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 .

# 30.1 Discussion

The algebra can mask the real implication of the three ...