Barreling the turn has been a hot topic lately in the BC, so I'm acting appropriately by asking about a fairly easy-to-understand scenario that teaches a whole lot about barreling.

We raise pre-flop and Villain calls. We see a heads-up flop OOP. We c-bet and Villain calls. The turn comes. We have a gutshot to the nuts, but no other outs. Suppose the turn pot is $3. If we bet $2 as a second barrel, how often does Villain have to fold for our barrel to be +EV?

For this example, let's assume that we never get raised on the turn, but that we also never make any extra money on the river the times we hit.