(a) Let A be the event that the student knows the answer, and C be the event that the student answers correctly. We are given:

P(A) = 3/5 (probability that the student knows the answer)

P(A') = 2/5 (probability that the student guesses)

P(C|A) = 1 (probability of answering correctly given the student knows the answer)

P(C|A') = 1/3 (probability of answering correctly given the student guesses)

We want to find P(A|C), the probability that the student knows the answer given that they answered correctly. Using Bayes' Theorem:

P(A|C) = [P(C|A) * P(A)] / [P(C|A) * P(A) + P(C|A') * P(A')]

P(A|C) = [1 * (3/5)] / [1 * (3/5) + (1/3) * (2/5)]

P(A|C) = (3/5) / (3/5 + 2/15)

P(A|C) = (3/5) / (9/15 + 2/15)

P(A|C) = (3/5) / (11/15)

P(A|C) = (3/5) * (15/11)

P(A|C) = 9/11

(b) Let X be the random variable representing the value of the prize. The possible values of X are ₹8, ₹4, and ₹2. The probabilities of drawing each prize are:

P(X = 8) = 2/10 = 1/5

P(X = 4) = 5/10 = 1/2

P(X = 2) = 3/10

The mean value of the prize, E(X), is calculated as:

E(X) = Σ [x * P(x)]

E(X) = (8 * 1/5) + (4 * 1/2) + (2 * 3/10)

E(X) = 8/5 + 4/2 + 6/10

E(X) = 1.6 + 2 + 0.6

E(X) = 4.2