(i) To find the mean of W, calculate the expected value:

Mean = \(\frac{1+1+1+2+3+3}{6} = \frac{11}{6} \approx 1.83\)

For the standard deviation, use:

SD = \(\sqrt{\frac{(1-1.83)^2 + (1-1.83)^2 + (1-1.83)^2 + (2-1.83)^2 + (3-1.83)^2 + (3-1.83)^2}{6}}\)

SD = \(\sqrt{\frac{29}{6}} \approx 0.898\)

(ii) The probability distribution table for X is calculated by considering all possible sums of two throws:

(iii) Given \(E(Y) = 8\), and \(Y\) follows a binomial distribution with \(p = \frac{1}{3}\), we have:

\(np = 8 \Rightarrow n = 24\)

Variance is given by \(Var(Y) = np(1-p) = 24 \times \frac{1}{3} \times \frac{2}{3} = \frac{16}{3} \approx 5.33\)