This is actually not difficult to calculate directly. We need to make some assumptions. In this ultra tight game, opponents will always enter the pot with AA, and fold otherwise. For Shania, you've decided it will be profitable to expand your portfolio of opening hands to also include KK. You've been dealt KK and you want to know the chance that someone has AA.
Let
a = number of players still to act
f = the number of players before you who have folded
P = probability that someone still to act has AA
Then the formula is
This can be expanded to any pocket pair.
Let
z = the number of overpairs to your pocket pair. For KK z=1, QQ z=2, ... 22 z=12
Then the probability that a player still to act has a higher pocket pair is
This can be plugged into an Excel spreadsheet as follows
You can get the preflop overpair calculator spreadsheet by e-mailing me. I used to have a spreadsheet online put the FTP site I was using is gone.
Plugging some numbers into the spreadsheet yields these results. Remember this model requires that anyone who acted before you does not have an overpair to your pocket pair and has folded.
UTG in a 10 handed game with KK, there is a 4.0% chance someone has AA.
UTG in a 9 handed game with KK, there is a 3.6% chance someone has AA.
UTG in a 6 handed game with KK, there is a 2.2% chance someone has AA.
UTG in a 10 handed game with QQ, there is a 7.9% chance someone has AA or KK.
UTG in a 10 handed game with 22, there is a 39.6% chance someone has 33-AA.
Folded to the button in a 10 handed game with KK there is a 0.9% chance someone has AA.
Folded to the button in a 10 handed game with QQ there is a 1.8% chance someone has AA or KK.
Folded to the button in a 6 handed game with 22 there is a 10.6% chance someone has 33-AA.
