The Host’s Last Door

On a game show, three closed doors hide one new car and two goats. The prize locations are chosen at random. You select one door but do not open it. The host knows what is behind every door and always follows the same rule: he opens one of the other doors that hides a goat, then offers you the chance to exchange your original choice for the single remaining closed door. If he has a choice of two goat doors, he selects either one at random. Your goal is simply to maximize your chance of winning the car. Should you keep your first choice, switch to the other unopened door, or does it make no difference?