Probability of the dominated hands in Texas Hold'em

When evaluating a hand before the flop, it’s useful to have some idea of how likely the hand is dominated. A dominated hand is a hand that is beaten by another hand (the dominant hand) and is extremely unlikely to win against it. Often the dominated hand has only a single card rank that can improve the dominated hand to beat the dominant hand (not counting straights and flushes.) For example, KJ is dominated by KQ—both hands share the king and the queen kicker is beating the jack kicker. Barring a straight or flush, the KJ will need a jack on the board to improve against the KQ (and will still be losing if a queen comes on the board also.) A pocket pair is dominated by a pocket pair of higher rank.

Pocket pairs

Barring a miracle straight or flush, a pocket pair needs to make three of a kind to beat a higher pocket pair.

To calculate the probability that another player has a higher pocket pair, first consider the case against a single opponent. The probability that a single opponent has a higher pair can be stated as the probability that the first card dealt to the opponent is a higher rank than the pocket pair and the second card is the same rank as the first. Where $r$ is the rank of the pocket pair (assigning values from 2–10 and J–A = 11–14), there are (14 − r) × 4 cards of higher rank. Subtracting the two cards for the pocket pair leaves 50 cards in the deck. After the first card is dealt to the player there are 49 cards left, 3 of which are the same rank as the first. So the probability of a single opponent being dealt a higher pocket pair is

P = (((14-r)x4)/50) x (3/49)

The following approach extends this equation to calculate the probability that one or more other players has a higher pocket pair.

Multiply the base probability for a single player for a given rank of pocket pairs by the number of opponents in the hand;
Subtract the adjusted probability that more than one opponent has a higher pocket pair. (This is necessary because this probability effectively gets added to the calculation multiple times when multiplying the single player result.)

Where $n$ is the number of other players still in the hand and $P ma$ is the adjusted probability that multiple opponents have higher pocket pairs, then the probability that at least one of them has a higher pocket pair is

P = ((84-6r)/1225) x n – P_ma.

The calculation for $P ma$ depends on the rank of the player’s pocket pair, but can be generalized as

P_ma = P₂ + 2P₃ + … + (n-1)P_n,

where $P 2$ is the probability that exactly two players have a higher pair, $P 3$ is the probability that exactly three players have a higher pair, etc. As a practical matter, even with pocket 2s against 9 opponents, $P 4 < 0.0015$ and $P 5 < 0.00009$ , so just calculating $P 2$ and $P 3$ gives an adequately precise result.

The following table shows the probability that before the flop another player has a larger pocket pair when there are one to nine other players in the hand.

Probability of facing a larger pair when holding	Against 1	Against 2	Against 3	Against 4	Against 5	Against 6	Against 7	Against 8	Against 9
KK	0.0049	0.0098	0.0147	0.0196	0.0244	0.0293	0.0342	0.0391	0.0439
QQ	0.0098	0.0195	0.0292	0.0388	0.0484	0.0579	0.0673	0.0766	0.0859
JJ	0.0147	0.0292	0.0436	0.0577	0.0717	0.0856	0.0992	0.1127	0.1259
TT	0.0196	0.0389	0.0578	0.0764	0.0946	0.1124	0.1299	0.1470	0.1637
99	0.0245	0.0484	0.0718	0.0946	0.1168	0.1384	0.1593	0.1795	0.1990
88	0.0294	0.0580	0.0857	0.1125	0.1384	0.1634	0.1873	0.2101	0.2318
77	0.0343	0.0674	0.0994	0.1301	0.1595	0.1874	0.2138	0.2387	0.2619
66	0.0392	0.0769	0.1130	0.1473	0.1799	0.2104	0.2389	0.2651	0.2890
55	0.0441	0.0862	0.1263	0.1642	0.1996	0.2324	0.2623	0.2892	0.3129
44	0.0490	0.0956	0.1395	0.1806	0.2186	0.2532	0.2841	0.3109	0.3334
33	0.0539	0.1048	0.1526	0.1967	0.2370	0.2729	0.3040	0.3300	0.3503
22	0.0588	0.1141	0.1654	0.2124	0.2546	0.2914	0.3222	0.3464	0.3633

The following table gives the probability that a hand is facing two or more larger pairs before the flop. From the previous equations, the probability $P m$ is computed as

P_m = P₂ + P₃ + … + P_n.

Probability of facing multiple larger pairs when holding	Against 2	Against 3	Against 4	Against 5	Against 6	Against 7	Against 8	Against 9
KK	< 0.00001	0.00001	0.00003	0.00004	0.00007	0.00009	0.00012	0.00016
QQ	0.00006	0.00018	0.00037	0.00061	0.00091	0.00128	0.00171	0.00220
JJ	0.00017	0.00051	0.00102	0.00171	0.00257	0.00360	0.00482	0.00621
TT	0.00033	0.00099	0.00200	0.00335	0.00504	0.00709	0.00950	0.01226
99	0.00054	0.00164	0.00330	0.00553	0.00836	0.01177	0.01580	0.02045
88	0.00081	0.00244	0.00493	0.00828	0.01253	0.01769	0.02378	0.03084
77	0.00112	0.00341	0.00689	0.01160	0.01758	0.02487	0.03351	0.04353
66	0.00149	0.00454	0.00918	0.01550	0.02353	0.03335	0.04503	0.05861
55	0.00191	0.00583	0.01182	0.01998	0.03040	0.04318	0.05840	0.07619
44	0.00239	0.00728	0.01480	0.02506	0.03821	0.05438	0.07371	0.09635
33	0.00291	0.00890	0.01812	0.03075	0.04698	0.06699	0.09099	0.11919
22	0.00349	0.01068	0.02180	0.03706	0.05673	0.08107	0.11034	0.14484

From a practical perspective, however, the odds of out drawing a single pocket pair or multiple pocket pairs are not much different. In both cases the large majority of winning hands require one of the remaining two cards needed to make three of a kind.

Hands with one ace

When holding a single ace (referred to as Ax), it is useful to know how likely it is that another player has a better ace—an ace with a higher second card. The weaker ace is dominated by the better ace. The probability that a single opponent has a better ace is the probability that they have either AA or Ax where x is a rank other than ace that is higher than the player’s second card. When holding Ax, the probability that the other player has AA is (3/50) x (2/49) ~ 0.00245. Where $x$ is the rank 2–K of the second card (assigning values from 2–10 and J–K = 11–13) the probability that a single opponent has a better ace is calculated by the formula

P = ((3/50) x (2/49)) + ((3/50) x (((13-x) x 4)/49) x 2) = (3/1225) + (12 x (13 – x))/1225 = (159 – 12x)/1225.

The probability (3/50) x (((13-x) x 4)/49) of a player having Ay, where y is a rank such that x < y <= K, is multiplied by the two ways to order the cards A and y in the hand.

This guide is licensed under the GNU Free Documentation License. It uses material from the Wikipedia.

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