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.