The critical branching process

Question

A particle produces 0, 1, or 2 offspring each generation, each outcome with probability 	frac13, independently across particles. The population starts from one particle. What is the probability of eventual extinction? Reconcile the answer with the fact that the population can clearly grow.

Accepted Answer

Let q be the extinction probability from one particle. Conditioning on its offspring count: q = \frac13\underbrace{(1)}{0	ext{ kids}} + \frac13\underbrace{(q)}{1	ext{ kid}} + \frac13\underbrace{(q^2)}{2	ext{ kids}}. Multiply by 3 and rearrange: q^2 - 2q + 1 = 0 \quad\Longrightarrow\quad (q - 1)^2 = 0. The only root is q = 1: extinction is certain. Reconciling with growth. The mean offspring is \mu = \frac{0 + 1 + 2}{3} = 1, so this is a critical branching process. The general theory says extinction is certain whenever \mu \le 1 (unless every particle has exactly one child with probability …

The critical branching process

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