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2. (Probability - 30 marks) Consider the following random experiment: Three coins are inde- pendently tossed; for each coin the probability that heads shows up is 2. (a) What is the sample space, denoted by $2? What are the probabilities for all elementary events in ? [6] (b) What is the probability that there are more heads than tails? What about the proba- bility that there are twice as many heads as tails? [4] (c) Find two events which are independent. Find another two events which are not inde- pendent. Justify! [6] (d) Construct a random variable X on ? and compute its expectation (showing the final result as a rational number suffices). [6] (e) Construct a random variable Y on S such that P(Y > 10) = 3