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# For a student to qualify, he must pass at least two out of three exams. The probability that he will pass the 1st exam is p. If he fails in one of the exams then the probability of passing in the next exam is p/2 otherwise it remains the same. Find the probability that he will qualify.

6 years ago
Hello Student,
Let us consider
E1 ≡ event of passing I exam
E2 ≡ event of passing II exam
E3 ≡ event of passing III exam
Then a student can qualify in anyone of following ways
1. He passes first and second exam.
2. He passes fires, fails in second but passes third exam.
3. He fails in first, passes second and third exam.
∴ Required probability
= P (E1) P (E2| E1) + P (E1) P (E2| E1) P (E3| E2) + P (E1) P (E2| E1) P (E3| E2)
[as an event is dependent on previous one]
= p. p + p. ((1 – p). p/2 + (1 – p) p/2 p
= p2 + p2/2 – p3/2 + p2/2 – p3/2 = 2p2 – p3

Thanks