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.
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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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