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Conference rusure::math

Title:Mathematics at DEC
Moderator:RUSURE::EDP
Created:Mon Feb 03 1986
Last Modified:Fri Jun 06 1997
Last Successful Update:Fri Jun 06 1997
Number of topics:2083
Total number of notes:14613

1390.0. "Boston Globe prob. problem (caused a commotion!)" by EAGLE1::BEST (R D Best, sys arch, I/O) Mon Feb 18 1991 17:35

I made a perfunctory search for this item under several headings and didn't
find it, so I'll post it.

This problem was posted in the Boston Globe Parade under the Mariliyn
vos Savant column a while ago (I don't have the original date) and apparently
created quite a controversy, drawing letters both attacking (many) and
supporting (few) the supplied solution from many people with
mathematical backgrounds.

The problem and another defence was posted in the 17-feb edition of the
magazine ibid..

Here it is, along with the Globe supplied solution (I've placed the solution
after a form feed so as not to spoil for those who want to try it
from scratch), but without commentary or detailed explanation.  I paraphrase,
but I think I've captured all of the essential aspects of the problem.

I think this is a good example of how devilish simple sounding problems in
probability can get.
------------------------------------------------------------------------
A contestant on a game show is shown 3 doors and is told that behind one is
a new car, and behind the other two are goats.

The contestant chooses door #1, but before opening the door, the host offers
the following help: the host opens door #3 revealing a goat, and allows the
contestant a last chance to change his/her choice to door #2.

The question is:

Should the contestant change his/her choice to door #2 ?

The supplied answer is (follows form feed)

Yes, the probability that the prize is behind door #2 is 2/3, while that of
it being behind door #1 is 1/3.
T.RTitleUserPersonal
Name
DateLines
1390.1see 1078.nn for a discussion of this and a more precisely formulated versionCSSE::NEILSENWally Neilsen-SteinhardtMon Feb 18 1991 18:360
1390.2Answer was wrong, though as good as any answer...CADSYS::COOPERTopher CooperMon Feb 18 1991 19:0538
    The problem, as stated here and in Parade, is indeterminant.  Certain
    additional assumptions are needed to solve it.  As this problem is
    usually stated the "host" is specifically Monty Hall and the show is
    specifically "Let's Make A Deal".  For those who know the show, this
    supplies enough information to complete the analysis -- giving
    Marilyn's counterintuitive result.  This is related both in form and
    in philosophy to the problem discussed in 1291.

    One obvious assumption that needs to be made is that there *is* one and
    only one good prize and the "goat" wasn't it.  You also must assume
    that the prize is not moved around in response to the contestant's
    actions: it either stays in one place or is switched around
    independently of the contestant's actions.  Similarly to 1291, the most
    interesting assumption you must make concerns the host's actions if
    circumstances were different than they turned out to be -- in other
    words what was the sampling process of which this was one sample of.

    The 1/3:2/3 answer comes from the assumption that the host knows where
    the "good" prize is and avoids revealing it if the contestant has not
    already chosen it.  The host's actions therefore reflect information
    which the contestant can take advantage of by "switching".

    If, on the other hand, the host chooses arbitrarily (i.e.,
    independently of the actual location of the "good" prize) than there
    is no information about the location reflected by his/her actions and
    there is no advantage (or disadvantage) to switching.

    More subtly -- if the contestant doesn't know whether or not the host
    has deliberately avoided revealing the prize or just chosen randomly
    than (assuming that these are the only possibilities) then they should
    switch.  The advantage will, however, be less than the 2:1 advantage
    if they know that the host avoids the prize.

    Of course, if the host always reveals the prize unless the contestant
    has already revealed it, then the contestant clearly should not switch
    if the host reveals a goat.

					    Topher
1390.3ELIS::GARSONV+F = E+2Mon Aug 19 1991 09:4115
re .2
    
    Sorry to wake a sleeping note but a friend of mine (non-DECcie) asked
    me about this problem over the weekend and I seemed to recall that it
    had already been covered here.
    
>   For those who know the show, this supplies enough information to complete
>   the analysis -- giving Marilyn's counterintuitive result.
    
    I'm not familiar with this show. Am I right in assuming that the actual
    behaviour of the host is that he does know where the car is and always
    chooses to open a door that he knows will not reveal the car? If the
    contestant has not picked the car then the host is forced in his choice
    of door. If the contestant has picked the car then the host can choose
    either of the remaining doors.
1390.4ALIEN::EDPAlways mount a scratch monkey.Mon Aug 19 1991 10:288
    Re .3:
    
    Yes, the correctly-stated problem includes those stipulations:  The
    host always opens a door not selected by the contestant to reveal a
    dud.
    
    
    				-- edp
1390.5CLT::TRACE::GILBERTOwnership ObligatesTue Aug 20 1991 15:132
    Actually, sometimes the host reveals what appears to be a 'dud', but
    when the curtain behind this 'dud' is raised, there is the big prize.