A jar contains $n$ marbles. Initially, all of the marbles are red. Every minute, a marble is drawn at random from the jar, and then replaced with a blue marble. Let $T$ be the number of minutes until the jar contains only blue marbles. Compute the probability generating function $G_T$.
I thought to write T as a sum of iid random variables $T = T_1 + \ldots + T_N$ where $N$ is also a random variable taking values $n, n+1, n+2, \ldots$ as it doesn't seem feasible to calculate $\mathbb{P}(T=n+k)$. But then I can't seem to see what to take $T_i$ as. Can I have some guidance?