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After the c/r, i think sets and flush draws/ pair + flush draw make up the majority of his range.
Text results appended to pokerstove.txt
22,770 games 0.005 secs 4,554,000 games/sec
Board: Th 8h 4h
Dead:
equity win tie pots won pots tied
Hand 0: 41.383% 40.79% 00.59% 9288 135.00 { KcKs }
Hand 1: 58.617% 58.02% 00.59% 13212 135.00 { QcQh, QdQh, QhQs, JcJh, JdJh, JhJs, TT, 88, KhQh, AcKh, AdKh, AhKc, AhKd, AhKs, AsKh, AcQh, AdQh, AhQc, AhQd, AhQs, AsQh }
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You have to shove over the check/raise. However, given your note that he likes to make big thin calls, i prefer checking the flop and betting non heart turn and rivers. His calling range on later streets would also be wider than his c/r'ing range on the flop. For example, he may look you up with hands like 9-9/J-J no heart (Ah-K/Qx?!) for stacks on the river if he is convinced you are betting with a heart draw that missed.
Without reads/notes i think betting the flop is best, but since villain likes to make big thin calls we can get bets in on the turn/river with substantially better equity. If a heart falls on the turn/river i would be trying to get to showdown as cheaply as possible - maybe a very small value bet on the river since we are IP. Note that a flush completes on later streets only about 1 time in 3, so we can use this line for >EV most of the time, and have to check to showdown/fold much less often.
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