A recent college graduate is planning to take the first 3 actuarial exams this s

Question

A recent college graduate is planning to take the first 3 actuarial exams this summer. If she passes the first exam, she will take the second. If she passes the second, she'll take the third. If she fails an exam, then she is not allowed to take any others. The probability that she passes the first exam is 0.9. If she passes the first exam, the conditional probability that she passes the second one is 0.8; if she passes both the first and the second exams, then the conditional probability that she passes the third exam is 0.7 What is the probability that she passes all three exams? Given that she did not pass all three exams, what is the conditional probability that she failed the second exam?

Explanation / Answer

let E=passing in exam

P(E1)=0.9

P(E2|E1)=0.8

P(E3|(E1 and E2)=0.7

P(E1 and E2 and E3)=P(E3|(E1 and E2)*P(E2|E1)*P(E1)=0.9*0.8*0.7=0.504

(b) failing in exam can be like this

fail in E1 + pass in E1 and fail in E2+pass in E1 and E2 and fail in E3

so conditional probability she failed in second exam=P(E1)*(1-P(E2)=0.9*(1-0.8)=0.18

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