(a) In answering a question on a multiple choice test, a student either knows the answer or guesses. Let 3/5 be the probability that he knows the answer and 2/5 be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability 1/3. What is the probability that the student knows the answer, given that he answered it correctly? OR (b) A box contains 10 tickets, 2 of which carry a prize of ₹8 each, 5 of which carry a prize of ₹4 each, and remaining 3 carry a prize of ₹2 each. If one ticket is drawn at random, find the mean value of the prize.

In a village of 8000 people, 3000 go out of the village to work and 4000 are women. It is noted that 30% of women go out of the village to work. What is the probability that a randomly chosen individual is either a woman or a person working outside the village?

Two dice are thrown. Defined are the following two events A and B: $A=\{(x,y):x+y=9\}$, $B=\{(x,y):x\ne3\}$ where (x, y) denote a point in the sample space. Check if events A and B are independent or mutually exclusive.

The probability distribution of a random variable X is given below : $$\begin{array}{|c|c|c|c|} \hline X & 1 & 2 & 3 \\ \hline P(X) & \frac{k}{2} & \frac{k}{3} & \frac{k}{6} \\ \hline \end{array}$$ (i) Find the value of $k$. (ii) Find $P(1\le X

For the vacancy advertised in the newspaper, 3000 candidates submitted their applications. From the data it was revealed that two third of the total applicants were females and other were males. The selection for the job was done through a written test. The performance of the applicants indicates that the probability of a male getting a distinction in written test is 0.4 and that a female getting a distinction is 0.35. Find the probability that the candidate chosen at random will have a distinction in the written test.