Chapter 16 Events and Probability Spaces548
However, the university’s lawyers argued that across the university as a whole,
the percentage of male candidates granted tenure was actuallylowerthan the per-
centage for women candidates. This suggests that if there was any sex discrimi-
nation, then it was against men! Surely, at least one party in the dispute must be
lying.
Let’s simplify the problem and express both arguments in terms of conditional
probabilities. To simplify matters, suppose that there are only two departments, EE
and CS, and consider the experiment where we pick a random candidate. Define
the following events:
AWWDthe candidate is granted tenure, FEEWWDthe candidate is a woman in the EE department, FCSWWDthe candidate is a woman in the CS department, MEEWWDthe candidate is a man in the EE department, MCSWWDthe candidate is a man in the CS department.Assume that all candidates are either men or women, and that no candidate be-
longs to both departments. That is, the eventsFEE,FCS,MEE, andMCSare all
disjoint.
In these terms, the plaintiff is making the following argument:
PrAjFEE<PrAjMEEand
PrAjFCS<PrAjMCS:
That is, in both departments, the probability that a woman candidate is granted
tenure is less than the probability for a man.
The university retorts thatoverall, a woman candidate ismorelikely to be granted
tenure than a man; namely that
PrAjFEE[FCS>PrAjMEE[MCS:
It is easy to believe that these two positions are contradictory, and the phe-
nomenon illustrated here is widely referred to as “Simpson’s Paradox.” But there is
no contradiction or paradox, and in fact, Table 16.1 shows a set of candidate statis-
tics for which the assertions of both the plaintiff and the university hold. In this
case, a higher percentage of men candidates were granted tenure in each depart-
ment, but overall a higher percentage of women candidates were granted tenure!
How do we make sense of this?