I am trying to solve the following problem. In a branching process the number offspring per individual has a binomial distribution with parameters 2, p. Starting with a single individual, calculate the extinction probability.
I believe the solution to such a problem is evaluated using the equation $z=P(z)$ where of course $p(z)$ is pgf of the size of the nth generation.
$$z = (p(z-1) + 1)^n$$
Due to lack of enough appropriate examples i am unsure how to proceed from here. I believe due to the convolution of each generation depending on the last we have a recursive equation.
So would it be correct to solve for $z_2$ using
$z_2 = (p(z_1-1) + 1)^2$ and $z_1 = (p(z_0-1) + 1)^1$.