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Re: The "two child solution"




On Mon, 28 Jul 1997 19:51:09 -0800 (PST) John Mallinckrodt
<ajmallinckro@CSUPomona.Edu> writes:
Richard Langer wrote:

The man has one of the following
older son older son
younger son younger daughter

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)
presented similar evidence and all went on to use these equal
probability outcomes to answer the question.

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.)

John
-----------------------------------------------------------------
A. John Mallinckrodt http://www.intranet.csupomona.edu/~ajm
Professor of Physics mailto:ajmallinckro@csupomona.edu
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Cal Poly Pomona fax:909-869-5090
Pomona, CA 91768-4031 office:Building 8, Room 223


I just reviewed probability, which I learned too late in life to be able
to retain on a permanent basis. I solved the problem about five
different ways - all agree with Marilyn. I did not need to go beyond
page 91 of Blum and Rosenstatt to find everything I thought I could
possibly use. Conceivably, some result concerning independence might
pop up, but I doubt it. One of the more strident "readers", Dave
Ferkinhoff of Middletown, RI, claimed Baye's Rule would reveal the
"truth", but it merely reaffirmed Marilyn. The poor lady can exchange
conditions with the man, but it costs 2/3 to do it. He spoke of some
"Dempster-Shafer theory of evidential reasoning". Now, I am extremely
skeptical about a theory of evidential reasoning. Logic is logic and a
special kind of logic (reasoning) is probably "scientology" at best. I
admit to extreme prejudice. If someone can look up that theory, what the
heck; but, if it ain't in Feller, it probably ain't. Boy, he sure do
talk fancy, but I believe he is just as far off as the rest.
Just one little concern (doubt?) remains. I haven't really seen a
conditional probability theory especially designed for events that have
already occurred, but, as elsewhere in life, what's important is when we
find out, not when it happened. New information can alter
probabilities. Strangely, the trick I did to compute probabilities in
China might be right.
Since I have so much trouble with probability, I decided to solve a
big problem a couple of years ago. I computed an algorithm to estimate
the expected value of a Texas State Lottery Ticket. Of course, it never
is worth a buck. But, amusingly, without leaving my apartment I
discovered two or three ways in which the State cheats. So, you can find
out things by doing math! The American Mathematically Monthly quite
properly turned it down and I have no idea what to do with it. It's a
very nice light little paper with a couple of tricks, but not much
originality. I would send it to anyone who is interested; however, I
don't expect any takers. I am not insulted. It's OK. I really
enjoyed writing it! I actually had to buy another meg for my TI PS-17
laser printer to print even a single page of math.
Very large factorials are troublesome. I used Stirlings formula
late in the game when I no longer needed it. Except with it I could
have gotten upper *and* lower bounds. I guess everyone knows that for
n around 50 million and k about 10, n! / (n-k)! is very close to n to
the power k.
Another cute little probability problem was circulated on another
listserver last December, but I can't find the mailing with the problem
statement. Perhaps I can back the problem out of my solution, which, in
this case, was right. The debate ensued over whether or not one could
choose the first of the three doors at random.
Well, that was fun. When do I get paid? Oh, I forgot, I'm an
amateur.

Regards to all and especially to the players / Tom