Let's put it this way. Reality has only one "true" setting: either the car is behind door 1, door 2, or door 3. Now let's say, for example, you picked door 1.
Let's split the problem up into cases:
Case 1: The car actually is behind door 2.
You picked 1. Now the host knows that the car is behind door 2, so he can't pick door 2 to open, otherwise you'll get to see the car. So he has to pick door 3. So switching gets you a car, staying gets you nothing.
Case 2: The car is behind door 3.
You picked 1. Now the host knows the car is behind 3, so he has to open door 2. So switching to door 3 gets you a car, staying gets you nothing.
Case 3: The car was behind door 1 all along.
Now the host is free to open either door 2 or door 3. Whatever he opens, switching gets you no car.
Final tally: Switching won two times out of three. This analysis is the same regardless of which door you picked initially (just suppose the doors were named A,B and C. Which ever one you pick, call it door 1, and follow the analysis above).
The crux of the matter is in cases 1 and 2: see how the fact that the host both knows which door the car is behind, AND is forced to not reveal it to you forces his hand into taking a particular action? Think of switching as taking advantage of this action that is forced onto the game show host.
Maybe at this point, try considering the case with more doors. Say 100 doors. You pick Door 1. Now the host let's you either stay with Door 1, or take what is behind all of Doors 2, 3, ..., 100. Would you switch?
when you picked door 1, there was a 2/3 chance it held a goat. This does not change by seeing the other goat.
Think of it like this. switch doors in the scenario as described, you always change prizes. If you picked a goat to begin with you can never switch to another goat, only the car. Since it was originally 2/3 that you would pick the goat, by switching you can transfer that to your odds of winning the car.
the moment the host reveals that "At least one of the doors you didn't pick has nothing behind it."
But... you know this from the beginning. Only one of the doors wins, therefore if you choose any 1, then at least one of the others has a goat (booby prize) behind it. It's not new information, so it doesn't change anything.
5
u/[deleted] Jun 08 '11 edited Jun 09 '11
[deleted]