No 1 in Gambling

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) x 4)/50) x 3/49 = (84 - 6r)/1225 .

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

1. Multiply the base probability for a single player for a given rank of pocket pairs by the number of opponents in the hand;
2. 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 P₂ is the probability that exactly two players have a higher pair, P₃ is the probability that exactly three players have a higher pair, etc. As a practical matter, even with pocket 2s against 9 opponents, P₄ < 0.0015 and P₅ < 0.00009, so just calculating P₂ and P₃ 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 + 2P₃ + ... + 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.

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