Three of the six vertices of a regular hexagon are chosen at random. The probability that the triangle with three vertices is equilateral equals

Five persons A, B, C, D and E are in queue of a shop. The probability that A and E are always together, is

If A, B and C are mutually exclusive and exhaustive events of a random experiment such that $$P(B) = {3 \over 2}P(A)$$ and $$P(C) = {1 \over 2}P(B)$$, then $$P(A \cup C)$$ equals

A student answers a multiple choice question with 5 alternatives, of which exactly one is correct. The probability that he knows the correct answer is $$p,0 < p < 1$$. If he does not know the correct answer, he randomly ticks one answer. Given that he has answered the question correctly, the probability that he did not tick the answer randomly, is