 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, sofor every finite initial size $i$. Conditioning on the first individual's outcome and applying the law of total probability givesThe equation factors asIts roots are $1$ and $q/p$. Since $p<1/2$, the second root exceeds one and cannot be a probability. Therefore ${\alpha }_1=1$ 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.