Amoeba Population

A pond holds one amoeba. Each minute it dies, stays the same, splits into two, or splits into three, each with probability one quarter, and every descendant behaves the same way independently. What is the probability the population eventually dies out?

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#A branching process

Track the family as a branching process. In one minute an amoeba leaves behind $0$ , $1$ , $2$ , or $3$ copies of itself, each with probability $\tfrac{1}{4}$ .

graph LR
  A(("amoeba")) -->|1/4| D["die (0)"]
  A -->|1/4| S["stay (1)"]
  A -->|1/4| T["two (2)"]
  A -->|1/4| H["three (3)"]

#The extinction equation

Let $q$ be the probability that the line from one amoeba dies out. The first minute splits the future into independent sub-lines, one per offspring, each extinct with probability $q$ . Conditioning on the number of offspring,

q = \tfrac{1}{4}\bigl(1 + q + q^2 + q^3\bigr). \tag{1}

The mean number of offspring is $\tfrac{0 + 1 + 2 + 3}{4} = \tfrac{3}{2} > 1$ , so the colony can run away and $q < 1$ .

#Solving

Clearing the quarter gives $q^3 + q^2 - 3q + 1 = 0$ . The value $q = 1$ always solves the extinction equation, and factoring it out leaves

(q - 1)(q^2 + 2q - 1) = 0. \tag{2}

The quadratic has a single root in $[0, 1)$ , namely

q = \sqrt{2} - 1 \approx 0.414. \tag{3}

So the colony escapes extinction and grows forever about $59\%$ of the time.