[Xhad]

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Thats my problem x. The thing is, although its the same mathematically, its a different situation.

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No it is not. If you KNOW he will call and you KNOW his hand is better than yours and you do not have pot odds to call a bet from him, you are still risking the same amount of money to win a pot of the same size. It is the same thing.

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If i check, he is correct in betting while i am incorrect in calling, and this would reflect if we played this out many times. However, if i bet out and he just calls, he makes a mistake according to the fundamental theorem of poker and therefore i gain what he misses, that is 1 BB.

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The problem is that you were incorrect to bet the worst hand in the first place under all the assumptions you're making. Look at it this way: If he would bet if you check, but only call if you bet, then betting or check/calling both result in you paying 1BB to see the next card. Thinking the EV in these two situations is any different shows a fundamental misunderstanding of the entire concept. There is no Poker God out to punish people for making a "mistake".

Put another way, you lose less when he calls vs. raising; you do not gain anything vs. checking.

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I think the reason it is "correct" to bet is because, in theory of course, we all face the EXACT SAME situations WITH THE EXACT SAME CARDS if we were to play out infinitely long. So while its true that he misses a bet and makes a mistake, i gain because when IM faced with a bet and hold the same KQ against JT, and I raise, i dont lose anything and therefore and 1 BB ahead of my opponent. If i were to just call as well both of our mistakes cancel eachother out in a sense.

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If he would call when you bet, but bet if you check, then betting is just doing his dirty work for him, regardless of whether or not he philosophically made a "mistake". If he would bet if you check, or raise if you bet, then you're intentionally putting money in with the worst hand which obviously can't be right.

The game does not "know" why money goes in, only that it does go in. If you bet when you know you're going to be called, it's the same thing as calling even if your opponent "should" raise.