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Subject: Re: New and final solution of the Monty Hall Dilemma

Author: Gerrit Reubold

Date: 14:30:11 09/27/02

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On September 27, 2002 at 17:07:57, Uri Blass wrote:

>On September 27, 2002 at 16:17:02, Gerrit Reubold wrote:
>
>>On September 27, 2002 at 16:04:45, Uri Blass wrote:
>>
>>>On September 27, 2002 at 15:40:07, Gerrit Reubold wrote:
>>>
>>>>On September 27, 2002 at 15:04:26, Uri Blass wrote:
>>>>
>>>>>On September 27, 2002 at 14:40:09, Gerrit Reubold wrote:
>>>>>
>>>>>>Uri,
>>>>>>
>>>>>>I think your model of the game is not a model of the situation which I am
>>>>>>discussing.
>>>>>>
>>>>>>Suppose you and me are playing the following game (f.ex. by email).
>>>>>>
>>>>>>1. I, the host, take an empty chessboard and put a single king one one of the
>>>>>>squares. Trust me that I don't cheat. You send me your guess which square this
>>>>>>might be.
>>>>>>
>>>>>>2. I assume you, the candidate, guess it is a1.
>>>>>>
>>>>>>3. I give you a hint:
>>>>>>The square is not one of
>>>>>>   b1 c1 d1 e1 f1 g1 h1
>>>>>>a2 b2 c2 d2 e2 f2 g2 h2
>>>>>>a3 b3 c3 d3 e3 f3 g3 h3
>>>>>>a4 b4 c4 d4 e4 f4 g4 h4
>>>>>>a5 b5 c5 d5 e5 f5    h5
>>>>>>a6 b6 c6 d6 e6 f6 g6 h6
>>>>>>a7 b7 c7 d7 e7 f7 g7 h7
>>>>>>a8 b8 c8 d8 e8 f8 g8 h8
>>>>>>
>>>>>>(Note that it is no difference if I suddenly forgot on which square the king
>>>>>>stands, I decide to look only on the given 62 squares and, surprise, all of them
>>>>>>are empty.)
>>>>>
>>>>>It is important if you know where the king is.
>>>>
>>>>No.
>>>>
>>>>>
>>>>>Suppose for the discussion that you also do not know where the king is
>>>>>so you always expose all the squares except a1 g5 after
>>>>>I choose a1.
>>>>
>>>>No. I look at 62 random squares, excluding a1.
>>>>
>>>>>
>>>>>Suppose that we start to play 64000 games(most of them are
>>>>>not played because you discover the king
>>>>>in one of the 62 squares).
>>>>
>>>>No. We play only this single game. We might play again and my hint will likely
>>>>be different, maybe all squares except a1 and f8.
>>>>
>>>>Do you think it is a difference whether I forgot where the king is (I would have
>>>>told you "game over" if I accidently looked at a square with the king on it) or
>>>>whether I know the square of the king.
>>>
>>>Yes
>>>Suppose you do not know the square of the king.
>>>
>>>Suppose that for every square x(not a1) we play 64000 games when you choose
>>>square x.
>>>In these games the king is in square y in 1000 games for every square y.
>>>
>>>I claim that I can expect 62000 games out of every 64000 games to be canceled
>>>because the king is exposed(not in x and not in a1).
>>>
>>>I expect to win 1000 games out of 64000 games(the 1000 games when the king is in
>>>a1) and I expect to lose 1000 games out of 64000(the 1000 games when the king is
>>>in square x).
>>>
>>>The same hapen for everyone of the 63 squares that you can choose and I expect
>>>to win 63000 games and to lose 63000 games.
>>>
>>>Uri
>>
>>One question:
>>
>>Do you agree: If the host knows the solution (the door with the car, the
>>position of the king, ...), the candidate should switch?
>>
>>Greetings,
>>Gerrit
>
>Yes

If (_only if_, and this is the only situation which I discuss) the situation
arises, that the host opened all the wrong doors, what should the candidate do:
should he switch or not? He can't read the mind of the host, maybe the host
knows the winning square/door, maybe it was just good luck. How should the
candidate seperate these cases: I think it is not possible and he should switch.

(obviously, I enjoy these discussion, sorry about wasting bandwith of those who
are not interested, this thread is of course off topic...)

Greetings,
Gerrit



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