In a Galton-Walton branching process, the offspring distribution is given by

where

Find the probability of extinction (assuming )

i did:

find the generating function first G(s).

now to find e so i need to find smallest non negative solution of s=G(s)

and i get:

however the next part of the question asks you to find, supposing has a poisson distribution with mean . Supposing also that the probability of ultimate extinction is

I know that if then it will be

however i cannot see how to get another distribution into the answer