Probability Derivations For Making Rank-based Hands in Omaha Hold 'em - Derivations For Starting Hands With Four of A Kind

Derivations For Starting Hands With Four of A Kind

Starting hands with four of a kind (XXXX) can only improve to a full house or two pair. To make a full house, this hand needs to have two or three cards of the same rank appear on the board. To make two pair, another pair on the board is needed. Of course, any other hand holding a pair also makes at least a full house or two with either of these boards. The following table shows the derivations for making a full house, two pair or one pair when holding four of a kind.

**Derivations for rank type XXXX (four of a kind) on the flop**
Hand to make	Board	Combos	Probability	Odds
Full house	rrr	48	0.0027752	359.3 : 1
Two pair	rrs	3,168	0.1831637	4.5 : 1
One pair	rst	14,080	0.8140611	0.2 : 1

**Derivations for rank type XXXX (four of a kind) on the turn**
Hand to make	Board	Combos	Probability	Odds
Full house	rrr*	2,112	0.0108541	91.1 : 1
	rrrr	12	0.0000617	16,214.0 : 1
	Total	2,124	0.0109158	90.6 : 1
Two pair	rrss	2,376	0.0122109	80.9 : 1
	rrst	63,360	0.3256244	2.1 : 1
	Total	65,736	0.3378353	2.0 : 1
One pair	rstu	126,720	0.6512488	0.5 : 1

**Derivations for rank type XXXX (four of a kind) on the river**
Hand to make	Board	Combos	Probability	Odds
Full house	rrr**	45,408	0.0265187	36.7 : 1
	rrrr*	528	0.0003084	3,242.0 : 1
	Total	45,936	0.0268270	36.3 : 1
Two pair	rrsst	95,040	0.0555042	17.0 : 1
	rrstu	760,320	0.4440333	1.3 : 1
	Total	855,360	0.4995375	1.0 : 1
One pair	rstuv	811,008	0.4736355	1.1 : 1

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