0082415	120.3 : 1
	AAA**	${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{50-4r \choose 2}\end{matrix}}$	4,284	0.0025019	398.7 : 1	2,912	0.0017006	587.0 : 1
	AAB**	${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 2}\end{matrix}}$	154,224	0.0900681	10.1 : 1	122,304	0.0714266	13.0 : 1
	xxAAA	${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 3}\end{matrix}}$	28	0.0000164	61,152.7 : 1	32	0.0000187	53,508.5 : 1
	xxAA*	${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 2}{50-4r \choose 1}\end{matrix}}$	756	0.0004415	2,264.0 : 1	672	0.0003925	2,547.1 : 1
	xxA**	${\begin{matrix}{1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 1}{50-4r \choose 2}\end{matrix}}$	4,284	0.0025019	398.7 : 1	2,912	0.0017006	587.0 : 1
	xAAAA	${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 4}\end{matrix}}$	14	0.0000082	122,306.4 : 1	16	0.0000093	107,018.0 : 1
	xAAA*	${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 3}{50-4r \choose 1}\end{matrix}}$	1,008	0.0005887	1,697.7 : 1	896	0.0005233	1,910.1 : 1
	xAA**	${\begin{matrix}{1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}{50-4r \choose 2}\end{matrix}}$	12,852	0.0075057	132.2 : 1	8,736	0.0051019	195.0 : 1
	??***	${\begin{matrix}{4r-2 \choose 2}{50-4r \choose 3}\end{matrix}}$	354,960	0.2072996	3.8 : 1	204,204	0.1192569	7.4 : 1
	?****	${\begin{matrix}{4r-2 \choose 1}{50-4r \choose 4}\end{matrix}}$	91,800	0.0536120	17.7 : 1	34,034	0.0198761	49.3 : 1
	*****	${\begin{matrix}{50-4r \choose 5}\end{matrix}}$	8,568	0.0050038	198.8 : 1	2,002	0.0011692	854.3 : 1
	Total		659,784	0.3853194	1.6 : 1	407,056	0.2377241	3.2 : 1

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

Starting hands with a low three of a kind and a high card (xxx*) cannot make a low hand, so the only possibilities are having a low hand possible on the board or not. With one high card in the hand, there are $52-4r-1=51-4r$ high cards and $4r-4+1=4r-3$ low cards left in the deck. The following tables show the derivations when holding a low three of a kind and a high card.

Derivations for low hand shape xxx* (a low three of a kind and a high card) on the flop

Hand to make	Board	Derivation	8-high qualifier (r = 8)			9-high qualifier (r = 9)
			Combos	Probability	Odds	Combos	Probability	Odds
Miss low	ABC	${\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}\end{matrix}}$	2,240	0.1295097	6.7 : 1	3,584	0.2072155	3.8 : 1
	xAB	${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}\end{matrix}}$	336	0.0194265	50.5 : 1	448	0.0259019	37.6 : 1
	Total		2,576	0.1489362	5.7 : 1	4,032	0.2331175	3.3 : 1
No low	AAA	${\begin{matrix}{r-1 \choose 1}{4 \choose 3}\end{matrix}}$	28	0.0016189	616.7 : 1	32	0.0018501	539.5 : 1
	AAB	${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$	1,008	0.0582794	16.2 : 1	1,344	0.0777058	11.9 : 1
	xAA	${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 2}\end{matrix}}$	42	0.0024283	410.8 : 1	48	0.0027752	359.3 : 1
	??*	${\begin{matrix}{4r-3 \choose 2}{51-4r \choose 1}\end{matrix}}$	7,714	0.4459991	1.2 : 1	7,920	0.4579093	1.2 : 1
	?**	${\begin{matrix}{4r-3 \choose 1}{51-4r \choose 2}\end{matrix}}$	4,959	0.2867137	2.5 : 1	3,465	0.2003353	4.0 : 1
	***	${\begin{matrix}{51-4r \choose 3}\end{matrix}}$	969	0.0560245	16.8 : 1	455	0.0263067	37.0 : 1
	Total		14,720	0.8510638	0.2 : 1	13,264	0.7668825	0.3 : 1

Derivations for low hand shape xxx* (a low three of a kind and a high card) on the turn

Hand to make	Board	Derivation	8-high qualifier (r = 8)			9-high qualifier (r = 9)
			Combos	Probability	Odds	Combos	Probability	Odds
Miss low	ABCD	${\begin{matrix}{r-1 \choose 4}{4 \choose 1}^{4}\end{matrix}}$	8,960	0.0460479	20.7 : 1	17,920	0.0920958	9.9 : 1
	AABC	${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$	10,080	0.0518039	18.3 : 1	16,128	0.0828862	11.1 : 1
	ABC*	${\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}{51-4r \choose 1}\end{matrix}}$	42,560	0.2187275	3.6 : 1	53,760	0.2762874	2.6 : 1
	xABC	${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 3}{4 \choose 1}^{3}\end{matrix}}$	2,240	0.0115120	85.9 : 1	3,584	0.0184192	53.3 : 1
	xAAB	${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$	1,008	0.0051804	192.0 : 1	1,344	0.0069072	143.8 : 1
	xAB*	${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}{51-4r \choose 1}\end{matrix}}$	6,384	0.0328091	29.5 : 1	6,720	0.0345359	28.0 : 1
	Total		71,232	0.3660808	1.7 : 1	99,456	0.5111317	1.0 : 1
No low	AAAA	${\begin{matrix}{r-1 \choose 1}{4 \choose 4}\end{matrix}}$	7	0.0000360	27,796.1 : 1	8	0.0000411	24,321.5 : 1
	AAAB	${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$	672	0.0034536	288.6 : 1	896	0.0046048	216.2 : 1
	AABB	${\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}\end{matrix}}$	756	0.0038853	256.4 : 1	1,008	0.0051804	192.0 : 1
	AAA*	${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{51-4r \choose 1}\end{matrix}}$	532	0.0027341	364.8 : 1	480	0.0024669	404.4 : 1
	AAB*	${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 1}\end{matrix}}$	19,152	0.0984274	9.2 : 1	20,160	0.1036078	8.7 : 1
	xAAA	${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 3}\end{matrix}}$	28	0.0001439	6,948.3 : 1	32	0.0001645	6,079.6 : 1
	xAA*	${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 2}{51-4r \choose 1}\end{matrix}}$	798	0.0041011	242.8 : 1	720	0.0037003	269.3 : 1
	??**	${\begin{matrix}{4r-3 \choose 2}{51-4r \choose 2}\end{matrix}}$	69,426	0.3567993	1.8 : 1	55,440	0.2849214	2.5 : 1
	?***	${\begin{matrix}{4r-3 \choose 1}{51-4r \choose 3}\end{matrix}}$	28,101	0.1444187	5.9 : 1	15,015	0.0771662	12.0 : 1
	****	${\begin{matrix}{51-4r \choose 4}\end{matrix}}$	3,876	0.0199198	49.2 : 1	1,365	0.0070151	141.5 : 1
	Total		123,348	0.6339192	0.6 : 1	95,124	0.4888683	1.0 : 1

Derivations for low hand shape xxx* (a low three of a kind and a high card) on the river

Hand to make	Board	Derivation	8-high qualifier (r = 8)			9-high qualifier (r = 9)
			Combos	Probability	Odds	Combos	Probability	Odds
Miss low	ABCDE	${\begin{matrix}{r-1 \choose 5}{4 \choose 1}^{5}\end{matrix}}$	21,504	0.0125585	78.6 : 1	57,344	0.0334894	28.9 : 1
	AABCD	${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}$	53,760	0.0313963	30.9 : 1	107,520	0.0627926	14.9 : 1
	ABCD*	${\begin{matrix}{r-1 \choose 4}{4 \choose 1}^{4}{51-4r \choose 1}\end{matrix}}$	170,240	0.0994216	9.1 : 1	268,800	0.1569815	5.4 : 1
	AABBC	${\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$	15,120	0.0088302	112.2 : 1	24,192	0.0141283	69.8 : 1
	AAABC	${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$	6,720	0.0039245	253.8 : 1	10,752	0.0062793	158.3 : 1
	AABC*	${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}{51-4r \choose 1}\end{matrix}}$	191,520	0.1118493	7.9 : 1	241,920	0.1412833	6.1 : 1
	ABC**	${\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}{51-4r \choose 2}\end{matrix}}$	383,040	0.2236986	3.5 : 1	376,320	0.2197741	3.6 : 1
	xABCD	${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 4}{4 \choose 1}^{4}\end{matrix}}$	8,960	0.0052327	190.1 : 1	17,920	0.0104654	94.6 : 1
	xABC*	${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 3}{4 \choose 1}^{3}{51-4r \choose 1}\end{matrix}}$	42,560	0.0248554	39.2 : 1	53,760	0.0313963	30.9 : 1
	xAABC	${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$	10,080	0.0058868	168.9 : 1	16,128	0.0094189	105.2 : 1
	xAAB*	${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 1}\end{matrix}}$	19,152	0.0111849	88.4 : 1	20,160	0.0117736	83.9 : 1
	xAAAB	${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$	672	0.0003925	2,547.1 : 1	896	0.0005233	1,910.1 : 1
	xAABB	${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 2}{4 \choose 2}^{2}\end{matrix}}$	756	0.0004415	2,264.0 : 1	1,008	0.0005887	1,697.7 : 1
	xAB**	${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 2}{4 \choose 1}^{2}{51-4r \choose 2}\end{matrix}}$	57,456	0.0335548	28.8 : 1	47,040	0.0274718	35.4 : 1
	Total		981,540	0.5732277	0.7 : 1	1,243,760	0.7263663	0.4 : 1
No low	AAAAB	${\begin{matrix}{r-1 \choose 1}{4 \choose 4}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$	168	0.0000981	10,191.3 : 1	224	0.0001308	7,643.2 : 1
	AAAA*	${\begin{matrix}{r-1 \choose 1}{4 \choose 4}{51-4r \choose 1}\end{matrix}}$	133	0.0000777	12,873.5 : 1	120	0.0000701	14,268.2 : 1
	AAABB	${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 2}\end{matrix}}$	1,008	0.0005887	1,697.7 : 1	1,344	0.0007849	1,273.0 : 1
	AAAB*	${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 1}\end{matrix}}$	12,768	0.0074566	133.1 : 1	13,440	0.0078491	126.4 : 1
	AABB*	${\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}{51-4r \choose 1}\end{matrix}}$	14,364	0.0083887	118.2 : 1	15,120	0.0088302	112.2 : 1
	AAA**	${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{51-4r \choose 2}\end{matrix}}$	4,788	0.0027962	356.6 : 1	3,360	0.0019623	508.6 : 1
	AAB**	${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 2}\end{matrix}}$	172,368	0.1006644	8.9 : 1	141,120	0.0824153	11.1 : 1
	xAAAA	${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 4}\end{matrix}}$	7	0.0000041	244,613.9 : 1	8	0.0000047	214,037.0 : 1
	xAAA*	${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 3}{51-4r \choose 1}\end{matrix}}$	532	0.0003107	3,217.6 : 1	480	0.0002803	3,566.3 : 1
	xAA**	${\begin{matrix}{1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 2}{51-4r \choose 2}\end{matrix}}$	7,182	0.0041943	237.4 : 1	5,040	0.0029434	338.7 : 1
	??***	${\begin{matrix}{4r-3 \choose 2}{51-4r \choose 3}\end{matrix}}$	393,414	0.2297571	3.4 : 1	240,240	0.1403022	6.1 : 1
	?****	${\begin{matrix}{4r-3 \choose 1}{51-4r \choose 4}\end{matrix}}$	112,404	0.0656449	14.2 : 1	45,045	0.0263067	37.0 : 1
	*****	${\begin{matrix}{51-4r \choose 5}\end{matrix}}$	11,628	0.0067909	146.3 : 1	3,003	0.0017538	569.2 : 1
	Total		730,764	0.4267723	1.3 : 1	468,544	0.2736337	2.7 : 1

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

Starting hands with a low four of a kind (xxxx) cannot make a low hand, so the only possibilities are having a low hand possible on the board or not. With no high cards in the hand, there are $52-4r-0=52-4r$ high cards and $4r-4+0=4r-4$ low cards left in the deck. The following tables show the derivations when holding a low four of a kind.

Derivations for low hand shape xxxx (a low four of a kind) on the flop

Hand to make	Board	Derivation	8-high qualifier (r = 8)			9-high qualifier (r = 9)
			Combos	Probability	Odds	Combos	Probability	Odds
Miss low	ABC	${\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}\end{matrix}}$	2,240	0.1295097	6.7 : 1	3,584	0.2072155	3.8 : 1
No low	AAA	${\begin{matrix}{r-1 \choose 1}{4 \choose 3}\end{matrix}}$	28	0.0016189	616.7 : 1	32	0.0018501	539.5 : 1
	AAB	${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$	1,008	0.0582794	16.2 : 1	1,344	0.0777058	11.9 : 1
	??*	${\begin{matrix}{4r-4 \choose 2}{52-4r \choose 1}\end{matrix}}$	7,560	0.4370953	1.3 : 1	7,936	0.4588344	1.2 : 1
	?**	${\begin{matrix}{4r-4 \choose 1}{52-4r \choose 2}\end{matrix}}$	5,320	0.3075856	2.3 : 1	3,840	0.2220167	3.5 : 1
	***	${\begin{matrix}{52-4r \choose 3}\end{matrix}}$	1,140	0.0659112	14.2 : 1	560	0.0323774	29.9 : 1
	Total		15,056	0.8704903	0.1 : 1	13,712	0.7927845	0.3 : 1

Derivations for low hand shape xxxx (a low four of a kind) on the turn

Hand to make	Board	Derivation	8-high qualifier (r = 8)			9-high qualifier (r = 9)
			Combos	Probability	Odds	Combos	Probability	Odds
Miss low	ABCD	${\begin{matrix}{r-1 \choose 4}{4 \choose 1}^{4}\end{matrix}}$	8,960	0.0460479	20.7 : 1	17,920	0.0920958	9.9 : 1
	AABC	${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$	10,080	0.0518039	18.3 : 1	16,128	0.0828862	11.1 : 1
	ABC*	${\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}{52-4r \choose 1}\end{matrix}}$	44,800	0.2302395	3.3 : 1	57,344	0.2947065	2.4 : 1
	Total		63,840	0.3280913	2.0 : 1	91,392	0.4696886	1.1 : 1
No low	AAAA	${\begin{matrix}{r-1 \choose 1}{4 \choose 4}\end{matrix}}$	7	0.0000360	27,796.1 : 1	8	0.0000411	24,321.5 : 1
	AAAB	${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$	672	0.0034536	288.6 : 1	896	0.0046048	216.2 : 1
	AABB	${\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}\end{matrix}}$	756	0.0038853	256.4 : 1	1,008	0.0051804	192.0 : 1
	AAA*	${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{52-4r \choose 1}\end{matrix}}$	560	0.0028780	346.5 : 1	512	0.0026313	379.0 : 1
	AAB*	${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$	20,160	0.1036078	8.7 : 1	21,504	0.1105150	8.0 : 1
	??**	${\begin{matrix}{4r-4 \choose 2}{52-4r \choose 2}\end{matrix}}$	71,820	0.3691027	1.7 : 1	59,520	0.3058896	2.3 : 1
	?***	${\begin{matrix}{4r-4 \choose 1}{52-4r \choose 3}\end{matrix}}$	31,920	0.1640456	5.1 : 1	17,920	0.0920958	9.9 : 1
	****	${\begin{matrix}{52-4r \choose 4}\end{matrix}}$	4,845	0.0248998	39.2 : 1	1,820	0.0093535	105.9 : 1
	Total		130,740	0.6719087	0.5 : 1	103,188	0.5303114	0.9 : 1

Derivations for low hand shape xxxx (a low four of a kind) on the river

Hand to make	Board	Derivation	8-high qualifier (r = 8)			9-high qualifier (r = 9)
			Combos	Probability	Odds	Combos	Probability	Odds
Miss low	ABCDE	${\begin{matrix}{r-1 \choose 5}{4 \choose 1}^{5}\end{matrix}}$	21,504	0.0125585	78.6 : 1	57,344	0.0334894	28.9 : 1
	AABCD	${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}$	53,760	0.0313963	30.9 : 1	107,520	0.0627926	14.9 : 1
	ABCD*	${\begin{matrix}{r-1 \choose 4}{4 \choose 1}^{4}{52-4r \choose 1}\end{matrix}}$	179,200	0.1046543	8.6 : 1	286,720	0.1674469	5.0 : 1
	AABBC	${\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$	15,120	0.0088302	112.2 : 1	24,192	0.0141283	69.8 : 1
	AAABC	${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$	6,720	0.0039245	253.8 : 1	10,752	0.0062793	158.3 : 1
	AABC*	${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}{52-4r \choose 1}\end{matrix}}$	201,600	0.1177361	7.5 : 1	258,048	0.1507022	5.6 : 1
	ABC**	${\begin{matrix}{r-1 \choose 3}{4 \choose 1}^{3}{52-4r \choose 2}\end{matrix}}$	425,600	0.2485540	3.0 : 1	430,080	0.2511704	3.0 : 1
	Total		903,504	0.5276540	0.9 : 1	1,174,656	0.6860090	0.5 : 1
No low	AAAAB	${\begin{matrix}{r-1 \choose 1}{4 \choose 4}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$	168	0.0000981	10,191.3 : 1	224	0.0001308	7,643.2 : 1
	AAAA*	${\begin{matrix}{r-1 \choose 1}{4 \choose 4}{52-4r \choose 1}\end{matrix}}$	140	0.0000818	12,229.7 : 1	128	0.0000748	13,376.4 : 1
	AAABB	${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 2}\end{matrix}}$	1,008	0.0005887	1,697.7 : 1	1,344	0.0007849	1,273.0 : 1
	AAAB*	${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1}\end{matrix}}$	13,440	0.0078491	126.4 : 1	14,336	0.0083723	118.4 : 1
	AABB*	${\begin{matrix}{r-1 \choose 2}{4 \choose 2}^{2}{52-4r \choose 1}\end{matrix}}$	15,120	0.0088302	112.2 : 1	16,128	0.0094189	105.2 : 1
	AAA**	${\begin{matrix}{r-1 \choose 1}{4 \choose 3}{52-4r \choose 2}\end{matrix}}$	5,320	0.0031069	320.9 : 1	3,840	0.0022426	444.9 : 1
	AAB**	${\begin{matrix}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 2}\end{matrix}}$	191,520	0.1118493	7.9 : 1	161,280	0.0941889	9.6 : 1
	??***	${\begin{matrix}{4r-4 \choose 2}{52-4r \choose 3}\end{matrix}}$	430,920	0.2516609	3.0 : 1	277,760	0.1622142	5.2 : 1
	?****	${\begin{matrix}{4r-4 \choose 1}{52-4r \choose 4}\end{matrix}}$	135,660	0.0792266	11.6 : 1	58,240	0.0340127	28.4 : 1
	*****	${\begin{matrix}{52-4r \choose 5}\end{matrix}}$	15,504	0.0090545	109.4 : 1	4,368	0.0025509	391.0 : 1
	Total		808,800	0.4723460	1.1 : 1	537,648	0.3139910	2.2 : 1

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

Starting hands with two low ranks and two high cards (xy**) can make a low hand when at least three other low ranks beside x and y appear on the board. With two high cards in the hand, there are $52-4r-2=50-4r$ high cards and $4r-4+2=4r-2$ low cards left in the deck. The following tables show the derivations when holding two low ranks and two high cards.

Derivations for low hand shape xy** (two low ranks and two high cards) on the flop

Hand to make	Board	Derivation	8-high qualifier (r = 8)			9-high qualifier (r = 9)
			Combos	Probability	Odds	Combos	Probability	Odds
Make low	ABC	${\begin{matrix}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}$	1,280	0.0740056	12.5 : 1	2,240	0.1295097	6.7 : 1
Miss low	xAB	${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$	1,440	0.0832562	11.0 : 1	2,016	0.1165587	7.6 : 1
	xyA	${\begin{matrix}{2 \choose 2}{3 \choose 1}^{2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$	216	0.0124884	79.1 : 1	252	0.0145698	67.6 : 1
	Total		1,656	0.0957447	9.4 : 1	2,268	0.1311286	6.6 : 1
No low	AAA	${\begin{matrix}{r-2 \choose 1}{4 \choose 3}\end{matrix}}$	24	0.0013876	719.7 : 1	28	0.0016189	616.7 : 1
	AAB	${\begin{matrix}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}\end{matrix}}$	720	0.0416281	23.0 : 1	1,008	0.0582794	16.2 : 1
	xxx	${\begin{matrix}{2 \choose 1}{3 \choose 3}\end{matrix}}$	2	0.0001156	8,647.0 : 1	2	0.0001156	8,647.0 : 1
	xxy	${\begin{matrix}{2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}\end{matrix}}$	18	0.0010407	959.9 : 1	18	0.0010407	959.9 : 1
	xxA	${\begin{matrix}{2 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$	144	0.0083256	119.1 : 1	168	0.0097132	102.0 : 1
	xAA	${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}\end{matrix}}$	216	0.0124884	79.1 : 1	252	0.0145698	67.6 : 1
	??*	${\begin{matrix}{4r-2 \choose 2}{50-4r \choose 1}\end{matrix}}$	7,830	0.4527058	1.2 : 1	7,854	0.4540934	1.2 : 1
	?**	${\begin{matrix}{4r-2 \choose 1}{50-4r \choose 2}\end{matrix}}$	4,590	0.2653793	2.8 : 1	3,094	0.1788853	4.6 : 1
	***	${\begin{matrix}{50-4r \choose 3}\end{matrix}}$	816	0.0471785	20.2 : 1	364	0.0210453	46.5 : 1
	Total		14,360	0.8302498	0.2 : 1	12,788	0.7393617	0.4 : 1

Derivations for low hand shape xy** (two low ranks and two high cards) on the turn

Hand to make	Board	Derivation	8-high qualifier (r = 8)			9-high qualifier (r = 9)
			Combos	Probability	Odds	Combos	Probability	Odds
Make low	ABCD	${\begin{matrix}{r-2 \choose 4}{4 \choose 1}^{4}\end{matrix}}$	3,840	0.0197348	49.7 : 1	8,960	0.0460479	20.7 : 1
	AABC	${\begin{matrix}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^{2}\end{matrix}}$	5,760	0.0296022	32.8 : 1	10,080	0.0518039	18.3 : 1
	ABC*	${\begin{matrix}{r-2 \choose 3}{4 \choose 1}^{3}{50-4r \choose 1}\end{matrix}}$	23,040	0.1184089	7.4 : 1	31,360	0.1611676	5.2 : 1
	xABC	${\begin{matrix}{2 \choose 1}{3 \choose 1}{r-2 \choose 3}{4 \choose 1}^{3}\end{matrix}}$	7,680	0.0394696	24.3 : 1	13,440	0.0690718	13.5 : 1
	Total		40,320	0.2072155	3.8 : 1	63,840	0.3280913	2.0 : 1
Miss low	xxyA	${\begin{matrix}{2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1}\end{matrix}}$	432	0.0022202	449.4 : 1	504	0.0025902	385.1 : 1
	xxAB	${\begin{matrix}{2 \choose 1}{3 \choose 2}{r-2 \choose 2}{4 \choose 1}^{2}\end{matrix}}$	1,440	0.0074006	134.1 : 1	2,016	0.0103608	95.5 : 1
	xyAA	$$