Problem 30
Simpson's Paradox (1951)
Problem. Consider the three events X, Y, and Z. Suppose
Prove that it is still possible to have
Solution. Using the law of total probability, we have
where 
and 
. Therefore,
We now consider the sign of Eq. (30.3). Let 
, 
, and 
. Then
where . Therefore, is negative if .
The algebra can mask the real implication of the three ...
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