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.
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.
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.
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.
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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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.