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I'm trying to understand what you're saying but are you saying you check call this flop if it's deeper? If you c/r and he pushes are you folding?

If it's deeper it changes the hand considerably since he doesn't push when checked to and you actually do have folding equity vs. whatever he has assuming not a set.

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Against a big stack this might be pretty neutral EV.

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Why? The numbers don't change because of the deeper stack it's just more variance.

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And against a deep stack, I c/c here. If he has an overpair, he'll often get spooked by your call of a big bet and give you a free river. If I hit on the turn, I play accordingly. If I don't hit, I fold the turn, unless he's giving me the right odds to call.

Against a short stack, I just put him in.