 Solution

The eventual extinction probability is the smallest solution $\boxed{q\in [0,1]\text{ of }{F}_1(q)=q}$. This is the Galton-Watson extinction fixed point result for one initial individual. Equivalently,since having an empty generation is an absorbing event and the extinction-by-generation events increase. In particular extinction is certain when the smallest fixed point is one. With $p_0>0$, an offspring mean at most one implies this, whereas a mean greater than one gives a fixed point below one. The fixed-point statement itself requires no finite-mean assumption.