Two goats and a car

This is an interesting maths problem taken from Mark Haddon's excellent book "The Curious Incident of the Dog in the Night-time":

You are in a game show on television. On this game show the idea is to win a car as a prize. The game show host shows you three doors. He says that there is a car behind one of the doors and there are goats behind the other two doors. He asks you to pick a door. You pick a door but the door is not opened. Then the game show opens a door that you did not pick to show a goat (because he knows what is behind the doors). Then he says that you have one final chance to change your mind before the doors are opened and you get a car or a goat. So he asks you if you want to change your mind and pick the other unopened door instead. What should you do?

Are the odds of you winning the car higher, lower or the same if you change your mind and pick the other door?

I will supply the answer with a diagram as explanantion in about three days. In the meantime please explain your answers in the comments feature. I was pretty bowled over by the answer myself, and yet it was mathematically flawless.

If you need more time just let me know, too.