Quote
naddok
If everything is done completely randomly, with no knowledge by either party as to where the prize is, then 1/3 of the time, you will initially pick the door with the prize, 1/3 of the time the host will end the game by revealing the prize behind one of the doors, and 1/3 of the time the prize will be behind the door neither you nor the host picked. If the game is still going after the host reveals what's behind the door, then half the time the prize will be behind your door, and half the time it will be behind the other door. The odds are not better by switching. But of course, they're not worse either. Only if you know there is less than a 1/3 chance of the host revealing the prize can you conclude that switching is better, and that is where intention comes in. I think Rade included this in his explanation, assuming 0% for the chances of the host revealing the prize.
First of all, I stipulated that the door opened by the host did not contain the prize.
However, even in your example where the host is choosing the door to open at random, if there happens to be no prize behind his door then you're still better off switching.
Either he reveals the prize, in which case you're not better off switching (because you're not winning either way) or he doesn't reveal the prize, in which case you're better off switching.