Four independent fair coins are tossed. The coins are modelled using a sample space , the outcomes in which are written as the quadruples (c1, c2, c3, c4) so that e.g. coin 1 coming up heads is recorded as c1 = 1 and the same coin coming up tails is recorded as c1 = 0. (a) Using correct mathematical notation, write down such that you avoid using a single written word (i.e. write down only mathematical symbols). [4] (b) Let the event Ak, where k {1, 2, 3, 4}, denote the kth coin coming up heads. Also, define the event B to mean that all coins come up heads. Compute the probabilities P(Ak), P(B), P(B|Ak) and P(Ak|B). Are the events Ak and B independent? [5] (c) Define the random variable X to be the sum of indicator variables of the sets Ak, i.e. let X = P4 k=1 1Ak , where 1A() = 1 if A 0 otherwise for any event A . Similarly, let the random variable Y be given as Y = 1B. Compute E[X], E[Y ], Var(X), Var(Y ) and Var(X + Y ).