Poker Range of Hands Math

[GTSamIAm]

You're supposed to always put a player on a range of hands and an approximate percentage that the opponent has a certain hand. So you could calculate your equity weighting the chance of each poker hand. Or you could play the opponent on each specific hand according to the percentage that he has it. In the long run, shouldn't these two choices have the same expected value? Also, if you choose the hand that has the highest expected value for you, shouldn't this be more profitable than playing his range of hands? Or what about choosing the most probable hand?

[AaronBrown]

I'm not sure I understand. Let's take an oversimplified example. Say a player raises in first position if and only if he has:

(1) Any Ace-King or Ace-Queen
(2) Any suited Ace-x
(3) Any pair from 7's to Jacks

Of course, no one is this predictable. But if he was, there are 102 out of 1,326 starting hands that he would raise on if first position. You could compute the probability of any specific hand, say pocket Jacks, by taking the number of ways to form the hand (6) divided by 102. Of course, these computations will change based on the hands you hold.

In theory, you could compute your equity for any of the possible hands for the other player, then weight them by the probability of that hand. In practice, you use rough guesses of the probabilities of the most likely hands to form an idea of what your best action is.

It's silly to worry about which of his hands is best or worst for you, neither you nor he control what he has.