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. My textbook answer reads $2p^2 – p^3$. This is possible if only the below cases are considered:
- He passes first and second exam.
- He passes first, fails in second but passes third exam.
- He fails in first, passes second and third exam.
But I think this is wrong since at least two out of three exams means,passing in first, second and third exam is inclusive. Someone please solve this paradox.