Introduction

• Poker Probability Texas Holdem Wikipedia Game
• Texas Holdem Odds And Probabilities

In poker, the probability of many events can be determined by direct calculation. This article discusses computing probabilities for many commonly occurring events in the game of Texas hold 'em and provides some probabilities and odds for specific situations. In poker, players form sets of five playing cards, called hands, according to the rules of the game. Each hand has a rank, which is compared against the ranks of other hands participating in the showdown to decide who wins the pot. In high games, like Texas hold 'em and seven-card stud, the highest-ranking hands win.In low games, like razz, the lowest-ranking hands win. Using The 'Outs' To Calculate Texas Hold'em Poker Odds. We have already determined that you have nine 'outs'. Now there are 52 cards in a deck and two of those are in your hand, leaving 50. New players only. One offer per player. Max bonus bet £5. 1st Dep offer is 100% up to £250 + 100 Bonus spins on Aloha Cluster Pays. Spins expire after 10 days, funds after 30 days. Bonus funds separate to Cash. Wager Req of 35x Deposit + Bonus Poker Probability (texas Holdem) apply.

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.

1. 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.
2. 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.
3. 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.
4. 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 — 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

Probability of Domination — Non-PairsExpand

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

Methodology: 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

Poker Probability Texas Holdem Wikipedia Game

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.

Texas Holdem Odds And Probabilities

Written by: Michael Shackleford