warning: this is a very in-depth post and I only recommend you read it if you know a thing or two about probability. This was a hand I experienced last night at my local casino. I want answers to focus around the math I have presented to see if there are holes in it anywhere (or my assumptions for the math), as my gut still says this is a fold, but the math says otherwise.

Important information to note: x4-x5 bb has been the standard pre flop raise at the table all night (which sucked btw). See the attachment for a visual of the hand. players 1. and 2. had me covered, player 3 had ~$190 before the hand. Player 1 is tight, passive. I would assign his PFR ~ 5. player 2 is a regular, and is loose, fishy, but his 3 bet range in early position is all but JJ+ (maybe AKs, but we are discounting this for argument's sake). Player 3 is LAG, and has been known to cold call large raises with marginal

hands. My image at the table over 3 hours was tight. If I shove they will give me cred for a premium hand.

Looking at the attachment, you will take note that there is already $99 in the pot once it is my turn to act. There are 5 players behind me yet to act, and player 1 might re-raise. For these reasons, I decide I will either shove or fold. I know player 2 fairly well, and know he will not call my shove with anything but KK or AA, and I am confident that is true for player 1 as well. Player 3 is much more likely to call me with a worse pocket pair.

If we assume player 1 and 2 (as well as the people yet to act) will fold anything worse than AA or KK, and will call otherwise (reasonable assumptions might I add), then the probability that anyone of the 5 players behind me yet to act were dealt KK or AA is (2/221)*5, or 4.5%. The probability that player 1, with a PFR of 5%, raised with KK or AA is 13.1% (8 combinations for Aces and Kings, 17 combinations for 88-QQ, and 36 combinations for AJs, AQo, AKo, thus 8/61 = .131). Finally, the probability that player 2 3-bet with AA or KK is 61.5%(8/13 combinations).

Each of these probabilities are independent of each other, thus, the probability that neither the 5 people behind me were dealt KK/AA, player 1 didn't raise with AA/KK, and player 2 didn't 3bet with AA/KK is as follows:

(1-.615)*(1-.131)*(1-.045) = .32, or 32% chance I will take down the pot uncontested.

**Thus, .32($330) + .68(.18($560) + .81($-230)) = $105.6 + .68($-85.5) = **__+$47.46__.

The above equation explained: a 32% i will take down the pot uncontested and win $330. Add that to a 68% chance that I don't - now I am a 4.5:1 dog against KK or AA, but I can still suck out, and do, 18% of the time, and take down $560. The other 81% of the time I lose my $230 stack. Yet, if I did everything correctly, This still seems to be a +EV play.

thoughts, comments, questions? I would appreciate input if possible.