[sharpie]

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Assume the BB and all of the limpers will always call. If I raise, I'm investing $5 more to get $12 more into the pot. So, in a way, I'm giving myself 12-5 or 2.4-1 "odds" on the raise.

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This would be correct if you was going to either raise or fold, but it's between call or raise, so you only count the odds you're getting on the $3 raise.

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Now let's instead pretend that we're playing some bizarre structure where I post a live $5 blind but it still remains $3 to call preflop. If I "raise," I'm only putting in another $1 to get the same $12 into the pot, giving myself 12-1 "odds." In theory, it would be then profitable to raise with anything I'd be willing to take 12-1 odds on to strictly call.

Am I making the slightest bit of sense at all?

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This is true, but doesn't really relate to the actual hand.