(a) Suppose that in the population of college applicants, being good at baseball

Question

(a) Suppose that in the population of college applicants, being good at baseball is independent of having a good math score on a certain standardized test (with respect to some measure of “good”). A certain college has a simple admissions procedure: admit an applicant if and only if the applicant is good at baseball or has a good math score on the test.

Give an intuitive explanation of why it makes sense that among students that the college admits, having a good math score is negatively associated with being good at baseball, i.e., conditioning on having a good math score decreases the chance of being good at baseball.

(b) Show that if A and B are independent and C = AB, then A and B are conditionally dependent given C (as long as P (A B) > 0 and P (A B) < 1), with

P(A|B,C) < P(A|C).

This phenomenon is known as Berkson’s paradox, especially in the context of ad- missions to a school, hospital, etc.

UPDATE:

They're asking for an "intuitive explanation" in (a), and (b) seems pretty explicitly stated. What additional information is needed??

Explanation / Answer

(a) Suppose that in the population of college applicants, being good at baseball

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