Brian states: "The issue is how I can persuade anyone that my answer—or any particular answer—is correct."
Here's another attempt to provide an intuitive insight into why 2/3 is the correct answer...
Suppose there are 100 doors, and all other conditions of the original 3-door puzzle remain intact.
The player chooses one door.
Would everyone agree that the chances of the prize being behind one of the remaining 99 doors is 99/100 ? (And, that the door chosen by the player has only a 1 in a 100 chance of containing the prize?)
"Monty" opens 98 of the remaining 99 doors.
What do you think the chances are of the prize being behind that one door that "Monty" did not open ?
99 to 1, or 1:1 (50/50) ?
I think you'd have a strong feeling that the chances are 99:1, and that you should switch doors. Right ?
Apply that same logic ("feeling") to the 3-door situation and you are left with the choice:
2:1 or 1:1 ?
It seems then, applying the same logic, that the 2:1 odds apply to that remaining unopened door, and that switching doors would be in your best interest.
Hope this analogous problem helps some of you get by the mind-bending nature of the 3-door problem.
posted by Peter Rauch
August 25, 2008