From: bs@gauss.mitre.org (Robert D. Silverman)
Subject: Re: Marilyn Vos Savant's error?
Organization: Research Computer Facility, MITRE Corporation, Bedford, MA
Date: Tue, 15 Dec 1992 20:22:07 GMT

In article <BzBE0J.A6F@watdragon.uwaterloo.ca> deghare@daisy.uwaterloo.ca (Dave Hare) writes:
 
stuff deleted for brevity....

:>>   1) You meet a man on the street and ask him how many children he has.
:>>      He replies "two, and one is a boy."  What is the probability that
:>>      his other child is also a boy?
:>
:>It depends entirely on how your probability space is constructed.

etc.

:This is entirely bogus.  The question is clearly intended to be a particular
:probability problem phrased in familiar terms.  The intent of the problem is
:the basic probability exercise, not a problem in the sociology of people
:meeting on the street.  
 
If one really wants to be fussy about language, I can make a case that
the answer to the question should be ZERO.

The man replied "two, and one is a boy". He did NOT say:

"two, and at least one is a boy"

The first can be intrepreted as saying: number of boys = 1,
the second:				number of boys >= 1

The problem with many of these problems lies not in the mathematics of
condition probability, but rather in the sloppiness of English. Does
the phrase "one is a boy" imply "exactly one is a boy", or does it imply
"at least one is a boy"? It can be interpreted either way.
 
For those who disagree, look at this another way: Suppose the man had said
"seven, and 3 are girls". Would you not interpret this to mean that
exactly 3 are girls? So why does the phrase "two, and one is a boy"
imply that the number of boys is greater than or equal to 1 and not
exactly one??


I remember an old statistics course, in which the instructor had posed
a similar question on an exam. I put down the answer ZERO, just as above
and had it marked wrong. I went back and presented the above argument and
the professor agreed that the sloppiness in the way the question was worded
could lead to my intrepretation. Indeed, that a careful reading, rather than
a colloquial one would lead ONLY to my interpretation.

Let me give another example, taken from an article in Notices of the AMS
from a while back. The discussion was about the GRE exam. One question
gave an equation or equations and asked which of the following sets 
satisfied the equation(s). It then gave 4 answers plus (e) none of the
above. The trouble is, while one of the answers was clearly intended as
the correct one, none of the answers presented were in set notation.
That is to say, they were not sets, but conditions satisfied by sets.
[i.e.  1 < x < 3  is not a set, but: {x: 1 < x < 3} is].  The article went
on to say that for students who were taught to be careful about language
and careful about definitions, the only correct answer was (e), even though
(b) was the intended answer.

Too many of these problems are exercizes in clarifying the ambiguity in
English phrases, rather than in probability.
--
Bob Silverman
These are my opinions and not MITRE's.
Mitre Corporation, Bedford, MA 01730
"You can lead a horse's ass to knowledge, but you can't make him think"
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