What is the eventual extinction $\\xi$ for a branching process starting with one individual and with offspring distribution denoted $p_r$ for $r=0, 1, 2$ where:
\n$p_0 = \\var{a}$, $p_1 =\\var{b}$ and $p_2 =\\var{c}$ ? (please give exact expression/answer)
", "advice": "When the expected number of offspring is $\\leq 1$ then $\\xi = 1$. When this is not the case, we must find the smallest positive root of $s = G(s)$. In this case $G(s)$ is just a quadratic, and it is not too difficult to show that we obtain $\\xi = \\frac{p_0}{p_2}$.
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