From: mack@isl.Stanford.EDU (mack)
Subject: Monty Hall Paradox (was Re: Physics-1A Question)
Organization: Stanford University
Date: Wed, 7 Oct 92 21:18:40 GMT

seyyed@wuee1.wustl.edu (Seyyed Mahdavian) writes:

    [shahrokh's puzzle deleted]

>Wrong! "Given that #2 is empty", #1 and #3 are again equiprobable, and this is
>consistent with intuition, because your first choice (of #1) doesn't affect
>anything. For a simple proof of this, consider x1,x2,x3 as three random 
>variables representing the state of the doors. x1=0 means no prize behind
>door #1, x1=1 means there is a prize behind door #1, and so on. Obviously
>x1+x2+x3=1. Now P(x1=1,x3=0 | x2=0)=P(x1=1,x2=0,x3=0)/P(x2=0). P(x2=0) is
>the "apriori" probability, therefore P(x2=0)=1-P(x2=1)=2/3
>P(x1=1,x2=0,x3=0)=P(x1=1) because x1+x2+x3=1, therefore it equals 1/3.
>So the final "conditional" probability is equal to (1/3)/(2/3)=1/2, which
>is the desired result.

'What the hell' [(c) Zafar, 92] are you talking about? Why don't you
just 'shut up' [(c) Seyyed, 92] when you have no knowlegde of
probability.   :-)

This is the well known Monty Hall Paradox. I waited a couple of
days for some net.probability.god to shed some light on Seyyed's
reply, but there was none. So here we go:

The problem (a reworded version of Shahrokh's puzzle):

 => You are a participant on "Let's Make a Deal." Monty Hall shows you
 => three closed doors.  He tells you that two of the closed doors have a
 => goat behind them and that one of the doors has a new car behind it.
 => You pick one door, but before you open it, Monty opens one of the two
 => remaining doors and shows that it hides a goat.  He then offers you a
 => chance to switch doors with the remaining closed door.  Is it to your
 => advantage to do so?

For a detailed solution you can refer to the solution offered by Steve
Selvin in [American Statistician, Feb 1975, V. 29, No. 1, p. 67] under
the title ``A Problem in Probability.''

Surprisingly, he received several letters from the readers of American
Statistician contesting the accuracy of his solution (just like Seyyed
has done), so he responded in another article two issues later
[American Statistician, Aug 1975, V. 29, No. 3, p. 134], part of which
follows:

 => ...  The basis to my solution is that Monty Hall knows which box
 => contains the prize and when he can open either of two boxes without
 => exposing the prize, he chooses between them at random ...
 =>
 => Benjamin King pointed out the critical assumptions about Monty
 => Hall's behavior that are necessary to solve the problem, and
 => emphasized that ``the prior distribution is not the only part of
 => the probabilistic side of a decision problem that is subjective.''
 =>
 => Monty Hall wrote and expressed that he was not ``a student of
 => statistics problems'' but ``the big hole in your argument is that
 => once the first box is seen to be empty, the contestant cannot
 => exchange his box.''  He continues to say, ``Oh, and incidentally,
 => after one [box] is seen to be empty, his chances are not 50/50 but
 => remain what they were in the first place, one out of three.  It
 => just seems to the contestant that one box having been eliminated,
 => he stands a better chance.  Not so.''  I could not have said it
 => better myself.

'hAlA rAzi shodi' [(c) Seyyed, 92]?

>Your prize is a golden passe-gardani which will be given to you in MehrAbAd
>airport.

What a nice welcome that would be!

>Seyyed.

    --mack

--
She's closing her eyes, then she takes me away,
and she's pulling the strings, when she's playing with love.
She's playing with love.                       --Alphaville

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