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Texas Hold 'Em Dominated Hand Probabilities

Last update: Nov. 16, 2007

In Texas Hold 'Em a hand is said to be dominated if another player has a similar, and better, hand. No two sources seem to agree on an exact definition of when the two hole cards are dominated. To add to the confusion, I will present yet another definition. A two-card hand is dominated if any of the following are true:

  1. A pair is dominated if another player has a higher pair. Example: K,K is dominated by A,A.
  2. Two singletons are dominated if another player has a pair equal or greater to the lower singleton. Examples Q,J is dominated by JJ; K,10 is dominated by A,A.
  3. Two singletons are dominated if another player has one card matching in rank, and the other card is less than the other player's other card. Examples: J,9 is dominated by Q,9; Q,7 is dominated by Q,8.

That said, the following tables present the probability of every two-card hand being dominated, according to the total number of players.

Probability of Domination — Pairs
Cards 2 Players 3 Players 4 Players 5 Players 6 Players 7 Players 8 Players 9 Players 10 Players
2,2 0.05876 0.11412 0.16621 0.21501 0.26086 0.30428 0.34490 0.38343 0.41936
3,3 0.05385 0.10478 0.15293 0.19877 0.24160 0.28255 0.32120 0.35757 0.39216
4,4 0.04904 0.09553 0.14002 0.18212 0.22216 0.26018 0.29663 0.33085 0.3637
5,5 0.04409 0.08634 0.12661 0.16546 0.20200 0.23743 0.27104 0.30331 0.33423
6,6 0.03926 0.07694 0.11309 0.14813 0.18149 0.21375 0.24491 0.27453 0.30328
7,7 0.03426 0.06741 0.09951 0.1305 0.16038 0.18967 0.21749 0.24468 0.27105
8,8 0.02934 0.05795 0.08576 0.11288 0.13902 0.16466 0.18951 0.21366 0.23722
9,9 0.02456 0.04838 0.0719 0.09465 0.11712 0.13906 0.16046 0.18128 0.20169
10,10 0.01957 0.03883 0.05786 0.07642 0.09477 0.11269 0.13037 0.14764 0.16482
J,J 0.01468 0.02918 0.04357 0.05779 0.07194 0.08566 0.09927 0.11296 0.12629
Q,Q 0.00977 0.01957 0.02921 0.03876 0.04838 0.0579 0.06734 0.07676 0.08600
K,K 0.00487 0.00979 0.0147 0.01961 0.02443 0.02928 0.03421 0.03903 0.04402
A,A 0 0 0 0 0 0 0 0 0

Probability of Domination — Non-Pairs
Cards 2 Players 3 Players 4 Players 5 Players 6 Players 7 Players 8 Players 9 Players 10 Players
3,2 0.27175 0.47456 0.62453 0.73433 0.81409 0.87151 0.91225 0.94075 0.96061
4,2 0.25719 0.45285 0.60025 0.71077 0.79272 0.85306 0.89692 0.92864 0.95111
4,3 0.24734 0.43772 0.58326 0.69345 0.77659 0.83864 0.88459 0.91835 0.94286
5,2 0.24251 0.43041 0.57511 0.68575 0.76957 0.83273 0.87982 0.91458 0.93995
5,3 0.23271 0.4152 0.55729 0.66744 0.75205 0.81680 0.86575 0.90248 0.92986
5,4 0.22294 0.39962 0.53913 0.64847 0.73371 0.79985 0.85062 0.88944 0.9189
6,2 0.22775 0.40782 0.54925 0.65931 0.74484 0.81051 0.86055 0.89834 0.92686
6,3 0.21804 0.39203 0.53037 0.63980 0.72573 0.7927 0.84452 0.88432 0.91468
6,4 0.20825 0.37622 0.51154 0.61968 0.70572 0.77371 0.82718 0.86907 0.90147
6,5 0.19838 0.36056 0.49217 0.59901 0.68501 0.75404 0.80898 0.85269 0.88709
7,2 0.21312 0.38465 0.52211 0.6315 0.71819 0.78606 0.83912 0.88005 0.91151
7,3 0.20327 0.36878 0.50278 0.61083 0.69729 0.76614 0.82076 0.86367 0.8971
7,4 0.19347 0.35264 0.48292 0.58923 0.67582 0.74528 0.8012 0.8459 0.88131
7,5 0.18363 0.33638 0.46297 0.56744 0.65318 0.72323 0.7805 0.82678 0.86409
7,6 0.17376 0.32008 0.44254 0.54476 0.62985 0.70037 0.75855 0.80646 0.84557
8,2 0.19842 0.36113 0.49425 0.60226 0.68936 0.75933 0.81514 0.85914 0.89382
8,3 0.18865 0.34493 0.47393 0.58018 0.66671 0.73738 0.79441 0.84018 0.87677
8,4 0.17873 0.32856 0.45366 0.55744 0.64344 0.71418 0.77222 0.81965 0.85805
8,5 0.16906 0.31203 0.43273 0.53412 0.61914 0.69 0.74895 0.79756 0.83774
8,6 0.1593 0.2953 0.41137 0.51031 0.59388 0.66446 0.7241 0.77389 0.81552
8,7 0.14936 0.27858 0.38979 0.48557 0.56776 0.63785 0.69778 0.74882 0.79182
9,2 0.18372 0.33732 0.46536 0.57132 0.6588 0.73017 0.78848 0.83563 0.87337
9,3 0.17397 0.32078 0.44419 0.54785 0.63428 0.70583 0.76501 0.81351 0.85326
9,4 0.16414 0.30414 0.42302 0.52394 0.60892 0.68035 0.74009 0.78987 0.83104
9,5 0.15443 0.28725 0.40138 0.49924 0.58259 0.65358 0.71369 0.76448 0.80723
9,6 0.1445 0.27013 0.37937 0.47373 0.55545 0.6257 0.68586 0.73743 0.78125
9,7 0.13475 0.25314 0.35697 0.44794 0.52725 0.59629 0.6565 0.7084 0.75331
9,8 0.12496 0.23584 0.33409 0.42109 0.49819 0.56587 0.62526 0.67746 0.72327
10,2 0.16891 0.31301 0.43542 0.53879 0.62571 0.69841 0.759 0.80872 0.84978
10,3 0.15914 0.29624 0.41375 0.5139 0.59949 0.67158 0.73259 0.78349 0.82618
10,4 0.14939 0.27906 0.39154 0.48857 0.57201 0.64354 0.7046 0.75642 0.80037
10,5 0.13951 0.26193 0.36905 0.46246 0.54371 0.61402 0.67501 0.72735 0.7724
10,6 0.12981 0.2447 0.3463 0.4357 0.51428 0.58319 0.64362 0.69625 0.74199
10,7 0.11998 0.22736 0.32294 0.4082 0.48405 0.55111 0.61069 0.66305 0.70931
10,8 0.11019 0.20961 0.29936 0.38001 0.45256 0.51757 0.57575 0.62777 0.67409
10,9 0.10039 0.19201 0.27531 0.35114 0.42023 0.48263 0.53925 0.59034 0.63649
J,2 0.15425 0.2884 0.40456 0.50465 0.59045 0.6639 0.72618 0.77873 0.8228
J,3 0.14444 0.27122 0.38197 0.4784 0.56203 0.63441 0.69666 0.75 0.79513
J,4 0.13464 0.25395 0.35903 0.45167 0.53269 0.60356 0.66527 0.71893 0.76515
J,5 0.12483 0.23638 0.3357 0.4239 0.50212 0.57128 0.63222 0.68564 0.73259
J,6 0.11512 0.21879 0.31196 0.39579 0.47055 0.53745 0.59712 0.65014 0.69714
J,7 0.10517 0.20102 0.28791 0.36671 0.43788 0.50228 0.56031 0.61251 0.65915
J,8 0.09556 0.18306 0.26361 0.33708 0.40426 0.46545 0.52137 0.57217 0.6184
J,9 0.08574 0.16506 0.23852 0.30652 0.3693 0.4271 0.48051 0.52939 0.57458
J,10 0.07592 0.14694 0.21341 0.27542 0.33327 0.38725 0.43748 0.48431 0.52785
Q,2 0.13958 0.26335 0.37256 0.46873 0.55295 0.62637 0.69005 0.74485 0.79197
Q,3 0.12976 0.24582 0.34919 0.44104 0.52238 0.59395 0.65703 0.71199 0.7599
Q,4 0.11997 0.22814 0.32546 0.41263 0.49064 0.56021 0.62205 0.67687 0.72495
Q,5 0.11017 0.21036 0.30135 0.38353 0.45787 0.52487 0.58515 0.63894 0.68715
Q,6 0.10032 0.19242 0.27668 0.35374 0.42391 0.48799 0.54599 0.59868 0.64638
Q,7 0.09057 0.17429 0.25182 0.32318 0.38903 0.44944 0.5048 0.55579 0.60232
Q,8 0.08084 0.15618 0.22636 0.29186 0.35266 0.40919 0.46145 0.51016 0.55505
Q,9 0.07105 0.13781 0.20084 0.25976 0.3154 0.36735 0.41613 0.46176 0.50447
Q,10 0.0612 0.11934 0.17456 0.22698 0.27673 0.32375 0.36821 0.41054 0.45033
Q,J 0.05137 0.10083 0.14807 0.19349 0.2368 0.27842 0.31824 0.35636 0.39277
K,2 0.12478 0.23797 0.33956 0.43113 0.51297 0.58584 0.65014 0.7073 0.75695
K,3 0.11504 0.2201 0.31542 0.40191 0.48013 0.55037 0.61352 0.66984 0.71993
K,4 0.10514 0.2021 0.29087 0.37213 0.44603 0.51327 0.57457 0.62966 0.67951
K,5 0.09549 0.18397 0.26574 0.34129 0.41095 0.47462 0.53352 0.58707 0.63581
K,6 0.08562 0.16584 0.2403 0.30993 0.37458 0.43443 0.48998 0.54114 0.58874
K,7 0.07584 0.14737 0.21448 0.27754 0.33698 0.39236 0.44417 0.49266 0.538
K,8 0.06608 0.12886 0.18847 0.24471 0.29806 0.34843 0.396 0.44109 0.48348
K,9 0.05623 0.1101 0.16167 0.21081 0.258 0.30277 0.34548 0.38637 0.42517
K,10 0.04652 0.09139 0.13465 0.17642 0.21661 0.25533 0.29255 0.32837 0.36294
K,J 0.03671 0.07244 0.10727 0.14107 0.17397 0.20591 0.2371 0.2672 0.29658
K,Q 0.02697 0.05341 0.07948 0.105 0.13012 0.15482 0.17896 0.20268 0.22608
A,2 0.11015 0.21206 0.3057 0.39167 0.47022 0.54167 0.60645 0.66503 0.71711
A,3 0.10045 0.19382 0.28066 0.36097 0.43499 0.50318 0.56568 0.62265 0.67447
A,4 0.09063 0.17557 0.25506 0.32946 0.39856 0.4628 0.52237 0.57734 0.62801
A,5 0.08079 0.15716 0.22919 0.2972 0.36094 0.42069 0.4767 0.52909 0.57792
A,6 0.07107 0.13867 0.20294 0.26381 0.32187 0.37678 0.42865 0.47734 0.52354
A,7 0.06126 0.11987 0.17622 0.23009 0.28146 0.33084 0.37782 0.42257 0.46532
A,8 0.05141 0.10106 0.14912 0.19535 0.24005 0.28312 0.32446 0.36428 0.40282
A,9 0.04162 0.0821 0.12161 0.15986 0.19728 0.23338 0.26848 0.30278 0.33584
A,10 0.03181 0.06305 0.09365 0.12369 0.15304 0.18179 0.20987 0.23748 0.2645
A,J 0.02208 0.04377 0.06527 0.0865 0.10742 0.12811 0.14853 0.16859 0.18847
A,Q 0.01222 0.0244 0.03655 0.04853 0.06055 0.07248 0.08427 0.096 0.10777
A,K 0.00245 0.00488 0.00734 0.00978 0.01224 0.01469 0.01713 0.01960 0.02206

Methedology: These tables were created by a random simulation.

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