Some of the historical confusion on this problem revolves around the point "After you pick the door, a 3rd party (ie the gameshow host) opens up a door which does NOT have the prize."
If the gameshow host has a choice in whether they open a door or not, there is no way to know for sure what option is better. In the original gameshow the host did not always open the door, nor offer the option to switch.
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I would argue that if he doesn't open a door then this is no longer an interesting problem because you are just randomly trying to choose a prize out of 3 doors. The problem is only interesting to consider once he opens a door. In terms of your choice to switch or not, that would be a decision that you make after a door is opened (or in your scenario not opened), so you would only apply the logic described above if the host did open a door. Otherwise, you are just in a random guessing game.
Yes, definitely. But it's partly the variations on the problem wording which has created some of the confusion as to the probability. Often when people repeat the problem they don't clearly indicate the host must open a door.
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Some of the historical confusion on this problem revolves around the point "After you pick the door, a 3rd party (ie the gameshow host) opens up a door which does NOT have the prize."
If the gameshow host has a choice in whether they open a door or not, there is no way to know for sure what option is better. In the original gameshow the host did not always open the door, nor offer the option to switch.
I would argue that if he doesn't open a door then this is no longer an interesting problem because you are just randomly trying to choose a prize out of 3 doors. The problem is only interesting to consider once he opens a door. In terms of your choice to switch or not, that would be a decision that you make after a door is opened (or in your scenario not opened), so you would only apply the logic described above if the host did open a door. Otherwise, you are just in a random guessing game.
Yes, definitely. But it's partly the variations on the problem wording which has created some of the confusion as to the probability. Often when people repeat the problem they don't clearly indicate the host must open a door.