BB turns weak pair after flop is checked around

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"grunch"

Call. Villian may just have a pair of sixes and since you checked after taking a stab, he may be trying the same. If not, this is one of those times where paying for information is worth it. If he slow played top pair, limped in with a big pair, or hit the jack on the river, it would be worth one bet to see what he is willing to limp in with in fairly early position or see if he's capable of tricky play. Plus, he'll be less likely to bluff in the near future if he knows you'll make him show.

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i think this is OK since it's your first hand at the table (good way to find out who to 'value bet' later on)

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I check/fold. I don't think this is that great of a spot to be calling a river bet getting 4.5:1.

"Calling down for a read" is not a good reason to call here.