On Wednesday, 8 October 2014 at 01:22:49 UTC, Timon Gehr wrote:
The secret behind the monty hall scenario, is that the host is
actually
leaking extra information to you about where the car might be.
You make a first choice, which has 1/3 chance of being right,
then the
host opens another door, which is *always* wrong. This last
part is
where the information leak comes from. The host's choice is
*not* fully
random, because if your initial choice was the wrong door,
then he is
*forced* to pick the other wrong door (because he never opens
the right
door, for obvious reasons), thereby indirectly revealing which
is the
right door. So we have:
1/3 chance: you picked the right door. Then the host can
randomly choose
between the 2 remaining doors. In this case, no extra info is
revealed.
2/3 chance: you picked the wrong door, and the host has no
choice but to
pick the other wrong door, thereby indirectly revealing the
right door.
So if you stick with your initial choice, you have 1/3 chance
of
winning, but if you switch, you have 2/3 chance of winning,
because if
your initial choice was wrong, which is 2/3 of the time, the
host is
effectively leaking out the right answer to you.
The supposedly counterintuitive part comes from wrongly
assuming that
the host has full freedom to pick which door to open, which he
does not
But yes. He has. It makes no difference.
If he would ever open the right door, you would just take it too.
So if the win is behind the two doors you did not choose first,
you will always get it.
The problem with this explanation is simply that it is too long
and calls the overly detailed reasoning a 'secret'. :o)
So take this shorter explanation:
"There are three doors and two of them are opened, one by him
and one by you. So the chance to win is two out of three."
It doesn't matter if he uses his knowledge to open always a
false door. It only matters that you open your door AFTER him,
which allows you to react on the result of his door. If you
open the door first, your chance is only 1/3.
