THE LET'S MAKE A DEAL PARADOX

You are a contestant on a quiz show.  The quizmaster shows you three closed doors.   The grand prize is behind one of them.  You are invited to select one of the doors but you may not open it.  The quizmaster opens one of the two remaining doors and you see that there is nothing behind it.  He makes you the following offer:   You can keep whatever is behind the door you have selected or you can trade for whatever is behind the other unopened door.  What should you do?

This problem is more subtle than it appears.  Which course of action will maximize the probability that you will win the grand prize?  Hint.