There's a textbook treatment in "Quantum Probability",
Stanley P. Gudder, Academic Press 1988, ISBN 0-12-305340-4,
pages 102-106, which concludes it's not a problem.
John (forkosh@panix.com)
P.S. To see the "classical problem" consider, e.g., a college
with Law and Business schools, interested in its admissions
of Men vs. Women. It tabulates
#accepted/#applied=%accepted
for Men and Women at each school, finding %accepted is greater
for Women at both schools individually, but greater for Men
when the schools are combined. How's this possible? Consider...
Law School Business Combined
----------------------------------------------
Men 18/120=15% 180/240=75% 198/360=55%
----------------------------------------------
Women 24/120=20% 64/80 =80% 88/200=44%
The problem is that the "combining rule" is a/b,c/d --> (a+c)/(b+d)
which isn't a typical arithmetic operation, though it does model
the "word question" posed by the college.
Arithmetically, we have 18/120 < 24/120 and 180/240 < 64/80,
and we're intuitively concluding (18+180)/(120+240) < (24+64)/(120+80).
Substitute symbols, and a little algebra shows this isn't generally true.
(Note: It is true if the denominators at each school are equal,
e.g., multiply the Business Women by 3/3.)
Thus, ultimately, the "word question" isn't really well-posed
in terms of percentages, because division isn't linear in the sense
assumed by the problem.
############################################################################
This message was posted through the fuzzy mailing list.
(1) To subscribe to this mailing list, send a message body of
"SUB FUZZY-MAIL myFirstName mySurname" to listproc@dbai.tuwien.ac.at
(2) To unsubscribe from this mailing list, send a message body of
"UNSUB FUZZY-MAIL" or "UNSUB FUZZY-MAIL yoursubscription@email.address.com"
to listproc@dbai.tuwien.ac.at
(3) To reach the human who maintains the list, send mail to
fuzzy-owner@dbai.tuwien.ac.at
(4) WWW access and other information on Fuzzy Sets and Logic see
http://www.dbai.tuwien.ac.at/ftp/mlowner/fuzzy-mail.info
(5) WWW archive: http://www.dbai.tuwien.ac.at/marchives/fuzzy-mail/index.html