[Spicymoose]

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If I believe I have the best hand, shouldn't I continue betting? What about if I bet to make sure he doesn't get a free showdown? Maybe he'll fold after a river bet?

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If he folds to a river bet, it is with a worse hand, so betting and checking have the same result. You don't bet just because you have the best hand. You bet because he will call with a worse hand. Giving him a free showdown doesn't matter he has a worse hand that would of folded.

Edit: furthermore, he has you beat often, so you are costing yourself money here. He will call with a lot of better hands that he wont bet with.