Branching Processes - Binomial Distribution

Question

Let $(X_n)_{n \geq 0}$ be a branching process, with $X_0=1$

Given $Z \text{~} Binomial(2,\frac{1}{3})$. Find the probability that the branching process becomes extinct.

My Workings:

$G(S)= \mathbb{E}(s^Z) = (ps + q)^n = s$

$= (\frac{1}{3}s + \frac{2}{3})^2 = s$

$=\frac{1}{9}s^2 + \frac{4}{3}s + \frac{4}{9} = s$

$=\frac{1}{9}s^2 + \frac{1}{3}s + \frac{4}{9} = 0$

However, $S=1$ is then not a solution, which I thought it always had to be, so I think I have made a mistake / have misunderstood something?

Answer 1

$(\frac{1}{3}s + \frac{2}{3})^{2}$ should have been $(\frac{1}{3} + \frac{2s}{3})^{2}$ and the equation is $4s^{2}-5s+1=0$ of which $s=1$ is a solution.