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Re: Marilyn and puppy litters




On Tue, 12 Aug 1997 02:58:47 -0500 (CDT) "A. R. Marlow"
<marlow@beta.loyno.edu> writes:
On Mon, 11 Aug 1997, John Mallinckrodt wrote:

So, did anyone else think Marilyn blew it yesterday?

John

P.S. For the "Parade Magazine-deprived," Marilyn had the cover
article
this week and the lead question, printed on the cover was, "Your dog
has a
litter of four. Is it most likely that two are males and two are
females?"

Of the sixteen possibilities, 8 are of the 3 + 1 variety, while only 6
are
of the 2 + 2 variety, so it would seem that Marilyn is correct in her
answer that it is more likely that the litter of four would have three
of
one sex and one of the other sex than that it would be a 2-2 split.
(Always assuming both sexes equally likely on each birth, of course.)

A. R. Marlow E-MAIL:marlow@beta.loyno.edu
Department of Physics, Box 124 PHONE: (504) 865 3647
(Office)
Loyola University 865 2245 (Home)
New Orleans, LA 70118 FAX: (504) 865 2453


******************************************************************
My first letter to Marilyn follows:From: TWayburn
Full-Name: Thomas L Wayburn
To: rgrandy@ruf.rice.edu
Subject: Please, give up your meaningless work and join me in work
worthy of you.
X-Status: Unsent

--------- Begin forwarded message ----------From: TWayburnTo:
marilyn@parade.comCc: ajmallinckro@CSUPomona.EduSubject: Please, give up
your meaningless work and join me in work worthy of you.Date: Tue, 12 Aug
1997 02:02:17 Message-ID:
<19970812.020123.8335.3.TWayburn@juno.com>Thomas L. Wayburn, PhD,
Executive DirectorAmerican Policy Institute, Inc.2638 Yorktown Apt.
294Houston, TX 77056-4873713-963-8944 (Phone first for
faxing.)TWayburn@juno.com for e-mailAttachments may be sent to
ELee@UH.edu <Ellen Lee, Director>.Dear Marilyn,This is from a letter to
A. John Malinckrodt, a professor of physics, who has supported you when
you are right, which is almost always, but has found fault with the way
you pose certain problems, e.g., the two-child problem, which you got
right. (When a critic appeals to a theory of "judicial reasoning",
"stand not on the order of your leaving but leave immediately." -
Burroughs, *The Naked Lunch*.) Also, he, John, was very negative
concerning your insulting book on Fermat's Last Theorem. Perhaps certain
forms of criticism are best directed to the practitioners under
indictment. Also, such criticism is the responsibility of the concerned
scholars, in this case, mathematicians. Actually, though, I don't like
much of what is going on in math these days either. I think I have a
right to think that as "I could have been a
contender".**************************************************************************Here
is my answer (right or wrong is the least important factor in this
equation):I will now be conventional instead of slick (which got me the
wrong answer in the Two-Child Problem (but I said I thought it was wrong
when I posted it - even before I peeked at the other responses). This
time I wrote out the entire sample space and found (conventionally)
p(2,2) = 3/8, p(0,4) = p(4,0) = 1/16, and p(3,0) = p(0,3) = 1/4.
Therefore, although 2 males and 2 females is more likely than any other
particular outcome (sample), it is not more likely than *some* other
outcome (event). Could this be the entertainment feature in this
problem? I know I don't really know probability. I haven't read and
memorized the contents of both volumes of Feller, worked all the
exercises, and practiced on harder problems of my own invention or from
my work. I have a notion of what probability theory is like. I could
use it to solve many conventional (deterministic) problems and save
effort. I know measure theory (a little) better - or once did. Let's
see, a convergent sequence of measurable functions is "usually" uniformly
convergent, where "usually" means "good enough for physics". [I take it
back. Honest. It means that it is a "virtual certainty"?] So, I can't
get too concerned about whether I get this (probability exercise, which I
never read) right or not. But, I really think I'll write Marilyn and
ask her why she is throwing away her life on such meaningless work. I
can provide real problems the solutions of which will affect the survival
of life on this planet. She has a rich husband. Come and work at the
American Policy Institute, Inc., for no pay and make something of
yourself. The reason I can't pay a dollar a year is that it would
trigger a nonzero in my income tax statements, which, having all zeroes
as it currently does, can be completed on one page in less than 10
minutes. We do not wish to become embroiled in fiduciary matters unless
they are research topics that can achieve worthwhile goals - now and in
the far-distant future. If there is one.Regards / Tom P.S. Gee, I
practically have the letter written. All I need now is to send it.
*Maybe I
will.****************************************************************************Dear
Marilyn, Well, as it turns out, I have written and you have learned what
I would truly like to say to you in a somewhat unconventional manner. I
have just read your question, which, corroborating John Malinckrodt, is
poorly phrased. When you said, "Is it most likely that two are males
and two are females?", I am entitled to decide most likely out of what
additional possibilities. As it happened you chose the union of two
events (subsets of our sample space), whereas I might have decided to
divide the space differently. For example, "Is it more likely that three
and only three will be of the same gender or that at least two will be
female," Or, if I wish to be naive (and I am entitled to be naive within
the context of problem solving), I could have asked: " Which is more
likely three and only three will be of the same gender or at least one
will be either male or female?" By now, I hope you agree with me that
the problem statement is incomplete and that you have been mildly (and,
perhaps, innocently) deceptive, arbitrary, and, excuse me, unfair.
Actually, you have compared the likelihood of an "outcome" with the
likelihood of an "event" to use the language of probability systems. I
hope you see fit to understand me and print a retraction. (You needn't
mention my name.) Also, I hope you don't confuse me with those awful
(vindictive?) people who criticized you so foolishly (and maliciously)
subsequent to the two-child problem. That was the first time I had read
letters like that; and, frankly, I was shocked. Finally, if I have
violated any of the axioms for a *probability system*, please inform me
and you shall have my speedy and sincere apology - and, maybe, a little
present. A little known but beautiful book, say.With the kindest regards
and the highest hopes that you will discuss a change in career direction
with me before very much more time passes. I have some interesting and
crucially important problems for your consideration. Could saving nine
billion people from starving to death get your attention?I am yours
truly,Thomas Wayburn, PhD--------- End forwarded message ----------
*************************************************************
Here is my second letter to Marilyn:

From: TWayburn
Full-Name: Thomas L Wayburn
To: marilyn@parade.com
Cc: ajmallinckro@CSUPomona.Edu, rgrandy@ruf.rice.edu
Subject: Better explanation of your error (fault) in stating the
problem of the puppies
X-Status: Unsent

Dear Marilyn, Sorry to be a bother, but I can improve the following
paragraph. Also, I apologize for the mistakes in my previous posting -
so common in e-mail. When you said, "Is it most likely that two are
males and two are females?", I am entitled to suppose that you are
referring to a single (simple) set of *outcomes*, in the language of
probability systems. As it happened you chose the union of two events
(subsets of the sample space are called *events* in probability systems
theory). Whereas it is true that you may have been referring to a
simple event by accident and you intended to compare this event with more
complex events all along, that interpretation is unlikely. (Sorry about
the pun.) Now, if complex events are allowed, your answer is
definitely wrong. For example, "Is it more likely that three and only
three will be of the same gender or that at least two will be female,"
Or, if I wish to be naive (and I am entitled to be naive within the
context of problem solving), I could have asked: " Which is more likely
three and only three will be of the same gender or at least one will be
either male or female?" I shall now check that none of the
probability-system rules have been broken by our examples. If an event
A is in the event class E, it's complement is too. A_1 was precisely
three of the same gender = [A_2 precisely one male] union [A_3 precisely
one female], which implies comp(A_1) = [comp(A_2) intersection
[comp(A_3)]. All the complements and the intersection are in the event
class E. I grant that your example is valid. Let's take just my first
example: A_4 at least two female = UNION(A_5 two female; A_6 three
female; A_7 four female). A_4 a set of eleven outcomes is in E. And,
clearly, so are the other three sets as is their union. So far, so good.
I think it is fair to assume that our *finite* set possesses *countable
additivity*; it doesn't even have an infinite number of outcomes. Let's
not check that one unless you wish to do it on your own. (I don't want
to examine every sequence of pairwise disjoint sets in E, do you?) We
have already checked that our sample space S has a real-valued function
P such that P(S) =1. The function P such that (a) 0 is no greater than
P(A_i), which is no greater than 1, (b) P(S) = 1, and (c) countable
additivity (assumed) is called a probability measure. Measure theory is
very important in probability, which you probably (sorry) know. I think
you should have immediately employed such mathematical rigor as you
command to silence those two-child ruffians, however it is doubtful that
mathematical rigor will play in Peoria or, for that matter, on Broadway.
Please think about getting into another line of endeavor. With my
deepest apologies for botching the first e-mail,I am yours truly,Tom
Wayburn, PhD

*********************************************************

The point is not that of all complex events (unions, complements, and
intersections) two boys and two girls is not the most likely. The point
is we were set up to exclude such complex events. The event p(2,2) is a
simple event compared to [precisely three of one gender] = [precisely
three boys] union [precisely three girls]. But, my choices of complex
events had higher probability; so, why did she not consider them for her
reported solution? They are complex too, but the complex category had
been opened by her. -TLW

P.S. I did not want to be vindictive, though, like the "two-child"
dissenters, so I was easier on her than I might have been.
***********************************************************