Re: Simpson's Paradox and Quantum Entanglement

John Forkosh (forkosh@panix7.panix.com)
Sat, 20 Nov 1999 15:54:15 +0100 (MET)

ca314159@bestweb.net wrote:
: Simpson's Paradox:
: http://curriculum.qed.qld.gov.au/kla/eda/sim_par.htm
: Simpson's Paradox is a statistical artifact ...<snip>

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.

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