In a message posted on June 10th, 2003, under
the heading of "A deceptively simple test of
deductive capability," the following problem was
posed. Given the premises: (a) Most tall men wear
large-size shoes; and (b) Robert is tall. What is
the probability, P, that Robert wears large-size
shoes? The correct answer--which may come as a
surprise to some--is: P is indeterminate, that is, P
is either undefined or unknown. In other words, the
premises convey no information about P. The fact
that P is indeterminate calls into question the
validity of much of probability-based reasoning in
the realm of law.
To justify this answer, it is convenient to
consider a more general, generic version of the
problem. Given the premises: (a) QA's are B's or,
equivalently, Count(B|A) is Q); and (b) X is A,
where A and B are specified fuzzy subsets of a
universe of discoupe, U, and Q is a specified fuzzy
quantifier (number), e.g., most. What is the
probability, P, that X is B? It is understood that
in "X is A" and "X is B," A and B are possibility
distributions of X, that is, the fuzzy sets of
values which X may take.
There are two cases: (1) X is a non-random
variable, e.g., X=Robert, on the understanding that
Robert is a specified member of U, with U being a
collection of individuals; and (2) X is random
variable taking values in U with an unknown
probability distribution, implying that Robert is an
unspecified member of U. In case (1), it is not
meaningful to ask: What is the probability that X is
B, since X is not a random variable, and hence P is
undefined. In case (2), since the probability
distribution of X is not known, all that can be said
about the probability, P, that X is B, is that its
value lies between 0 and 1, implying that P is
unknown. Invocation of the maximum entropy principle
is not admissible because the principle is of
questionable validity and, in any case, is not
applicable when events and/or their probabilities
are fuzzy rather than crisp.
What else can be said? If we assume that X is a
random variable with a uniform probability
distribution and A, B and Q are fuzzy sets, then the
following can be established.
If (a) "Count(B/A) is Q" is interpreted as
"Sigma-count(B/A) is Q," where sigma-count (B/A) is
the relative count of the elements of B which are in
A; (b) the intersection of A and B is defined in
terms of the min norm; and (c) the probability of
the fuzzy event "X is B," is defined as a weighted
sum or integral (L.A. Zadeh, "Probability Measures
of Fuzzy Events, Journal of Math, Analysis and
Applications, vol. 22, pp. 421-427, 1968), then what
can be asserted is that "P is Q," meaning that Q is
the possibility distribution of P.
Returning to the original problem, what we see
is that the general, generic version cannot be
analyzed through the use of standard probability
theory. What can be analyzed is a crisp version,
e.g.: Given the premises: (a) Over 70% of men whose
height exceeds 180 cm wear shoes whose size exceeds
11; and (b) Robert's height is over 180 cm. What is
the probability that Robert wears shoes of size 11
or over? Assuming that Robert is chosen at random
from U with uniform probability, the answer is: P is
between 0.7 and 1.
A simpler crisp example is the following. Given:
(a) Over 99% of professors have a Ph.D. degree; and
(b) Robert is a professor. What is the probability,
P, that Robert has a Ph.D. degree? The correct
answer is that P is indeterminate. And yet, most
people, including those with scientific training,
would say that P is over 0.99. This answer is
correct only if it is assumed that Robert is drawn
at random from U with uniform probability. In
general, there is no valid justification for the
assumption. This is why the usual modes of
probability-based reasoning in the realm of law may
be open to challenge in legal proceedings.
Returning to the general, generic version of the
Robert example: Given the premises (a) QA's are B's;
(b) X is A; and the question: What is the
probability, P, that X is B, my claim that the
correct answer is "P is indeterminate," is not
likely to be accepted without challenge. Can anyone
point to an analysis of the example in question in
the literature of probability theory?
- --
Professor in the Graduate School
Director, Berkeley Initiative in Soft Computing(BISC)
Computer Science Division, Department of EECS
University of California
Berkeley, CA 94720-1776
Tel(office): (510) 642-4959
Fax(office): (510) 642-1712
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