An applicant hasa 0.26 probability of passing a test when they take it for thefi

Question

An applicant hasa 0.26 probability of passing a test when they take it for thefirst time, and if the pass it they can move on to the next stage.However, if they fail the test the first time, they must take thetest a second time, and when an applicant takes the test for thesecond time there is a 0.43 chance that they will pass and beallowed to move on to the next stage. The applicant is rejected ifthe test is failed on the second attempt.

a)      What is the probability that anapplicant moves on to the next stage but needs two attempts at thetest?

b)      What is the probability that anapplicant moves on to the next stage?

c)      If an applicant moves on to thenext stage, what is the probability that they passed the test onthe first attempt?

Explanation / Answer

First, lets name our events:

P(A) = P(The student passes exam atfirst try) = 0.26

P(A') = P(The student does not passexam at first try) = 0.74

P(B) = P(Student passes the exam atsecond try) = 0.43

P(B') = P(Student does not pass theexam at second try) = 0.57

a) P (The student passes exam,butinsecond try) = P (A')P(B) = (0.74)(0.43) = 0.3182

b) P(The student passes exam) = P(A B) = P(A) + P(A')P(B) = 0.26 + [(0.74)(0.43)] = 0.5782

c) P(A | A B) = P(A (A B)) / P(A B) = P(A) / P(A B) = 0.26 / (0.5782) =0.449671

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