sandwich's Comments
It seems to me that after the host has opened one of the doors, each remaining door now has equal odds of being the correct choice. Now, when the host asks whether you want to switch, he is effectively asking you to choose again, but now the choice is between two doors, rather than three. Choosing the same door you did in the first place after the "choose again" prompt is effectively the same as having not switched with the "Would you like to switch?" prompt. So the chance is 50/50. Am I wrong with this interpretation?
Of course, if the game is being manipulated in response to the player's choice, (ie, the car switches doors, the player is only given this choice if they chose correctly, etc) then questions of probability are irrelevant.
Of course, if the game is being manipulated in response to the player's choice, (ie, the car switches doors, the player is only given this choice if they chose correctly, etc) then questions of probability are irrelevant.
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When you first choose, there is a 66% chance you picked the wrong door. This means there is a 66% chance that the correct door is one of the other two.
Then one of those two is eliminated. This does not change the fact that there is a 66% chance that one of those two is correct, except now you know one of them that isn't the correct door. So there is a 66% chance that the other door is the right one. You should switch.
I think the hang up for most people (certainly for me) is that it seems like the choice between two doors is independent of the original choice between three doors. What I just realized is that they aren't independent, because your first choice helps determine what doors will be available for the second choice.