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Re: The "two child solution"



At 07:51 PM 7/28/97 -0800, you wrote:
Richard Langer wrote:

The man has one of the following
older son older son
younger son younger daughter
=20
The woman has one of the following
older son older son older daughter
younger son younger daughter younger son

Others (e.g., Stanley McCaslin, Maurice Barnhill, David Bowman)=20
presented similar evidence and all went on to use these equal=20
probability outcomes to answer the question.

Nevertheless, despite the fact that this line of reasoning is=20
absolutely *correct* (neglecting the very minor "real world"=20
considerations that some mentioned and the philosophical concerns=20
that Leigh and I have expressed), I do not find it at all=20
*convincing* at a gut level as Maurice--one of very few who responded=20
to my request and expressed a level of confidence in his answer--also=20
confessed. Moreover, I think the response from Marilyn's readers=20
proves my point. To them (and, obviously, to some if not most of=20
us), whether or not the known son is older *seems* irrelevant; the=20
*only* question in either case seems to be, "What is the gender of=20
the other child?" (I find it interesting to note that this *is* the=20
only question *only* because the order of birth info is *not*=20
irrelevant and only *when* the order of birth info is given.)


I too am bothered by the fact that the "correct" answer is NOT convincing.
Why is it correct?

By specifying that one child is a boy, you have excluded part of the
possible sample space (the two girl portion).

By specifying that the older child is a boy, you have excluded even more of
the sample space.

It is the selective exclusion of part of the sample space that gives rise
to the different, "interesting" effects. To me it is not "intuitively
obvious" that the two DIFFERENT conditions exclude different amounts of
sample, but with a little thought, one sees that they do.

Leigh brings up the fascinating question of
<quoting>
"I ask the straightforward question: How can knowledge
of a fragmentary *a posteriori* nature affect the *a
priori* probability of either person having two sons?"
<end quote>
I respond - by having "=E0 posteriori" knowledge, we are limiting that
portion of the "=E0 priori" universe that we consider. If we choose to
exclude two DIFFERENT portions of the universe, then we will get two
different answers to the same question (with different conditions)
Richard M. Langer Gateway High School,
Physics Teacher 5101 McRee Ave.
rlanger@dtd1.slps.k12.mo.us St. Louis, MO 63110 =1A