Re: The "two child solution"

[David Bowman <dbowman@tiger.gtc.georgetown.ky.us>]

Concerning the solution of the "two child problem" John Mallinckrodt wrote:

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

Regarding the philosophical objections about the probability of
after-the-fact events, I think such objections may be allayed by a simple 
rewording of the question so it refers to potential events instead of those
which have already happened.  Also, such objections lose their force anyway if
only a more Bayesian interpretation of the probabilities involved is used
(which I prefer, BTW).  After all, the whole point of the problem is to
illustrate the effect of how conditional probabilities depend on additional
information.

Regarding the point about the relevancy/irrelevancy of the nature of the
birth order information, it is not the birth order information *per se* which
is the relevant item, but rather the *distingushing* nature that that 
information provides.  For instance, if instead of the birth order some other
uniquely distinguishing property (which is uncorrelated with gender) were
provided, (maybe relative birth weights, relative number of hairs on the
children's head at birth, etc.?) the result would be the same.  This 
information serves to make the children *distinguishable*.  Actually, come to
think of it, what could be done is that the arbitrary labels 'A' and 'B'
could *randomly* assigned to each of the parents' children so that each
parent would have one 'A' child and one 'B' child.  Then the information that
the man's 'B' child is a boy is equivalent for the problem at hand to the
information that the man's oldest child is a boy.  This is more information
about the situation than that at least one of the woman's children is a boy.
The reason that the distinguishing feature of the information is relevant is
that it distinguishes the two mixed gender cases, i.e. the
(('A'=boy)and('B'=girl)) case from the (('B'=boy)and('A'=girl)) case.
Whether different cases are distinguishable or not makes a difference--as
anyone familiar with the effects of quantum indistinguishability on the
statistical behavior of ensembles of identical particles knows.  (And John
was worried, I expect, that the 'two child problem' didn't relate to physics.
Just about everything relates to physics.)

David Bowman
dbowman@gtc.georgetown.ky.us