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