A student has to take a multiple-choice exam with n possible answers in each question (only one being correct in each case). The probability that the student knows the correct answer to a question is p (0 < p < 1). If the student does not know the answer to a particular question, he chooses completely at random among the n possible options.

(i) What is the proportion of correct answers given by the student in the exam?

(ii) If the student gives the correct answer to a particular question, what is the probability that he knew the answer?

(iii) The course lecturer wants to ensure that a student who has studied carefully with p at least 80%, when he/she answers a question correctly, the probability that he/she knew the answer is at least 95%. Find the smallest number n of possible answers the lecturer should assign for each question.