Probability Derivations For Making Rank-based Hands in Omaha Hold 'em - Derivations For Starting Hands With One Pair

Derivations For Starting Hands With One Pair

Starting hands with one pair (XXYZ) can improve to three of a kind, a full house or four of a kind when either an X card is on the board or when two or three of the remaining Y or Z cards (YY or YYY) is on the board. They also improve to a full house if three or more of another rank is on the board (rrr or rrrr), although any other hand holding a pair also makes a full house with this board. These hands make two pair if another pair (rr) appears on the board. The following tables show all the ways for XXYZ to make four of a kind, a full house, three of a kind, two pair or one pair on the flop, turn and river.

Derivations for rank type XXYZ (one pair) on the flop
Hand to make Board Derivation Combos Probability Odds
Four of a kind XX* 46 0.0026596 375.0 : 1
YYY 2 0.0001156 8,647.0 : 1
Total 48 0.0027752 359.3 : 1
Full house XYY 12 0.0006938 1,440.3 : 1
Xrr 120 0.0069380 143.1 : 1
YYZ 18 0.0010407 959.9 : 1
rrr 40 0.0023127 431.4 : 1
Total 190 0.0109852 90.0 : 1
Three of a kind XYZ 18 0.0010407 959.9 : 1
XYr 480 0.0277521 35.0 : 1
Xrs 1,440 0.0832562 11.0 : 1
YYr 240 0.0138760 71.1 : 1
Total 2,178 0.1259251 6.9 : 1
Two pair YZr 360 0.0208141 47.0 : 1
Yrr 360 0.0208141 47.0 : 1
rrs 2,160 0.1248844 7.0 : 1
Total 2,880 0.1665125 5.0 : 1
One pair Yrs 4,320 0.2497687 3.0 : 1
rst 7,680 0.4440333 1.3 : 1
Total 12,000 0.6938020 0.4 : 1
Derivations for rank type XXYZ (one pair) on the turn
Hand to make Board Derivation Combos Probability Odds
Four of a kind XX** 1,035 0.0053191 187.0 : 1
YYY* 90 0.0004625 2,161.0 : 1
Total 1,125 0.0057817 172.0 : 1
Full house XYYZ 36 0.0001850 5,404.0 : 1
XYYr 480 0.0024669 404.4 : 1
XYrr 720 0.0037003 269.3 : 1
Xrrs 4,320 0.0222017 44.0 : 1
YYZZ 9 0.0000463 21,619.0 : 1
YYZr 720 0.0037003 269.3 : 1
rrr* 1,760 0.0090451 109.6 : 1
rrrr 10 0.0000514 19,457.0 : 1
Total 8,055 0.0413969 23.2 : 1
Three of a kind XYZr 720 0.0037003 269.3 : 1
XYrs 8,640 0.0444033 21.5 : 1
Xrst 15,360 0.0789393 11.7 : 1
YY?? 4,680 0.0240518 40.6 : 1
Total 29,400 0.1510947 5.6 : 1
Two pair YZ?? 7,020 0.0360777 26.7 : 1
Yrrs 12,960 0.0666050 14.0 : 1
rrss 1,620 0.0083256 119.1 : 1
rrst 34,560 0.1776133 4.6 : 1
Total 56,160 0.2886216 2.5 : 1
One pair Yrst 46,080 0.2368178 3.2 : 1
rstu 53,760 0.2762874 2.6 : 1
Total 99,840 0.5131051 0.9 : 1
Derivations for rank type XXYZ (one pair) on the river
Hand to make Board Derivation Combos Probability Odds
Four of a kind XX*** 15,180 0.0088652 111.8 : 1
YYY** 1,980 0.0011563 863.8 : 1
XXYYY −2 −0.0000012 −856,153 : 1
Total (see #1 below) 17,158 0.0100204 98.8 : 1
Full house XYYZZ 18 0.0000105 95,127.0 : 1
XYYZr 1,440 0.0008410 1,188.1 : 1
XYY?? 9,360 0.0054663 181.9 : 1
XYZrr 1,080 0.0006307 1,584.5 : 1
XYrrs 25,920 0.0151375 65.1 : 1
Xrrss 3,240 0.0018922 527.5 : 1
Xrrst 69,120 0.0403667 23.8 : 1
YYZZr 360 0.0002102 4,755.4 : 1
YYZ?? 14,040 0.0081995 121.0 : 1
rrr** 37,840 0.0220989 44.3 : 1
rrrXX −40 −0.0000234 −42,808.6 : 1
rrrr* 440 0.0002570 3,890.6 : 1
Total (see #2 below) 162,818 0.0950871 9.5 : 1
Three of a kind XYZrs 12,960 0.0075687 131.1 : 1
XYrst 92,160 0.0538222 17.6 : 1
Xrstu 107,520 0.0627926 14.9 : 1
YYrrs 12,960 0.0075687 131.1 : 1
YYrst 46,080 0.0269111 36.2 : 1
Total 271,680 0.1586634 5.3 : 1
Two pair YZrrs 19,440 0.0113531 87.1 : 1
YZrst 69,120 0.0403667 23.8 : 1
Yrrss 9,720 0.0056766 175.2 : 1
Yrrst 207,360 0.1211000 7.3 : 1
rrsst 51,840 0.0302750 32.0 : 1
rrstu 322,560 0.1883778 4.3 : 1
Total 680,040 0.3971491 1.5 : 1
One pair Yrstu 322,560 0.1883778 4.3 : 1
rstuv 258,048 0.1507022 5.6 : 1
Total 580,608 0.3390800 1.9 : 1

Read more about this topic:  Probability Derivations For Making Rank-based Hands In Omaha Hold 'em

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