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The player on the button doesn't seem very aggressive overall, so i wouldn't count on him firing out another bet on the turn without a pair of Kings or better. I definitely wouldn't expect him to call more money into the pot with worse. Basically your options for a non club turn are going to be much worse than they are on the flop, and a non club turn is most turns.
What % of his range hits the flop (A-K, A-Qs, 10-10+)?
18 combinations of A-Ko, A-Ks
6 combinations of A-Qs
18 combinations of 10-10, J-J, Q-Q
(though with J-J/Q-Q i would expect BTN to check behind a good % of the time, probably close to 100%)
12 combinations of K-K, A-A
Given a reasonable c-bet range, you probably only fold out about 1/3rd of his likely holdings by check/shoving the flop. Though against the range of hands that would call a shove (A-Ko, A-Ks, K-K+), you do have about 35% equity.
Board: Kc 3c 6h
Dead:
equity win tie pots won pots tied
Hand 0: 65.138% 65.14% 00.00% 9673 0.00 { KK+, AKs, AKo }
Hand 1: 34.862% 34.86% 00.00% 5177 0.00 { AcJc }
An EV analysis:
Hero wins 50BB 100% of the time, 33% of the time. (50 * 1.00 * 0.33) = 16.5
Hero loses 120BB 65% of the time, 33% of the time. (120 * 0.65 * 0.33) = -25.7
Hero wins 120BB 35% of the time, 33% of the time. (120 * 0.35 * 0.33) = 13.9
50BB = [(10xBB preflop 3bet * 3players) + (20BB c-bet)], Hero check/shoving the flop and winning the pot uncontested.
120BB = Hero check/shoving the flop and losing/winning the hand when BTN calls.
(16.5 + (-25.7) + 13.9) = 4.7BB
It's +EV given that, due to the relatively shallow stacks, you win a nice sized pot the times villain folds to your check/shove.
This doesn't mean your preflop decision was correct, though. Even with a weak player in the 3bet pot, calling with a hand like A-Js can put you in some tough spots on the flop (like when you hit TP and are faced with a bet).
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