ranges and probabilities

Since Pokerstove is a favorite topic around here, I wanted to suggest something that I do (on a more vague, instictive level) in my live games that will take Pokerstove users to the next level in terms of equity.

As I understand it, after you place a player on a range, you can input that range (along with the board and the pot and bet sizes) into Pokerstove and it will tell you how much equity you have in the pot. This is obviously a hugely helpful tool. But there is a fallacy in this process. If you tell Pokerstove that the Villain's range is AA, AK, KK, or QQ, Pokerstove will value your equity based on the assumption that the Villain is equally likely to have those hands-- or, slightly more accurately, that Villain is likely to have those hands in direct proportion to the relative probability of those hands being dealt from the deck (i.e., AK is more probable than AA or KK).

But that's not, of course, actually how a range works. Rather, when I put a player on a range, especially based on limited information, I am attaching probabilities to the various hands within that range based on the information the player has given me.

For example, let's suppose that I raise pre-flop with KK from the button, and the small blind 3-bets me. He's been a tight player and I know he has a limited raising range, but I don't know how limited. His range for that 3-bet could be TT-AA and AK, or it might be as tight as AA KK.

So what do I do? Well, what I do is treat this as a probability exercise. In other words, not having a complete read on his pre-flop 3-betting range, I assign probabilities to what he has. I figure that I know that he will 3 bet AA or KK, but there are only 6 possible AA hands and 1 possible KK hand (as I have the other two kings). It is somewhat possible that he will 3-bet QQ (6 combinations), less possible that he will 3-bet JJ (6 combinations) or AK (12 combinations), less possible still that he will 3-bet TT (6 combinations).

Thus, based on the information I have, I might think of the problem as follows:

Hand Probability that it is in player's range Relative probability of occurrence
AA 100 16
KK 100 2.5
QQ 50 16
JJ 30 16
AK 30 33
TT 15 16

When we multiply these out, we get the following ratios:
AA 16
KK 2.5
QQ 8
JJ 4.8
AK 10
TT 2.4

So now, let's approximate these to a 100 point scale (rounded off):
AA 37 percent
KK 6 percent
QQ 18 percent
JJ 11 percent
AK 23 percent
TT 5 percent

Now we have a much more accurate estimation of a player's range. Now, obviously, I don't expect anyone to be doing this while they are deciding whether to call or raise a bet. But there is an important thing we learned here. You don't need me to tell you a three bet from a tight player could very likely mean aces. But note that in many instances, AK, rather than QQ or JJ, is the second most likely hand you are looking at. And note further that you are ahead of 57 percent of the likely hands.

Now this, like all such calculations, is all dependent on your actual reads. But the suggestion I am making is that when making reads, the first level of thinking is "what is this player's range in this situation?", and the second level of thinking is "how probable is each hand that I placed in the player's range?". This thought process will lead you to more accurate estimations about ranges and probabilities of various hands in the range, and thus more accurate betting decisions.