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Villain was playing a little bit erratic. He drew 2 on the last draw twice in his first two hands, but wasn't out of line since. Maybe he was advertising, I'm not sure.

Poker Stars $2/$4 Limit 2-7 Triple Draw Lowball - 4 players - View hand 1563540
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Pre Draw: (1.5 SB) Hero is BTN with T 7 7 3 2
CO folds, Hero raises, SB 3-bets, BB folds, Hero calls

First Draw: (7 SB) (2 players)
SB draws 2, Hero draws 2
Hand: K J 7 3 2
SB bets, Hero calls

Second Draw: (4.5 BB) (2 players)
SB draws 2, Hero draws 2
Hand: J 7 7 3 2
SB checks, Hero checks

Third Draw: (4.5 BB) (2 players)
SB draws 1, Hero draws 1
Hand: J 7 3 3 2
SB checks, Hero checks

On the turn I was expecting him to bet upon improving, given that he bet the flop unimproved.

Should I bet the turn? If he calls and draws one (or two, I guess), do I snow or draw one to the Jack?

i bet intending to draw 1 here. the FE from that bet should be great enough to outweigh the fact that we have a J draw for the times he c/c's and draws 1 to whatever.

Turn check is good, your misplay was holding the jack. Equity and implied odds are both better drawing 2 on the end (run some simple twodimes.net sims if you like)

i dont think id ever check here unless V c/c's alot of 1cds and c/r's every pat hand. i agree that as played drawing 2 may be best though.

fwiw when I am villain here vs decent players I will check the turn with my entire range, and CR the crap out of people who bet every 4 card ten and jack...

ok, but noone does that, so betting T's and J's is still exploitative. seems like a cool line OOP though. do you c/r 1cds as well as pats?

Turn check is good, your misplay was holding the jack. Equity and implied odds are both better drawing 2 on the end (run some simple twodimes.net sims if you like)

Agree with this completely. Your hand is often face up as a very weak 1-c draw with this line and it allows villain to play pretty well vs you on the river, value betting you very thin with no consequences.

I didn't know that drawing 2 there has better equity. Learned something new.

DD, tell me more about this check-raising business, please.

i welcome me to the forums

id like to hear more about c/raising turn? 1cd? pats? DD?

As a starting point (don't have time to write too long now and think this is best thought about on your own some also, and finally I might be wrong as a lot of good regs don't do this) - assume you check turn OOP in 2:2, if villain bets every time he improves to a 4 card nine, any pat jack or better, and then some random snows, he is betting in that spot A LOT, you can punish this a lot better by allowing him to do so and then balancing your reactions. Also the worst case scenario of missing a bet is not that bad when its a 2:2 turn spot because usually that occurs after the flop goes check, check.