Most of the time, how well a student does on a test depends on how long they stu

Question

Most of the time, how well a student does on a test depends on how long they studied for it. Assume 5% of students who studied for a short amount of time got an A, 30% of students who studied a moderate amount of time got an A, and 60% of students who studied for a long period of time got an A. Typically, 35% of students study a short amount of time, 50% of students study a moderate amount of time, and 1 5% of students study a long amount of time. If a student got an A, what is the probability that they studied a long amount of time?

Explanation / Answer

Let

S, M, L = shirt, moderate, long period of studying, respectively

A = got an A

Hence, by Bayes' Rule,

P(A) = P(S) P(A|S) + P(M) P(A|M) + P(L) P(A|L) = 0.35*0.05 + 0.50*0.30 + 0.15*0.60 = 0.2575

Hence,

P(L|A) = P(L) P(A|L)/P(A) = 0.15*0.60/0.2575 = 0.349514563 [ANSWER]

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