(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 = 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 Berkson’s paradox, especially in the context of admissions to a school, hospital, etc.

Explanation / Answer

B&C SOLVED TOGETHER)

HERE IN THIS QUESTION THE CRITERIA OF THE COLLEGE TO GIVE ADMISSION ON THE BASIS OF MATHS SCORE OR HIS ABILITY TO PLAY BASEBALL BUT THE CRITERIA IS NEGATIVELY SET BY THE COLLEGE AS THE TWO ARE NOT INDEPENDENT OF THE OTHERS BECAUSE ONE BEING GOOD IN MATHS MAY WILL LEAD TO BE LESS GOOD IN THE BASEBALL

SIMILARLY BEING GOOD AT BASEBALL WILL AFFECT BEING GOOD IN MATHS AT THE SAME TIME SO THIS WILL AFFECT THE OVERALL ADMISSION

ALSO C= A/B

C = P(A INTERSECTION B) / P(B)

NOW WE KNOW THAT P(A INTERSECTION B) = VERY LESS AS VERY LESS NUMBER OF STUDENTS WILL BE AVAILABLE WHO WILL BE GOOD AT BOTH MATHS AND THE BASEBALL

HENCE THE C WILL BE DEPENDENT ON BOTH A AND B

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