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

Chapter 2, Problem 36

(choose chapter or problem)

(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 = A[B, 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 Berksons paradox, especially in the context of admissions to a school, hospital, etc.