Re: AQ....brain is mush.
Given Button's numbers, his 3-betting range in this spot looks alot like TT+, and AK.
You beat 12 combos of AK and 6 combo of JJ.
You lose to 16 combos of AA,KK,QQ,TT.
So you are a little bit ahead of button here. Is he going to bet AK in this spot? If so I would go for a check raise to protect my hand.
If he can take the free card in this spot, then I like betting out, if we get 1 caller, he won't get too tricky and if he raises we will know that we need to improve in order to beat his hand. Also betting out makes sure that money does go into the pot on the flop when we are ahead.
We might even get UTG+1 to raise with something like KQ.
Against this villain, I feel betting out might be better than check raising since his numbers suggest that he is more likely to check behind with AK in this spot. Is this a reasonable assumption?
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