Is the MIT Problem In 21 really right

NickTri

Aspirant
The problem is that you have 3 doors and behind one of these doors is a prize and behind the other 2 is nothing. Once you pick your door the host leaves it closed and instead opens ONE of the doors that you DID NOT pick and shows that there is nothing behind it. After you are left with 2 doors and the host offers you two choices: Stick with the door you have OR switch to the door that you did not pick. Many people say that it doesnt matter and that the choice is 50/50 but is it really? I wrote out all the possibilities and they show that the choice is indeed 50/50. Please tell me what you think. Here is my test.

$=money is behind this door
*=this is the door you pick
X=this is the door that the host opens and shows nothing is behind
#= nothing is behind this door
[1], [2], [3]= door labeled 1 , 2 or 3

We are looking at what happens when you pick door Number 3

-X-- $-- *
[1] [2] [3] === you should SWITCH here because you picked the wrong door

-$-- X-- *
[1] [2] [3] === you should SWITCH here because you picked the wrong door

-X-- #-- *
----------$
[1] [2] [3] === you should STICK WITH your door because the money is behind the same door you picked

-#-- X-- *
----------$
[1] [2] [3] === you should STICK WITH your door because the money is behind the same door you picked

as you can see 50% of the time you should switch and 50% of the time you should stay. SO what to believe???
 

koupparis

Carpe noctum
You're counting the last case twice, which you shouldn't. Its only once that the money is behind door #3, not twice. It doesn't matter which door the host opens.
 
or for a quick and dirty visualization, consider a gazillion doors, with money behind just 1 of them. You choose 1 door, say the millionth door knowing fully well that its an exercise in futility.
Then the host starts opening all doors starting for door 1, keeps going, leaves the door you picked closed, and just one more door, the 9,000,053rd door closed. He opens all other doors.
Would you switch?
 
A quick an simple explanation, he can't open the one with the money behind it. Because of that it's not random. If you chose the wrong door (which is likely), he only has 1 door to pick from.
 
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