5 Keira has two unbiased coins. She tosses both coins. The number of heads obtained by Keira is denoted by \(X\).
- Find the probability generating function \(\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } )\) 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 \(Y\). - Find the probability generating function \(\mathrm { G } _ { Y } ( \mathrm { t } )\) of \(Y\).
The random variable \(Z\) is the total number of heads obtained by Keira and Hassan. - Find the probability generating function of \(Z\), expressing your answer as a polynomial.
- Use the probability generating function of \(Z\) to find \(\mathrm { E } ( Z )\).
- Use the probability generating function of \(Z\) to find the most probable value of \(Z\).