Most of the recent postings about the Monty Hall problem
(and Marilyn's treatment thereof) have utterly missed the
point.
We must consider
A) "THE" Monty Hall problem writ large, which comprises
B) quite a number of variations.
The optimal strategy for "THE" game is very sensitive to
the details of the rules, including what Monty is required
and/or allowed to do, and whether he is your ally or your
adversary. The usual statement of "THE" Monty Hall problem
is severely underspecified. Therefore this problem is a
poster child for ill-posed problems.
There are right ways and wrong ways to handle ill-posed
problems! This is important! This is something people
ought to learn in school ... but usually don't. This
falls in the category of "critical thinking skills".
Such things are all-too-often not listed on the syllabus,
but they are IMHO far more important than most of the
things that are listed.
If you want to get some idea for how to handle such a
problem, treating it as an underspecified problem, see
see the table about 2/3rds of the way down in the article http://en.wikipedia.org/wiki/Monty_Hall_problem
(Search the page for the word "unspecified".)
I mention this because several recent postings have referred
in a general way to "THE" Monty Hall problem and/or to
Marilyn's "original" posting on the subject ... all of
which are grossly underspecified. As such the correct
answer must be a _solution set_ rather than a singleton
solution. To say the same thing another way: I insist
that for any underspecified problem, it is seriously
*wrong* to give a simple numerical answer -- or yes/no
answer -- without saying what conditions and assumptions
lead to that answer ... or (better!) (much better!)
doing a case-by-case analysis with a different numerical
answer for each case.
Marilyn's statement that "Very few [of her readers] raised
questions about ambiguity" is (a) a sad reflection on her
readership, and (b) an example of arrogant lawyering that
does not excuse the fact that she did not address the
ambiguities either, not until much later, and not
systematically even then.