Problem

A bag contains 7 red balls and 3 blue balls. Kieran selects 2 balls at random, without replacement. The number of red balls selected by Kieran is denoted by $X$, and the number of different colours present in Kieran’s selection is denoted by $Y$.

(a) Find the probability generating functions, $G_X(t)$ of $X$ and $G_Y(t)$ of $Y$.

The random variable $Z$ is the sum of the number of red balls and the number of different colours present in Kieran’s selection. Kieran claims that the probability generating function of $Z$ is equal to $G_X(t) \times G_Y(t)$.

(b) Explain why Kieran is incorrect.

(c) Find the probability generating function of $Z$, expressing your answer as a polynomial in $t$.

(d) Use the probability generating function of $Z$ to find $E(Z)$.