Newsgroups: sci.math
Followup-To: poster
From: hoey@AIC.NRL.Navy.Mil (Dan Hoey)
Date: Wed, 24 Nov 1993 22:26:55 GMT
Subject: Re: MvS erred on Monty Hall. was: dumping on Marilyn vos Savant

I should note that this problem appears in sci.math and rec.puzzles
with unpleasant frequency--I know I've been watching it bob up every
few months for the past five years--so I have strong reservations
about adding more discussion on the topic.  This time, though, my
concern about the accuracy of the discussion has overcome my usual
willingness to let the problem die of its own accord.  The sci.math
readership should not, however expect me to inflict\\\\\\\favor you
with such clarification often; in particular, followup posts are not
invited and will generally not elicit my response.  I am considerably
more forthcoming in email.

Just so we know what we're discussing, the problem as printed in
Marilyn vos Savant's column goes:

] "Suppose you're on a game show, and you're given a choice of three
] doors.  Behind one door is a car; behind the others, goats.  You pick
] a door--say, No.1--and the host, who knows what's behind the doors,
] opens another door--say, No.3--which has a goat.  He then says to you,
] 'Do you want to pick door No.2?'  Is it to your advantage to switch
] your choice?"

tycc...@kronecker.mit.edu (Timothy Y. Chow) writes:

> I have yet to meet someone who unwittingly chooses unusual tacit
> assumptions, correctly reasons from those tacit assumptions to a
> probability of 1/2, and is unaware that the tacit assumptions most
> people make lead to a probability of 2/3....  Marilyn vos Savant
> certainly did not provide enough context from the letters she quoted
> to enable readers to tell which way those Ph.D.'s were thinking, and
> extrapolating from my own experience I strongly suspect that the
> Ph.D.'s were way off base.

Well, I am someone who wittingly refuses to ascribe tacit assumptions
to the problem, whether usual or unusual.  I have had to conclude that
the problem as stated does not determine a probability of winning by
switching, because it does not specify whether the game show host
always offers the contestant a chance to switch or not, and if not
under what circumstances the offer is made.

I will say that my understanding of the problem statement is that
there is indeed one door with a valuable prize and two without, and
that the host will award whatever is behind the door the contestant
ends up picking.  Perhaps that is tacit, but it is the way I
understand the everyday English of the problem.  I deny that my
understanding of this is comparable to the assumption that the host
always offers a chance to switch.

In August, ruln...@nimbus.seas.ucla.edu (John M. Rulnick) posted nine
letter-excerpts from Marilyn vos Savant's column taking issue with
her analysis.  Even if we reject the one that suggested a probability
textbook and the one that blamed her mistake on her gender, there
are still seven for which the excerpt printed is consistent with an
objection on these grounds.

> In short, the problem was indeed ambiguously worded....

Once again, I deny that the problem was ambiguously worded.  It is an
unambiguous presentation of a problem that does not provide enough
information to determine the kind of answer you and Marilyn are
looking for.  Reread the problem statement: there is no support for
the assumption that the host's offer of a second choice would have
been made had the contestant chosen a different door.  Hundreds of
people posting arguments and programs with this tacit assumption do
not provide such support.

f...@cc.gatech.edu (Gary Peterson) goes on

> In addition, some people assume the host does not want to give out
> the big prize (which is the opposite of how game shows really work)
> and therefore suspect the offer to switch is made more frequently
> when your initial choice contains the big prize.

First, it has been repeatedly noted that the problem is inconsistent
with the way the ``Let's Make A Deal'' game show worked.  On that game
show the contestant was never offered a second chance to choose a
previously rejected prize (and if you are talking about some other
game show, you had best say which one.)  So arguments about ``how game
shows really work'' are not relevant to the problem.

Second, you have misstated the position of everyone I know who
mentions the possibility of the host ``not wanting to give out the
big prize''.  We are not making the assumption that the host has that
strategy.  We are simply trying to make it clear to you that the
assumption you make has changed the problem.  There are other possible
strategies of the host, including the strategy of always offering a
choice as you have assumed.  There are many other strategies, as well.
It is just not specified which possible strategy occurs in the case of
the problem as stated.

I will note, however, that the contestant can guarantee a 1/3
probability of winning the prize by choosing a door at random and
refusing all offers to switch.  Any contestant strategy that includes
accepting an offer to switch will have a lower payoff under some host
strategies.  Thus, refusing to switch maximizes the minimum payoff.
There is no way to maximize the average payoff without information
about the probability distribution of host strategies.

> The discussion of odds after this point becomes absurd.

Well, I discussed probability in the previous paragraph, and I don't
happen to think it is absurd.

> Hence: Always shows goat and always makes offer are critical
> for many people's understanding of the problem. (Although they may
> choose to ignore this data.)

There is no logical support for your conclusion.  Just because you
can't compute probability without changing the problem doesn't mean
the problem really means what you are changing it into.  The problem
as stated simply does not determine the probability of winning by
switching.

> That it still comes up, and still people both state it and solve
> it wrong despite even having the FAQ now, is terribly depressing.

It is also depressing having a FAQ that has the answer wrong.  Note
that at least the rec.puzzles archive article (to which the sci.math
FAQ refers) mentions that the problem is underspecified, and notes the
assumptions used to derive the probabilistic solution.

> (But the most depressing aspect are the TOTAL BOZOS who post
> programs to do simulations to "prove" their answer....)

I have to agree.  It convinces them of an incorrect answer, based on
an unfaithful model of the problem.

Dan Hoey
Hoey@AIC.NRL.Navy.Mil