= Solution
This is a <Galton-Watson process> with offspring count $0$ with <probability> $q=1-p$ and $2$ with <probability> $p$. The descendants of distinct initial individuals evolve independently, so
$$
\alpha_i=\alpha_1^i
$$
for every finite initial size $i$. Conditioning on the first individual's outcome and applying the <law of total probability> gives
$$
\alpha_1=q+p\alpha_2=q+p\alpha_1^2.
$$
The equation factors as
$$
(\alpha_1-1)(p\alpha_1-q)=0.
$$
Its roots are $1$ and $q/p$. Since $p<1/2$, the second root exceeds one and cannot be a <probability>. Therefore \b[$\alpha_1=1$ and $\alpha_i=1$ for every finite initial size]. The population reaches the absorbing state zero almost surely; no choice between two admissible fixed-point <probabilities> remains in this subcritical case.
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