Amoeba extinction probability

Question

An amoeba population starts with a single amoeba. Each minute, every amoeba independently does one of four things, each with probability $\tfrac14$ : dies, stays the same, splits into $2$ , or splits into $3$ .

What is the probability that the population eventually goes extinct?

Accepted Answer

Let p be the extinction probability starting from a single amoeba. The offspring of one amoeba found independent lineages, and the whole population dies iff every lineage dies. Conditioning on the first step: p = \frac14\underbrace{(1)}{	ext{dies}} + \frac14\underbrace{(p)}{	ext{1 copy}} + \frac14\underbrace{(p^2)}{	ext{2 copies}} + \frac14\underbrace{(p^3)}{	ext{3 copies}}. Multiply by 4 and rearrange: p^3 + p^2 - 3p + 1 = 0. Spot the root p = 1 (always a root, "extinct forever" is a fixed point) and factor: p^3 + p^2 - 3p + 1 = (p - 1)(p^2 + 2p - 1) = 0. The quadratic gives p = -1 \pm \…

Amoeba extinction probability

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