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

Derivations For Starting Hands With Three of A Kind

To make a full house or three or four of a kind, starting hands with three of a kind (XXXY) need to either catch the case (last) X or catch two or three of the remaining Y cards (YY or YYY). They also improve to a full house if three or more of another rank appears on the board (rrr or rrrr), although any other hand holding a pair also makes a full house with this board. Three of a kind makes two pair if either a Y card or another pair appears on the board. The following tables show all the ways for XXXY 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 XXXY (three of a kind) on the flop
Hand to make Board Derivation Combos Probability Odds
Four of a kind YYY 1 0.0000578 17,295.0 : 1
Full house XYY 3 0.0001735 5,764.3 : 1
Xrr 66 0.0038159 261.1 : 1
rrr 44 0.0025439 392.1 : 1
Total 113 0.0065333 152.1 : 1
Three of a kind XYr 132 0.0076318 130.0 : 1
Xrs 880 0.0508788 18.7 : 1
YYr 132 0.0076318 130.0 : 1
Total 1,144 0.0661425 14.1 : 1
Two pair Yrr 198 0.0114477 86.4 : 1
rrs 2,640 0.1526364 5.6 : 1
Total 2,838 0.1640842 5.1 : 1
One pair Yrs 2,640 0.1526364 5.6 : 1
rst 10,560 0.6105458 0.6 : 1
Total 13,200 0.7631822 0.3 : 1
Derivations for rank type XXXY (three of a kind) on the turn
Hand to make Board Derivation Combos Probability Odds
Four of a kind YYY* 45 0.0002313 4,323.0 : 1
Full house XYYr 132 0.0006784 1,473.1 : 1
XYrr 198 0.0010176 981.7 : 1
Xrrs 2,640 0.0135677 72.7 : 1
rrr* 1,936 0.0099496 99.5 : 1
rrrr 11 0.0000565 17,688.1 : 1
Total 4,917 0.0252698 38.6 : 1
Three of a kind XYrs 2,640 0.0135677 72.7 : 1
Xrst 10,560 0.0542707 17.4 : 1
YY?? 2,838 0.0145853 67.6 : 1
Total 16,038 0.0824237 11.1 : 1
Two pair Yrrs 7,920 0.0407031 23.6 : 1
rrss 1,980 0.0101758 97.3 : 1
rrst 47,520 0.2442183 3.1 : 1
Total 57,420 0.2950971 2.4 : 1
One pair Yrst 31,680 0.1628122 5.1 : 1
rstu 84,480 0.4341659 1.3 : 1
Total 116,160 0.5969781 0.7 : 1
Derivations for rank type XXXY (three of a kind) on the river
Hand to make Board Derivation Combos Probability Odds
Four of a kind YYY** 990 0.0005782 1,728.6 : 1
Full house XYY?? 2,838 0.0016574 602.3 : 1
XYrrs 7,920 0.0046253 215.2 : 1
Xrrss 1,980 0.0011563 863.8 : 1
Xrrst 47,520 0.0277521 35.0 : 1
rrr** 41,624 0.0243088 40.1 : 1
rrrr* 484 0.0002827 3,536.8 : 1
Total 102,366 0.0597826 15.7 : 1
Three of a kind XYrst 31,680 0.0185014 53.1 : 1
Xrstu 84,480 0.0493370 19.3 : 1
YYrrs 7,920 0.0046253 215.2 : 1
YYrst 31,680 0.0185014 53.1 : 1
Total 155,760 0.0909652 10.0 : 1
Two pair Yrrss 5,940 0.0034690 287.3 : 1
Yrrst 142,560 0.0832562 11.0 : 1
rrsst 71,280 0.0416281 23.0 : 1
rrstu 506,880 0.2960222 2.4 : 1
Total 726,660 0.4243756 1.4 : 1
One pair Yrstu 253,440 0.1480111 5.8 : 1
rstuv 473,088 0.2762874 2.6 : 1
Total 726,528 0.4242985 1.4 : 1

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

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