Screw the experiment. The page you found has a better explanation:
"The probability of picking the wrong door in the initial stage of the game is 2/3. If the contestant picks the wrong door initially, the host must reveal the remaining empty door in the second stage of the game. Thus, if the contestant switches after picking the wrong door initially, the contestant will win the prize. The probability of winning by switching then reduces to the probability of picking the wrong door in the initial stage which is clearly 2/3."
TM
