Keira has two unbiased coins. She tosses both coins. The number of heads obtained by Keira is denoted by \(X\).
(a) Find the probability generating function \(\mathrm{G}_{X}(t)\) of \(X\).

Hassan has three coins, two of which are biased so that the probability of obtaining a head when the coin is tossed is \(\frac{1}{3}\). The corresponding probability for the third coin is \(\frac{1}{4}\). The number of heads obtained by Hassan when he tosses these three coins is denoted by \(Y\).
(b) Find the probability generating function \(\mathrm{G}_{Y}(t)\) of \(Y\).

The random variable \(Z\) is the total number of heads obtained by Keira and Hassan.
(c) Find the probability generating function of \(Z\), expressing your answer as a polynomial.

(d) Use the probability generating function of \(Z\) to find \(\mathrm{E}(Z)\).

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