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Texas Hold ’Em Dominated Hand Probabilities

Last Update: Feb. 24, 2012

Update Notice

In February, 2012, one of my readers wrote to questioned my definition of what constituted a "dominated" hand. In the process of trying to answer his concerns I questioned my own definition. So, I'm redoing the whole page. I have expanded my definition of a "dominated hand," which is why my numbers are different than what they used to be.

Main Content Begins here.

In Texas Hold 'Em a hand is said to be dominated if another player has a similar, and better, hand. To be more specific, a dominated hand is said to rely on three or fewer outs (cards) to beat the hand dominating it, not counting difficult multiple-card draws. There are four types of domination, as follows.

A pair is dominated by a higher pair. For example J-J is dominated by Q-Q. Only two cards help the J-J, the other two jacks.
A non-pair is dominated by a pair of either card. For example, Q-5 is dominated by Q-Q or 5-5. In the case of 5-5, three cards only will help the Q-5, the other three queens.
A non-pair is dominated by a pair greater than the lower card. For example, Q-5 is dominated by 8-8. Only three cards will help the Q-5, the other three queens.
A non-pair is dominated by another non-pair if there if there is a shared card, and the rank of the opponent's non-shared card is greater the dominated non-shared card. For example Q-5 is dominated by K-5 or Q-7. In the former case (K-5 over Q-5) only three cards can help Q-5, the other three queens.

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	0.0588	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

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	0.2742	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

Methedology: These tables were created by a random simulation. Each cell in the table above for pairs was based on 7.8 million hands, and 21.7 million for the non-pairs.

2-Player Formula

The probability of domination in a two player game is easy to calculate. For pairs it is 6×(number of higher ranks)/1225. For example, the probability a pair of eights is dominated is 6×6/1225 = 0.0294, because there are six ranks higher than 8 (9,T,J,Q,K,A).

For non-pairs the formula is (6+18×(L-1)+12×H)/1225, where

L=Number of ranks higher than lower card
H=Number of ranks higher than higher card

For example, the probability that J-7 is dominated is (6+18×(7-1)+12×3)/1225 = 150/1225 = 0.1224.