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 his passing in the next exam, is p/2 otherwise it remains the same.Find the probability that he will qualify.

Explanation

Let us consider the following events.
`{:(E_(1):"event of passing I exam",),(E_(2):"event of passing II exam",),(E_(3):"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 first, fails in second but passes third exam.
3. He fails in first, passes second and third exam.
Therefore, the required probability is
`P(E_(1))P(E_(2)//E_(1))+P(E_(1))P(E'_(2)//E_(1))P(E_(3)//E'_(2))`
`" "+P(E'_(1))P(E_(2)//E'_(1))P(E_(3)//E_(2))`
[as an event is dependent on previous one]
= p p `+p(1-p)p/2+(1-p)p/2p`
`=p^(2)+(p^(2))/(2)-(p^(3))/(2)+(p^(2))/(2)-(p^(3))/(2)=2p^(2)-p^(3)`