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
An intuitive way of getting it is making the number of doors way higher.
Imagine you have 10,000 doors in front of you. You choose one. Then someone opens 9,998 doors, revealing that they are empty.
Two doors now remain:
– the one you chose facing 10,000 closed doors,
– and one out of the 9,999 other doors, all of the others being empty.
Do you keep your initial door, and trust your unreal luck (1 chance out of 10,000) or do you change to the only door that is left closed?
Pretty instinctively, you’ll want to change, right?
Think of it this way :
When you choose you door, the probability that it contains the prize is 1 out of 10,000.
Correlatively the probability that the prize is in the set {all other doors} is 9,999 out of 10,000.
When the doors are opened, the set {all other doors} still exists, you just have the information that 9,998 doors in it are empty, except for one.
It is as if the remaining closed door captures the 9,999/10,000 probability by itself. It has “become” the set {all other doors}.
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