= Extinction probability of a branching process
{title2=$q=\mathbb P(\exists j:Z_j=0)$}
= Branching extinction probability
{synonym}
For a <Galton-Watson process> with one ancestor, the <branching extinction probability> is the <probability> that some generation is empty:
$$
q=\mathbb P\left(\bigcup_{n\geq0}\{Z_n=0\}\right)=\lim_{n\to\infty}\mathbb P(Z_n=0).
$$
The equality uses the <absorbing state> at zero. The complementary <branching survival probability> $1-q$ is the <probability> that every generation is nonempty. The <Galton-Watson extinction fixed point> computes $q$ from the <offspring distribution>.
For $p_0=p_1=1/4$, $p_2=1/2$, one ancestor has $q=1/2$. Starting instead from $m$ ancestors with <independent> descendant families gives
$$
\mathbb P(\text{eventual extinction}\mid Z_0=m)=q^m,\qquad \mathbb P(Z_n=0\mid Z_0=m)=q_n^m.
$$
All $m$ families must die out. With three ancestors, eventual extinction has <probability> $1/8$ and survival has <probability> $7/8$; by generation two, extinction has <probability> $(11/32)^3=1331/32768\approx0.040619$. Extra ancestors change the extinction <probability> without changing the <offspring distribution>. https://www.uni-muenster.de/Stochastik/lehre/WS1011/SpezielleStochastischeProzesse/Ch_1.pdf[University of Münster branching-process notes, Lemma 1.2 and equation (1.10)]{ref}
For finite offspring <mean> $\mu$, the extinction classification is:
* If $\mu<1$, extinction has <probability> $1$. For example $p_0=3/4$, $p_2=1/4$ gives $\mu=1/2$ and $F(s)=3/4+s^2/4$; its only <fixed point> in $[0,1]$ is $1$.
* If $\mu=1$ and $p_1\ne1$, extinction also has <probability> $1$. For example $p_0=p_2=1/2$ gives $F(s)-s=(s-1)^2/2$ and $q=1$, despite <expected generation size in a Galton-Watson process> remaining one.
* If $1<\mu<\infty$, extinction has <probability> $q<1$. The value can be zero: if $p_0=0$, every individual has at least one child, so an initially nonempty family never becomes empty. In particular $p_2=1$ gives $Z_n=2^n$ and $q=0$.
The critical exception is $p_1=1$: every individual has exactly one child, $F(s)=s$, and $Z_n=1$ forever from one ancestor. Thus $\mu=1$ but $q=0$; a blanket claim that every critical <Galton-Watson process> becomes extinct would be false. These cases concern the <Galton-Watson process> with its common <offspring distribution> and <independence> assumptions. https://www.uni-muenster.de/Stochastik/lehre/WS1011/SpezielleStochastischeProzesse/Ch_1.pdf[University of Münster branching-process notes, Section 1.4, trivial cases and Theorem 1.7]{ref}
This <probability> answers whether any lineage persists, which the offspring <mean> alone cannot quantify in the supercritical case. A <branching process conditioned on extinction> uses the same <fixed point> to reweight reproduction and describe families known eventually to die out.
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