Texas Hold'em Hands with a Pocket Pair
By Jim Magee
Matching the pocket pair:
C42 * C483 = 17,296
4 of a kind in the 5 up cards:
C121 * C44 * C481 = 576
Total ways = 17,296 + 576 = 17,872
Ways to make a Full House:
Trips matching the pocket pair:
- * {C
Trips in the 5 up cards:
| {C121 * C43} * {C442} = 45,408 | (Includes full house up. Excludes trips matching the pocket pair. Excludes 4 of a kind.) |
Trips both on board and a card matching pocket pair.
{C21 * C121} * {C43 * C111} * {C41} = 4,224
Total ways = 136,224 + 45,408 + 4,224 = 185,856
Ways to make trips:
Trips can only be made by matching the pocket pair. If trips comes up on the board, it is a full house.
C21 * C484 = 389,160
Exclude full houses created by matching pocket pair and an up pair.
389,160 - 45,408 = 343,752
Exclude flushes.
Flush includes one of the pocket pair.
- * C
Flush includes the trip on board
- * C
Flush in the remaining suit
- * C
(includes up to 5 straight flushes)
Total number of flushes while holding trips.
- + 495+ 495 = 1980 (includes up to 20 straight flushes)
Exclude straights
When trips gets beat by a straight, the straight has to be on board and include the third card of the trips. It can be in any of 5 positions within the straight (adjust for 2-4, J-K & A).
| {C11 * 5} * C41 * C41 * C41 * C41 = 256 | (Includes 1 straight flush) |
Total number of trips.
343,752 - 1,980 -256 +1 = 341,517
Conclusions
When holding a pocket pair, the following occurs:
Trips:
2,118,760 (all possible draws) / 341,517 (draws that make trips) = 1 out of 6.20 times.
Full house:
2,118,760 (all possible draws) / 185,856 (draws that make full house) = 1 out of 11.40 times.
4 of a kind:
2,118,760 (all possible draws) / 17,872 (draws that make 4 of a kind) = 1 out of 118.55 times.
© Copyright 1999-2006 Kenneth R. Churilla