TITLE: Probability (usage of recursion)
QUESTION [5 upvotes]: In an hour, a bacterium dies with probability $p$ or else splits into two. What is the probability that a single bacterium produces a population that will never become extinct?

REPLY [2 votes]: \begin{align}
r & =\Pr(\text{posterity lives forever}) \\[10pt]
& = \Pr(\text{split})\cdot\Pr(\text{at least one offspring lives forever}) \\[10pt]
& = (1-p)\cdot(1-\Pr(\text{both lines eventually die out})) \\[10pt]
& = (1-p)\cdot(1-(1-r)^2).
\end{align}
So
$$
r = (1-p)(1-(1-r)^2).
$$
That's a quadratic equation in the variable $r$.  One solution is obviously $r=0$, so the polynomial $(1-p)(1-(1-r)^2)-r$ can be factored as
$$
(1-p)(1-(1-r)^2)-r = r\cdot\text{something}.
$$
Then write $\text{something}=0$ and it's not even quadratic.