Survival of the species (Solution)

Let $$e$$ be the probability that the species goes extinct. This is also the probability that one microorganism’s family line goes extinct.

$e = (1-p) + pe^2$

There’s a $$1-p$$ chance that the first organism does not multiply, in which case the species is extinct. With probability $$p$$, the organism multiplies and there are now two microorganisms. The probability that they both go extinct is simply $$e^2$$.

Solving for $$e$$ gives two solutions:

$e = \begin{cases} 1 \\ \frac{1 - p}{p} \end{cases}$

If $$p <= \frac{1}{2}$$, both solutions are $$>= 1$$, meaning that the species will definitely go extinct. If $$p > \frac{1}{2}$$, the solution $$\frac{1 - p}{p}$$ gives values for $$e$$ that are $$< 1$$, meaning the species has a chance.

For a better explanation of how to know which solution for $$e$$ to choose based on the value of $$p$$, look here.