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Hand Dominance Probabilities in Texas Hold'Em

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Introduction

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In Texas Hold'Em, a hand is considered dominated when another player possesses a comparable yet superior hand. To clarify, a hand is deemed dominated if it can rely on three or fewer outs—cards that can improve it—without considering challenging multi-card draws. There are four distinct types of hand domination.

  1. A pair can be dominated by a superior pair. For instance, the hand J-J is overshadowed by Q-Q. In this case, only two cards can improve J-J, which are the other two jacks.
  2. A non-paired hand is dominated by any existing pair of either card. For example, Q-5 is outmatched by Q-Q or 5-5. With 5-5, only three cards can assist Q-5—namely the remaining three queens.
  3. A non-paired hand may be overshadowed by a pair that consists of a higher card than its lower ranking card. For instance, Q-5 is dominated by 8-8, and it can only be improved by three cards—the three queens.
  4. A non-paired hand can also be dominated by another non-paired hand if they share a card, and the rank of the opponent's non-shared card is higher than that of the dominated non-shared card. For example, Q-5 is dominated by either K-5 or Q-7. In the case of K-5 over Q-5, only three cards—the remaining queens—will be able to assist Q-5.

With that in mind, the subsequent tables outline the likelihood of each two-card hand being dominated, factoring in the total number of participants in the game.

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Probability of Domination — PairsExpand

Cards 2 Players 3 Players 4 Players 5 Players 6 Players 7 Players 8 Players 9 Players 10 Players
2,2 0.0588 0.1142 0.1659 0.2150 0.2609 0.3044 0.3449 0.3835 0.4195
3,3 0.0540 0.1049 0.1532 0.1983 0.2419 0.2826 0.3212 0.3576 0.3922
4,4 0.0489 0.0956 0.1400 0.1820 0.2220 0.2602 0.2966 0.3313 0.3640
5,5 0.0441 0.0862 0.1265 0.1653 0.2021 0.2376 0.2710 0.3031 0.3345
6,6 0.0392 0.0767 0.1133 0.1481 0.1816 0.2136 0.2448 0.2745 0.3036
7,7 0.0344 0.0675 0.0996 0.1306 0.1605 0.1895 0.2177 0.2447 0.2709
8,8 0.0295 0.0581 0.0858 0.1129 0.1391 0.1648 0.1894 0.2138 0.2369
9,9 0.0246 0.0485 0.0720 0.0947 0.1173 0.1391 0.1604 0.1813 0.2017
T,T 0.0196 0.0389 0.0578 0.0765 0.0947 0.1126 0.1300 0.1478 0.1649
J,J 0.0147 0.0293 0.0435 0.0577 0.0719 0.0856 0.0992 0.1132 0.1262
Q,Q 0.0098 0.0195 0.0292 0.0389 0.0483 0.0579 0.0674 0.0766 0.0861
K,K 0.0049 0.0098 0.0147 0.0196 0.0245 0.0294 0.0341 0.0391 0.0439
A,A 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

Non-Pair Domination ProbabilityExpand

Cards 2 Players 3 Players 4 Players 5 Players 6 Players 7 Players 8 Players 9 Players 10 Players
3,2 0.2742 0.4785 0.6289 0.7389 0.8187 0.8753 0.9156 0.9438 0.9629
4,2 0.2645 0.4634 0.6124 0.7227 0.8036 0.8626 0.9049 0.9350 0.9562
4,3 0.2496 0.4417 0.5877 0.6986 0.7815 0.8433 0.8888 0.9220 0.9459
5,2 0.2546 0.4487 0.5956 0.7060 0.7881 0.8489 0.8934 0.9255 0.9486
5,3 0.2399 0.4263 0.5701 0.6805 0.7645 0.8279 0.8754 0.9108 0.9367
5,4 0.2253 0.4036 0.5439 0.6539 0.7393 0.8050 0.8556 0.8937 0.9227
6,2 0.2450 0.4338 0.5786 0.6885 0.7718 0.8344 0.8809 0.9152 0.9403
6,3 0.2302 0.4110 0.5525 0.6620 0.7470 0.8118 0.8614 0.8986 0.9266
6,4 0.2154 0.3881 0.5254 0.6344 0.7199 0.7869 0.8394 0.8796 0.9105
6,5 0.2008 0.3647 0.4975 0.6047 0.6911 0.7599 0.8146 0.8581 0.8919
7,2 0.2350 0.4186 0.5611 0.6709 0.7550 0.8191 0.8676 0.9042 0.9311
7,3 0.2204 0.3955 0.5340 0.6430 0.7285 0.7948 0.8461 0.8854 0.9155
7,4 0.2057 0.3724 0.5065 0.6138 0.7000 0.7681 0.8220 0.8642 0.8971
7,5 0.1910 0.3484 0.4776 0.5833 0.6693 0.7388 0.7951 0.8402 0.8761
7,6 0.1763 0.3244 0.4478 0.5510 0.6365 0.7071 0.7651 0.8128 0.8514
8,2 0.2255 0.4034 0.5434 0.6526 0.7375 0.8032 0.8536 0.8923 0.9213
8,3 0.2105 0.3800 0.5157 0.6237 0.7095 0.7771 0.8300 0.8714 0.9034
8,4 0.1959 0.3563 0.4870 0.5932 0.6791 0.7481 0.8037 0.8478 0.8828
8,5 0.1812 0.3323 0.4574 0.5614 0.6467 0.7168 0.7743 0.8208 0.8586
8,6 0.1666 0.3078 0.4272 0.5277 0.6122 0.6829 0.7416 0.7904 0.8311
8,7 0.1518 0.2829 0.3952 0.4922 0.5750 0.6453 0.7056 0.7563 0.7992
9,2 0.2156 0.3878 0.5250 0.6338 0.7194 0.7862 0.8388 0.8793 0.9104
9,3 0.2010 0.3643 0.4968 0.6039 0.6895 0.7583 0.8130 0.8564 0.8904
9,4 0.1862 0.3402 0.4674 0.5720 0.6577 0.7274 0.7843 0.8300 0.8668
9,5 0.1714 0.3157 0.4371 0.5388 0.6234 0.6937 0.7523 0.8003 0.8398
9,6 0.1569 0.2911 0.4061 0.5036 0.5868 0.6573 0.7167 0.7667 0.8088
9,7 0.1419 0.2658 0.3734 0.4669 0.5476 0.6174 0.6776 0.7289 0.7730
9,8 0.1274 0.2403 0.3400 0.4282 0.5061 0.5742 0.6342 0.6867 0.7320
T,2 0.2057 0.3722 0.5066 0.6143 0.7005 0.7688 0.8229 0.8654 0.8987
T,3 0.1910 0.3485 0.4772 0.5831 0.6691 0.7387 0.7950 0.8402 0.8762
T,4 0.1764 0.3240 0.4474 0.5501 0.6352 0.7055 0.7638 0.8111 0.8499
T,5 0.1617 0.2995 0.4163 0.5153 0.5991 0.6696 0.7286 0.7784 0.8196
T,6 0.1470 0.2742 0.3843 0.4790 0.5606 0.6305 0.6904 0.7413 0.7847
T,7 0.1323 0.2487 0.3512 0.4411 0.5196 0.5881 0.6478 0.6996 0.7448
T,8 0.1176 0.2227 0.3169 0.4008 0.4754 0.5418 0.6009 0.6532 0.6993
T,9 0.1030 0.1965 0.2817 0.3586 0.4286 0.4923 0.5492 0.6010 0.6473
J,2 0.1960 0.3566 0.4877 0.5944 0.6808 0.7505 0.8063 0.8508 0.8862
J,3 0.1813 0.3324 0.4578 0.5617 0.6476 0.7180 0.7757 0.8227 0.8610
J,4 0.1665 0.3078 0.4271 0.5275 0.6120 0.6828 0.7419 0.7911 0.8317
J,5 0.1519 0.2827 0.3954 0.4916 0.5741 0.6441 0.7042 0.7549 0.7976
J,6 0.1371 0.2573 0.3621 0.4537 0.5336 0.6026 0.6625 0.7143 0.7590
J,7 0.1223 0.2314 0.3284 0.4142 0.4901 0.5572 0.6164 0.6688 0.7145
J,8 0.1077 0.2050 0.2931 0.3725 0.4442 0.5083 0.5658 0.6174 0.6638
J,9 0.0931 0.1785 0.2571 0.3289 0.3948 0.4553 0.5100 0.5601 0.6061
J,T 0.0783 0.1515 0.2199 0.2837 0.3427 0.3979 0.4493 0.4967 0.5409
Q,2 0.1862 0.3406 0.4685 0.5739 0.6604 0.7312 0.7886 0.8352 0.8727
Q,3 0.1713 0.3161 0.4379 0.5402 0.6255 0.6968 0.7557 0.8044 0.8445
Q,4 0.1568 0.2910 0.4062 0.5045 0.5880 0.6590 0.7189 0.7696 0.8119
Q,5 0.1422 0.2658 0.3736 0.4671 0.5482 0.6180 0.6783 0.7299 0.7744
Q,6 0.1273 0.2400 0.3400 0.4280 0.5055 0.5734 0.6333 0.6857 0.7312
Q,7 0.1126 0.2139 0.3048 0.3868 0.4600 0.5254 0.5835 0.6357 0.6818
Q,8 0.0979 0.1875 0.2691 0.3435 0.4113 0.4730 0.5289 0.5800 0.6257
Q,9 0.0833 0.1606 0.2321 0.2983 0.3600 0.4166 0.4689 0.5173 0.5619
Q,T 0.0687 0.1332 0.1940 0.2516 0.3052 0.3557 0.4032 0.4480 0.4894
Q,J 0.0540 0.1055 0.1547 0.2020 0.2474 0.2902 0.3313 0.3707 0.4082
K,2 0.1763 0.3246 0.4491 0.5532 0.6395 0.7111 0.7702 0.8185 0.8579
K,3 0.1616 0.2998 0.4178 0.5178 0.6027 0.6740 0.7343 0.7848 0.8269
K,4 0.1469 0.2745 0.3851 0.4808 0.5633 0.6343 0.6948 0.7466 0.7908
K,5 0.1322 0.2491 0.3517 0.4422 0.5211 0.5904 0.6509 0.7037 0.7494
K,6 0.1175 0.2230 0.3171 0.4013 0.4763 0.5431 0.6025 0.6550 0.7016
K,7 0.1029 0.1964 0.2814 0.3586 0.4285 0.4918 0.5490 0.6007 0.6473
K,8 0.0881 0.1697 0.2447 0.3139 0.3777 0.4367 0.4905 0.5397 0.5853
K,9 0.0734 0.1423 0.2069 0.2675 0.3238 0.3765 0.4259 0.4720 0.5148
K,T 0.0588 0.1146 0.1678 0.2183 0.2665 0.3120 0.3555 0.3961 0.4350
K,J 0.0441 0.0866 0.1277 0.1671 0.2058 0.2426 0.2780 0.3125 0.3452
K,Q 0.0294 0.0582 0.0865 0.1141 0.1414 0.1679 0.1940 0.2195 0.2444
A,2 0.1665 0.3086 0.4294 0.5316 0.6177 0.6901 0.7505 0.8009 0.8425
A,3 0.1517 0.2835 0.3970 0.4949 0.5791 0.6509 0.7120 0.7641 0.8080
A,4 0.1372 0.2578 0.3636 0.4565 0.5376 0.6082 0.6695 0.7227 0.7684
A,5 0.1224 0.2318 0.3294 0.4164 0.4934 0.5618 0.6223 0.6754 0.7225
A,6 0.1077 0.2054 0.2940 0.3741 0.4462 0.5115 0.5702 0.6228 0.6701
A,7 0.0931 0.1787 0.2575 0.3300 0.3963 0.4572 0.5129 0.5638 0.6101
A,8 0.0783 0.1516 0.2200 0.2837 0.3428 0.3983 0.4498 0.4976 0.5418
A,9 0.0637 0.1241 0.1810 0.2352 0.2866 0.3347 0.3804 0.4237 0.4647
A,T 0.0490 0.0959 0.1411 0.1847 0.2264 0.2664 0.3049 0.3417 0.3770
A,J 0.0343 0.0677 0.1003 0.1320 0.1629 0.1931 0.2223 0.2507 0.2784
A,Q 0.0195 0.0389 0.0582 0.0769 0.0956 0.1140 0.1320 0.1500 0.1676
A,K 0.0049 0.0098 0.0147 0.0195 0.0243 0.0292 0.0340 0.0388 0.0436

Data Collection Method: The data presented in these tables was derived from a random simulation process. Each cell in the above table for pairs is based on a total of 7.8 million hands, while 21.7 million hands were analyzed for non-pairs.

2-Player Formula

Calculating the probability of domination in a heads-up game is straightforward. For pairs, it amounts to 6 × (number of higher ranks)/1225. For instance, if you want to find out the domination probability of a pair of eights, it would be calculated as 6 × 6/1225 = 0.0294, since there are six ranks higher than an 8 (9, T, J, Q, K, A).

For non-paired hands, the formula is (6 + 18 × (L - 1) + 12 × H)/1225, with the following variables defined as:

L = Count of ranks that are higher than the lower card
H = Count of ranks that are higher than the upper card

For example, to calculate the probability of J-7 being dominated, you would use the formula (6 + 18 × (7 - 1) + 12 × 3)/1225 = 150/1225 = 0.1224.

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