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Classic Problems of Probability by Prakash Gorroochurn

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Problem 30

Simpson's Paradox (1951)

Problem. Consider the three events X, Y, and Z. Suppose

img

Prove that it is still possible to have

img

Solution. Using the law of total probability, we have

(30.1) equation

(30.2) equation

where img and img. Therefore,

(30.3) equation

We now consider the sign of Eq. (30.3). Let img, img, and img. Then

img

where . Therefore, is negative if .

30.1 Discussion

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

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