A bag contains four balls numbered 1, 2,3 and 4. In a game, a ball is drawn at random from the bag and then a fair coin is tossed a number of times that is equal to the number shown on the ball drawn. The random variable \(X\) is the number of heads recorded.

(i) Show that \(P (X=0)=\frac{15}{64}\). Find \(P (X=x)\) for all other possible values of \(x\).

[5]

(ii) Denoting the expectation and variance of \(X\) by \(\mu\) and \(\sigma^{2}\) respectively, find \(P (X>\mu)\) and show that \(\sigma^{2}=\frac{15}{16}\).

[3]

Adam plays this game 10 times.

(iii) Find the probability that there are at least two games with at least 2 heads recorded.

[2]

Bill plays this game 50 times.

(iv) Using a suitable approximation, estimate the probability that the average number of heads recorded is less than 1 .

[3]

(ii) $\frac{23}{64}$ or $0.359375$

(iii) $0.923(3$ s.f.)

(iv) $0.0339$ (3 s.f.)