6 A bag contains 4 red balls and 6 blue balls. Rassa selects two balls at random, without replacement, from the bag. The number of red balls selected by Rassa is denoted by \(X\).
- Find the probability generating function, \(\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } )\), of \(X\).
Rassa also tosses two coins. One coin is biased so that the probability of a head is \(\frac { 2 } { 3 }\). The other coin is biased so that the probability of a head is \(p\). The probability generating function of \(Y\), the number of heads obtained by Rassa, is \(\mathrm { G } _ { Y } ( \mathrm { t } )\). The coefficient of \(t\) in \(\mathrm { G } _ { Y } ( \mathrm { t } )\) is \(\frac { 7 } { 12 }\). - Find \(\mathrm { G } _ { Y } ( \mathrm { t } )\).
The random variable \(Z\) is the sum of the number of red balls selected and the number of heads obtained by Rassa. - Find the probability generating function of \(Z\), expressing your answer as a polynomial.
- Use the probability generating function of \(Z\) to find \(\mathrm { E } ( Z )\).
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