# martingales – Ameoba extinction probability

This has been asked several times on this website but I am trying to come up with a formal approach to justify why the extinction probability cannot be 1.

The problem statement is as follows. You start off with one amoeba in a petri dish. Every minute every ameoba in the dish either dies, does nothing, splits into two or splits into three. All these events have equal probability of $$1/4$$. What happens to ameobas are independent of each other. What is the probability that the ameobas eventually die out?

Let $$A_n$$ be the number of ameobas at minute $$n$$. $$A_0 = 1$$. Define $$tau := inf{n geq 1: A_n = 0}$$ and $$p := P{tau < infty}$$. By conditioning on what happens to the first ameoba, we can write

$$p = frac{1}{4}(1 + p + p^2 + p^3)$$
This has two positive solutions, namely $$sqrt{2}-1$$ and $$1$$. Based on some intuitive reasoning, we rule out $$1$$ as the solution. I want to make that intuition formal. I can write

$$E(A_{n+1} mid mathcal{F_n}) = frac{3}{2}A_n$$
$$E(A_n) = left(frac{3}{2}right)^n$$

I don’t see how these contradict $$tau < infty$$ a.s. $$A_n$$ is a submartingale. I tried stopping it at $$tau$$ but I don’t see where to go from there.