Small Stakes Hold Em question

[maxsick]

In the section on "Pot Odds" (pgs. 27-31), Ed Miller describes a scenario where you are playing heads up in a $2/$4 game. Your opponent has four outs to the winning hand against your top pair. The pot contains $44 on the turn. The bet size is $4. If you bet, your opponent will be correct to call with 12-to-1 pot odds. If you don't bet, you are offering your opponent infinite odds (by allowing him to check behind for free). Is there are way to profit in this scenario?