Probability Derivations For Making Rank-based Hands in Omaha Hold 'em - Derivations For Starting Hands With No Pair

Derivations For Starting Hands With No Pair

Starting hands with no pair (XYZR) can improve when two or three of the remaining X, Y, Z or R cards (XX or XXX) appears on the board. These hands can make two pair or a full house when two of more ranks from the hand appear (XY or XXY). They also can make three of a kind or a pair if two or three other ranks (ss or sss) appear, although these boards are likely to improve other hands at least as much. The following tables show all the ways for XYZR to make four of a kind, a full house, three of a kind, two pair, one pair or no pair (high card) on the flop, turn and river.

**Derivations for rank type XYZR (no pair) on the flop**
Hand to make	Board	Combos	Probability	Odds
Four of a kind	XXX	4	0.0002313	4,323.0 : 1
Full house	XXY	108	0.0062442	159.1 : 1
Three of a kind	XXs	432	0.0249769	39.0 : 1
	sss	36	0.0020814	479.4 : 1
	Total	468	0.0270583	36.0 : 1
Two pair	XYZ	108	0.0062442	159.1 : 1
	XYs	1,944	0.1123959	7.9 : 1
	Total	2,052	0.1186401	7.4 : 1
One pair	X??	7,560	0.4370953	1.3 : 1
	sst	1,728	0.0999075	9.0 : 1
	Total	9,288	0.5370028	0.9 : 1
No pair	stu	5,376	0.3108233	2.2 : 1

**Derivations for rank type XYZR (no pair) on the turn**
Hand to make	Board	Combos	Probability	Odds
Four of a kind	XXX*	180	0.0009251	1,080.0 : 1
Full house	XXYY	54	0.0002775	3,602.3 : 1
	XXYZ	324	0.0016651	599.6 : 1
	XXYs	3,888	0.0199815	49.0 : 1
	Total	4,266	0.0219241	44.6 : 1
Three of a kind	XX??	7,560	0.0388529	24.7 : 1
	Xsss	432	0.0022202	449.4 : 1
	ssss	9	0.0000463	21,619.0 : 1
	ssst	1,152	0.0059204	167.9 : 1
	Total	9,153	0.0470398	20.3 : 1
Two pair	XYZR	81	0.0004163	2,401.2 : 1
	XYZs	3,888	0.0199815	49.0 : 1
	XY??	34,020	0.1748381	4.7 : 1
	Total	37,989	0.1952359	4.1 : 1
One pair	Xsst	20,736	0.1065680	8.4 : 1
	Xstu	64,512	0.3315449	2.0 : 1
	sstt	1,296	0.0066605	149.1 : 1
	sstu	24,192	0.1243293	7.0 : 1
	Total	110,736	0.5691027	0.8 : 1
No pair	stuv	32,256	0.1657724	5.0 : 1

**Derivations for rank type XYZR (no pair) on the river**
Hand to make	Board	Combos	Probability	Odds
Four of a kind	XXX**	3,960	0.0023127	431.4 : 1
Full house	XXYYZ	324	0.0001892	5,283.9 : 1
	XXYYs	1,944	0.0011353	879.8 : 1
	XXYZR	324	0.0001892	5,283.9 : 1
	XXYZs	11,664	0.0068119	145.8 : 1
	XXY??	68,040	0.0397359	24.2 : 1
	Total	82,296	0.0480616	19.8 : 1
Three of a kind	XX???	85,680	0.0500378	19.0 : 1
	XYsss	1,944	0.0011353	879.8 : 1
	Xssst	13,824	0.0080733	122.9 : 1
	ssss*	396	0.0002313	4,323.0 : 1
	sss??	17,856	0.0104281	94.9 : 1
	Total	119,700	0.0699058	13.3 : 1
Two pair	XYZRs	2,916	0.0017030	586.2 : 1
	XYZ??	68,040	0.0397359	24.2 : 1
	XYsst	93,312	0.0544950	17.4 : 1
	XYstu	290,304	0.1695400	4.9 : 1
	Total	454,572	0.2654739	2.8 : 1
One pair	Xsstt	15,552	0.0090825	109.1 : 1
	Xsstu	290,304	0.1695400	4.9 : 1
	Xstuv	387,072	0.2260533	3.4 : 1
	ssttu	36,288	0.0211925	46.2 : 1
	sstuv	193,536	0.1130267	7.8 : 1
	Total	922,752	0.5388950	0.9 : 1
No pair	stuvw	129,024	0.0753511	12.3 : 1

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