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I have to say i hate this situation because you really feel that he has a flush, and if he shows down a flush and you lose, you say "dammit i knew it" and lost yourself another bet on the river...But if you think about the math of your expected pay off compared to the fraction of the time he actualy has the flush, i think the right move is to bet.
I lead out 3$ or so on the river. Thats 3$ into a 7$~ or so pot, 10$ if he calls. A little better than 3:1 if he calls, and a tiny bit of folding equity. If he made a flush, you're getting raised and can get away from it, at least you know you folded to a better hand. Checking gives him free reign to take the pot down with nothing. I think more than 25% of the hands in his range of hands that play this way are two non-hearts. And when you're getting 3:1 by betting 3$ on the river, you only need him to not have a flush 1/4th of the time or more.
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