Branching process extinction (critical)

HardProbability Stochastic Processes~6m

Each individual has $0$ offspring with probability $1/2$ and $2$ offspring with probability $1/2$ . Starting from one individual, what is the extinction probability?

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Solve $q = \tfrac12 + \tfrac12 q^2$ for the smallest root.
Mean offspring exactly $1$ is the critical case.

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1 - extinction is certain (critical mean = 1)

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Branching process extinction (critical)

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