Probability (usage of recursion) 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?

Let $x$ (for "extinction") be the probability that the colony dies out. This can happen in two ways: either the first bacterium dies immediately (with probability $p$) or it splits and then neither of its children has infinite descendants.

This means we can write $x$ as

$$x = p + (1-p)x^2$$

which is a quadratic with two real roots. One solution is $x=1$ and the other is $x=\frac{p}{1-p}$. The latter is a valid probability (ie, it's between 0 and 1) only when $p \in [0,1/2]$ when the graph looks like this:

!Graph of p/\(1-p\) from 0 to 1/2

So extinction is guaranteed for $p \geq 1/2$ and then decreases as shown in the graph above as $p$ decreases. Survival is guaranteed, of course, if $p=0.$