= Solution
\b[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,
$$
q=\lim_{n\to\infty}F_n(0),
$$
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.
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