[W. Deranged]

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An ace normally check-calls or check-raises.

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I don't know. I've never been in a live low limit game where a c/r on the flop was normal.

I'd estimate the bettor has worse than an ace 65% of the time, and the combined probability of no ace in 3 random hands is also around 65%. Since their hands should be at least slightly better than random, lets bring that % down to 50. That leaves you good on the flop .50 * .65 = 33% of the time. Do as you wish. You are getting 1:9 on a call and 2:9 +/- implied on a raise.

If the table is passive enough, I might raise and check behind to get to the river for two small bets. If it is a bit more aggressive I'd probably muck as you did.

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Raising the flop and then checking behind the turn with a made hand that has little chance of improving is almost never correct.

In fact, it makes me vomit a little.

I see poor low limit players do this all the time: they don't want to risk a lot so they try this kind of raise-check line, not realizing that if they have enough equity to get to the river for 2 SB they usually have enough equity to get there for 2 BB and checking the turn can have disastrous consequences when you give a free card to beat you or miss out on value agianst a draw.

In general, if I'm trying to get to the river cheap I either call down or raise the flop and bet the turn with the intent of checking the river behind. But this sort of raise the flop check the turn [censored] is generally just bad poker.