Eric has three identical coins, each of which is biased so that the probability of obtaining a head when it is thrown is \(\frac{1}{3}\). The random variable \(X\) is the number of heads obtained when Eric throws the three coins at the same time.
(a) Find the probability generating function \(\mathrm{G}_{X}(t)\) of \(X\).

Eric also has two fair 6 -sided dice with faces numbered 1 to 6 . The random variable \(Y\) is the number of sixes obtained when Eric throws the two dice at the same time. It is given that the probability generating function of \(Y\) is \(\frac{25}{36}+\frac{10}{36} t+\frac{1}{36} t^{2}\).

Eric throws the three coins and the two dice. The random variable \(Z\) is the sum of the number of heads obtained and the number of sixes obtained.
(b) Find the probability generating function \(\mathrm{G}_{Z}(t)\) of \(Z\), expressing your answer as a polynomial in \(t\).

ს EXWE
(c) Use \(\mathrm{G}_{Z}(t)\) to find \(\mathrm{E}(Z)\) and \(\operatorname{Var}(Z)\).