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People come up with the incorrect result because with 2 doors remaining, it would initially appear that they both have equal chance of containing the car. Becau

People come up with the incorrect result because with 2 doors remaining, it would initially appear that they both have equal chance of containing the car. Because it is unclear HOW we arrived at the final two doors. If we arrived there randomly, the probability is in fact 50-50. If there was some manipulation - i.e. if the host knows where the car is AND deliberately opens doors that do NOT contain the car - then the probability is not 50-50, and switching away from the initially-chosen door is advantageous.

defterGoose 6y ago
I find it a little more intuitive in a visual way with the 100 doors. If the 98 doors that the host subsequently eliminates leaves one in the middle of the pack (e.g. #52), most rational people would look at that door and say "hmm its probably in there rather than in the one i picked, #1"
- thaumasiotes 6y ago
  Eh, the formulation I like goes: 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?
- listenallyall 6y ago
  Except how many people, when presented with option of picking one of a hundred doors, picks Door #1? 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.
galaxyLogic 6y ago
I came up with a slightly different reasoning why you should switch: When the host is about to open a door there are 2 possibilities, he can either choose between 2 goats, OR he can choose between a car and a goat. 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?

defterGoose 6y ago

I find it a little more intuitive in a visual way with the 100 doors. If the 98 doors that the host subsequently eliminates leaves one in the middle of the pack (e.g. #52), most rational people would look at that door and say "hmm its probably in there rather than in the one i picked, #1"

thaumasiotes 6y ago
Eh, the formulation I like goes: 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?
listenallyall 6y ago
Except how many people, when presented with option of picking one of a hundred doors, picks Door #1? 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.

galaxyLogic 6y ago

I came up with a slightly different reasoning why you should switch: When the host is about to open a door there are 2 possibilities, he can either choose between 2 goats, OR he can choose between a car and a goat. 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?