6 Tanji has a bag containing 4 red balls and 2 blue balls. He selects 3 balls at random from the bag, without replacement. The number of red balls selected by Tanji is denoted by \(X\).
- Find the probability generating function \(\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } )\) of \(X\).
Tanji also has two coins, each biased so that the probability of obtaining a head when it is thrown is \(\frac { 1 } { 4 }\). He throws the two coins at the same time. The number of heads obtained is denoted by \(Y\). - Find the probability generating function \(\mathrm { G } _ { Y } ( \mathrm { t } )\) of \(Y\).
The random variable \(Z\) is the sum of the number of red balls selected by Tanji and the number of heads obtained. - Find the probability generating function of \(Z\), expressing your answer as a polynomial.
- Use the probability generating function of \(Z\) to find \(E ( Z )\) and \(\operatorname { Var } ( Z )\).
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