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20 hands on villain, no read. Played a few hands so it's hard to say but did not seem in the extremes, i.e. not a nit nor loose.

Poker Stars $0.02/$0.05 No Limit Hold'em - 6 players - View hand 1761081
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Hero (SB): $13.16
BB: $5.38
UTG: $12.20
MP: $5.31
CO: $6.33
BTN: $3.97

Pre Flop: ($0.07) Hero is SB with A J
1 fold, MP raises to $0.20, 2 folds, Hero raises to $0.65, 1 fold, MP calls $0.45

Flop: ($1.35) T 9 Q (2 players)
Hero bets $1.10, MP raises to $2.50, Hero ??????

The question is what do I do next, not should I have folded pre.
To his raise,
I can fold
I can call but we are all in on the turn.
I can shove but I doubt villain folds to a shove.

Its a fold IMO, your basically drawing dead to a K, if an A come hes most likely made two pair AQ or maybe A10, or he has a set just QQs 10s Worst you can hope for is AKs in which case your still beat I dont think he's ever doing this with worse, And as you say its all going in on the turn.

You need to do some work in pokerstove to see how your hand and this board does against his range, how are you doing against a bare FD, set, AQ, KQ, etc. Then you will know the answer

We should never call given stack sizes, and the only reason we want to stay in the pot is to spike our straight (so we want to see two cards not just one).

EV (fold) = 0 by definition

Board: Th 9s Qs
Hand 0: 28.892% { AdJs }
Hand 1: 71.108% { QQ, TT-99, AQs, AsTs, KJs, Q9s+, JTs, T9s, AQo, KJo, QTo-Q9o }

EV (shove) = (.289[our equity])*($10.62[total pot]) - (.711[villain's equity])*($3.56[what we still have to risk]) = $0.54

So shoving vs this calling range with no fold equity is profitable, but is very high variance. Even if you take out, AQ, ATs, JTs, QJs our equity stays close to the same and is still profitable

The EV calc isn't right, you're mixing up a couple of the methods. You can't let us win our equity in the whole pot including our bet while only allowing us just to lose villain's equity from our bet. We trade our entire bet for our share in the total pot.

Yep, should be (making the assumptions that you clearly intend to make re him never folding etc.):

EV(shove) = (.289 *$7.06 - .711*$3.56) = -$0.49

Yep, should be (making the assumptions that you clearly intend to make re him never folding etc.):

EV(shove) = (.289 *$7.06 - .711*$3.56) = -$0.49

Agreed! Given the change in EV, it's a fold!!!

Fold the flop. he has too many value hands and we have zero FE given the SPR.

I'm just curious but why are you 3 betting with AJ off out of position? IMO I don't think AJ is a good enough hand to 3 bet with out of position, especially if you aren't 100% sure what your opponents tendencies are. Why not just call and see a flop??? Also this is a very draw heavy board, and usually good players will check-raise you here. A check might have been better for pot control.