Why does the probability “lock in” and not change if you remove a variable but doubles for the other one?
I read it multiple times and it still doesnt make any logical sense to me.
In: Mathematics
The probability doesn’t change because nothing about the situation has changed.
When you pick a door, you know you have a 1/3 chance of being right, and a 2/3 chance of being wrong. In other words, there is a 2/3 chance that the prize is *somewhere in the other two doors that you didn’t pick.*
You also know something for certain about those other two doors: at least one of them is empty. If you picked the right door to begin with, then the other two doors are both empty, but if you picked wrong, then one of the other two doors is empty and one has the prize. Either way, there’s at least one empty door.
When the host opens one of those other two doors, it will always be empty, because he knows where the prize is.
He has shown you at least one door that is empty. You already knew that door was there…you just didn’t know which one it was. The dope showed it to you.
So there are still two sets of doors: the door you picked, and the sum total of the other two doors, with the known property that at least one is empty. There was a 1/3 chance the prize was in your door, and a 2/3 chance the prize was somewhere in the other two doors. That hasn’t changed! But since you now know which door is definitely empty, you now know which door has the 2/3 chance of the prize.
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