Not convinced? Consider the situation before the MC opens the door. There are 3 possibilities each equally probable:
Your choice............Door 2................Door 3
Goat.......................Goat...................Car
Goat.......................Car.....................Goat
Car.........................Goat...................Goat
After the MC opens a door the situation has changed to the following, with each line still having equal probability:
Your choice............Door 2................Door 3
Goat.......................Open..................Car
Goat.......................Car.....................Open
Car.........................Goat/Open.........Open/Goat
Let's see what happens if you stick with your initial choice. You will get the car only in case 3 ie your chance is 1/3.
If you switch you will get the car in the first two cases but not in the third, so you have a 2/3 chance.
"OK," you say, "that is the logical solution, but it doesn't make sense!"
The key to understanding why switching doors doubles your chances is that the MC does not choose which door to open in a random way. He will always open the door that hides a goat. If he had chosen the door to open randomly (and still revealed a goat) then your chances would be unaffected by switching.
If someone were to arrive after the door was opened and they were only told that one of the remaining doors hid a car, the other a goat, then their chances would be equal for each door. The difference is that our knowledge has been upgraded because we know that the MC always shows a goat.
Another way to see the correctness of the above solution is to increase the number of doors to 10,000. Imagine that you had initially picked door number 7,259 and that the MC opened every door except yours and door number 2,190, revealing a goat in every one. Would you switch? You sure would, as the chance that your initial choice was correct is only 1/10,000. That small probability is in no way increased by the MC showing you 9,998 goats, as he can always do so regardless of which door you initially chose.
Returning to the 3-door version, the same logic applies, as your initial choice of door 1 only has a 1/3 chance of being right, both before and after the MC opens a door. "But there are only 2 doors left!" you cry. Surely it is 1/2 now? No, it isn't. The reason is that the two doors have been selected in a different way. Your initial choice was random, but the host's choice to not open the door hiding the car was not. So that other door, the one with the car, is 'favoured' due to the way the situation has been set up.
A way to see this is that the MC's action is such as to collapse two possibilities into one in the first two cases, ie he has shown you which door has the car if yours does not have it. Hence by switching you double your chances.
My favourite solution was provided by my partner: "I'd smell the doors or slip a bit of carrot underneath them."
Now are you convinced? Let me know if you still think it is 1/2 each way.
I think this is a brilliant puzzle because a) it is very simple b) it is very hard, and c) it is profoundly counter-intuitive, forcing us to think hard about the nature of probability.
The key issue is that probability is not built into reality but that it is merely a measure of our rational expectations given the knowledge that we have. Thus a 1/2 chance of coming up heads is not built into every coin. In practice, if I threw up a coin 100 times and it was heads each time then we would strongly incline to the view that the coin was biased. It would take a brave man to lay equal odds on tails for the next toss.
In the Monty Hall problem the probability is not an inherent part of the situation but merely a measure of our expectations given what we have been told. In practice a coin always lands one way up or the other, and the car is always behind a single, specific door. Probability calculation refers to what we think will happen if a supposedly random process is repeated a large number of times, ie if we played the goat game 1000 times, say. Then we would find that we got the car roughly 667 times, provided we switched. Someone actually ran the scenario in a computer and got the expected results.
Cognitive psychologist Massimo Piattelli-Palmarini commented, "no other statistical puzzle comes so close to fooling all the people all the time," and "even Nobel physicists systematically give the wrong answer, and that they insist on it, and they are ready to berate in print those who propose the right answer." Pigeons repeatedly exposed to the problem show that they rapidly learn always to switch, unlike humans.