[milesdyson]

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. . . I understand the number Gris was using, but I will avoid trying to analyze the betting vs. checking question for now as I'm feeling too lazy.

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On this board, in this situation I can't imagine a case where we bet a hand that our opponent will call with (or raise with), but will check behind if we check. And betting stops him from continuing to bluff with his KJ or 54 UI.

On this board, I think I agree that we probably aren't ahead 67% of the time, but it's close given OP thinks the villain might play AA or KK or AQ this way. Give us some higher hearts, unpair the board, even make the '3' something higher, and I think we're there (well, certainly if we unpair the board.) It's a useful comparison number, though probably 64-65%, given the chance of an overplaying river 3-bet, is better.

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How could we not be good 67% of the time? I'd guess more like 80%.