On the Monty Hall game show there is a three door pick-a-door for a prize segment; it goes like this: The person picks a door. Immediately another door is opened which does not have a prize behind it. Now you are asked it you want to change your choice of door, with only the two remaining doors unopened. Part of the Mythbusters test was to see if people would stick with their original choice, or change. The claim is that most people will stick to their original choice. In fact, in the test, 100% decided to stick with their first choice, confirming that part of the test.

The part that confuses me is the second part. Mythbusters states that it is statistically wiser to switch doors. How can that be? What have I missed? It seems like with two doors remaining, you have a 50/50 chance of winning no matter which door you pick. Why is it that they state that statistically you should always switch doors for a better chance?

Are they saying that there is a 1/3 chance in the initial selection and that once one door is opened, it makes leaves the remaining door with a 2/3 chance and the original door still has a 1/3 chance? This is very difficult to see.

Besides simulation, is there a good way to prove it? It would be useful in creating gambling propositions, if it holds true.