In more than 50% of the cases the host encounters the situation where he must choose between a car and and goat. Why because in > 50% of the cases the user's initial choice does NOT have car behind it, so the host must choose between a goat and a car in > 50% of the cases.
And of course he chooses the goat. So in > 50% of the cases the remaining door will have the car behind it,
So by switching, you have > 50% chance of winning the car. Is this the correct answer? It would seem to me your chance to win if you switch is 1/2 x 2/3. Is this correct?
What is counter-intuitive is why the action by the host should give us any information about which door has the car. And it does not do so always. It only gives us information in 2/3rds of the cases. If the host chooses between two goats that should not give us any information at all should it? But that is good enough for us since we play the odds, and switching seems to increase our odds.
But, now I have my doubts. This kind of probabilistic reasoning would seem to apply only if we repeat the experiment many times. But no contestant gets to play this game more than once. Probability theory does not tell us anything about what happens in a single experiment. So is it rational for the player to think they should switch if they can play this game only once?
How much should they be willing to pay for the option of being able to switch the door, if they know they can only play this game once?
Further, the problem with the 100-door (or 1000, or million, etc) rationalization is that it did not appear in the initial Parade Magazine article which generated all the negative responses. Sure, today you know the secret and can explain the problem well, but it isn't fair to judge the people whose only exposure to the issue was the first, original Parade Magazine article.
You pick a door.
The host runs up and knocks on one of the other doors, without opening it or saying anything. He just indicates a door.
Now you have the option to stick with your one original door, or to take both of the other two doors.
Which is better?
And then: how is that scenario different from the one where he opens the special door and reveals a goat?
The probability that car is behind the door NOT selected by the host is 1/2 - because the host had only two choices which are equally good for him.
Probability that car is behind the door initially selected by the user is 1/3 because that is what happens when you randomly choose one out of three.
We have to think in terms of two different probabilities: a) That user selects the correct door initially and b) That door not selected by the host has the car.
But yes it is counter-intuitive. I can't quite point out which step in the flawed reasoning of no benefit from switching occurs. How do people come up with the incorrect result, and what are the flawed assumptions they make that cause them to arrive at the wrong answer? Where exactly does the logical error happen?