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TT min-bets the flop, gets called. check check on turn. you value bet the river after being checked to, he looks you up about 50% of the time

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FYP, and you just won a massive $3 pot. Calling that minbet is effectively checking behind on that flop.

If TT min bets and you raise, TT is probably calling. Turn Check/check, river check/call, you just won an extra $4 bucks (double the pot).

Edit re. TT calling the flop raise: in my experience, a min bet means absolutely nothing more than a check. Villain checked the flop. If he raises TT (unlikely) then you fold, shucks, we just folded the best hand on a [censored] flop. move on, [censored] happens.

If AQ min bets and you raise, he is either 3betting (easy fold) or smooth calling and leading turn (easy fold). If a queen doesnt bet the turn, make a note on the stupid donkey and move on.