Originally Posted by Paul
For anyone else, like me, who finds it hard to understand why the non-intuitive solution to the Monty Hall problem (that one should choose to switch doors with a 2/3 chance of getting the prize) is correct, it might help to think of it this way.
Suppose you choose door one. Then Monty Hall eliminates as an option one of the other doors, and gives you the choice of switching to the remaining door, or staying put. You take that option and switch to the other unopened door. This is equivalent to Monty saying, "you can stick with door one (1/3 chance of winning the prize), or I can open both doors two and three (2/3 chance of winning the prize)."
That's very close to the way it was explained to me that made the correct solution "click" in my head. Another way of saying it is that by allowing you to switch from door one, Monty is in effect allowing you to choose both doors two and three.
The probabilities always have to add up to 1 (i.e., there's never NOT a prize behind one of the three doors). When you initially choose, each door has a 1/3rd probability of being the correct one. After Monty opens one of the doors with a goat, the door you originally picked still has a 1/3rd probability, but the goat door now has a 0/3rd probability. Since the probabilities still have to add up to 1, the remaining door now has a 2/3rd probability.
Incidentally, Battlestar Galactica is an awesome show (maybe even the best show on TV now that The Wire is over), but even as a thumbnail description "humans fighting robots" is highly misleading given that at this point there aren't enough humans left to fill most baseball stadiums.