I have QQ in a moderately raised pot in LP. Flop is QT8 with 2 diamonds. My image at this point is quite tight (possibly WT) because I haven't caught many (if any) hands. It's checked around to me. Ok, I figure that anyone who doesn't bet their straight on that board is a fool, so I pot it.
Now, villain in ultra-deep stack check-raises pot. This guy is very LAG-ish. I've seen him check some nut hands, but I've also seen him buy quite a few pots. He has over $800 in a 1/2 game, and he's been quiet for quite some time (as an aside, this hand was the first "play" in a series of about 6 hands where he bluffed off his entire deep stack. But prior to this hand, he seemed pretty civilized for a long stretch--not sure whether the subsequent obvious bluffing was because he was upset that I folded vs. his nut straight with redraws or whether he had just gotten in the mood to start bluffing).
In any case, my pot-sized bet was around $35, and he makes it $135 or so to go. I feel like I can either push or fold and opted for folding.
The EV question for me is how often he needs to be bluffing for me to push here. Let's say p is the probability not of bluffing but of having some kind of J9. In that case, I'm going to say that I'm on average a 2:1 underdog. Flush draws are irrelevant. Worst case scenario would be something like TTJ9 because it blocks my boating outs, probably putting me at about 2.5:1 or so, I'd guess. But I think the average of 2:1 is fairly realistic.
How about if I am ahead? Ok, he can have a naked TT, in which case he's something like 20:1. If he has a flush draw with it, he's maybe 2.5:1 underdog. If he has something like AKJ with a flush draw, he's probably somewhere in the coinflip range. So, I'll put him at maybe 2:1 if he's behind. I don't know whether that's realistic or not. A lot of the time that he's behind, he's probably close to a coinflip, but I do think there's SOME chance that he's drawing near dead with just 2 pair or a lower set and not much of anything else. Those cases, drive the odds very much in my favor. So, I'll just call it 2:1 for me if I'm ahead (would be interested in opinions on that).
So, what does p (probability of his having the straight) have to be for me to push here?
Case 1:
He has the straight, and I'm 2:1 underdog. When I win, I win $35 + $35 + $200 = $270.
When I lose, I lose $165. So, it's (270-330)/3 = -20.
Case 2:
I'm ahead and 2:1 favorite. Winnings and losses same as before, so it's now (540 - 165)/3 = 385/3 = 130.
EV is thus 130*(1-p) - p*20 = 130 - 150*p.
130 - 150*p > 0 iff
130 > 150*p iff
7/8 > p
In other words, if he's bluffing at least 1/8 of the time, it's profitable for me to push.
Oh, well, I guess that means I should've pushed...
