Question

The probability that a student passes a certain exam is 0.9, given that he studied. The probability that he passes the exam without studying is 0.2. Assume that the probability that the student studies for an exam is 0.75. Given that the student passed the exam, what is the probability that he studied?​

Answer 1

The probability that he studies is P(B1/D) = 0.9310

Solution:

• Let B1 be the event that he studied for the exam
• P(B1) = 0.75
• Let B2 be the event that he has not studied for the exam
• P(B2) = 0.25
• Now P(B1) + P(B2) = 0.75 + 0.25 = 1
• Let D be the event that the student has passed the exam.
• The probability is
• P(D/B1) = 0.9
• The probability of passing the exam without studying is
• P(D/B2) = 0.2
• The probability of passing the exam is
• P(B1/D) = P(B1) P(D/B1)  / P(D)
• P(B1/D) = P(B1) P(D/B1) / P(B1) P(D/B1) + P(B2) P(D/B2)

                    = 0.75 x 0.9 / 0.75 x 0.9 + 0.25 x 0.2

                    = 27/29 = 0.9310

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