A 6-sided dice, \(A\), with faces numbered 1, 2, 3, 4, 5, 6 is biased so that the probability of throwing a 6 is \(\frac14\). The random variable \(X\) is the number of 6s obtained when dice \(A\) is thrown twice.

(a) Find the probability generating function of \(X\).

A second dice, \(B\), with faces numbered 1, 2, 3, 4, 5, 6 is unbiased. The random variable \(Y\) is the number of 6s obtained when dice \(B\) is thrown twice.

The random variable \(Z\) is the total number of 6s obtained when both dice are thrown twice.

(b) Find the probability generating function of \(Z\), expressing your answer as a polynomial.

(c) Find \(\operatorname{Var}(Z)\).

(d) Use the probability generating function of \(Z\) to find the most probable value of \(Z\).