The stupid 3-doors problem

[DWarrior]

Someone threw this at me a year ago, and I just rememberd it now again.

You're on a show and you have to pick one door out of three. Behind 1 of the prize, behind the other 2 is nothing. The agreement is that you pick your door, then the show host reveals one empty door and you have the option to switch to another.

Should you switch?

Obviously the odds of picking the first time are 1/3, but my friend and the CS professor claimed that the second pick is actually 2/3, because the remaining door somehow "combines" probability with the door that the host opened.

I argued that there is no way to know what's behind the second door, so as soon as the host opens the empty door, you have the option of choosing one door out of 2, so your odds are 1/2, which means that there is really no difference between switching and staying the on the same door.

So, 3 doors, A B and C, you pick A. Game host reveals door B as the empty one. Are the odds of door C having the prize 1/2 or 2/3 (B and C combined)?

Back then, I was awed by my CS professor, and considered him a wise man, so I did not argue (though I kept my theory), but a year later, I still think the odds are 1/2 on both doors.