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> Opening the first door obviously gives you a 1/3 chance, which remains unaffected by the host opening another door.

You had a 1/3 chance if you opened the second door too. Why does the second door's chance change, but the first door's chance is somehow locked by you having chosen it?

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The chance is locked in because, by the rules of the game, the host can't open that one and show you anything -- regardless of what it contains. You know that there's at least one door that the host can open (of the two remaining doors), so them opening a door changes nothing about the door you've chosen.

First door as in the first door to be selected. When you select it, it has a 1 in 3 chance of being the correct door. The door presented to you obviously has a 0 in 3 chance, so the remaining one must be 2 in 3.

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