Solution
Amoeba Extinction
Show the problem again
An amoeba each minute either dies, stays the same, splits into two, or splits into three, each with probability 1/4. Offspring behave independently and identically. What is the probability the population eventually dies out? Give an exact expression or a decimal rounded to 3 decimal places.
Worked solution
The answer is √2 − 1 ≈ 0.414. Let p be the extinction probability starting from one amoeba. Conditioning on the first minute: p = (1/4)(1 + p + p² + p³). Rearranging gives p³ + p² − 3p + 1 = 0, which factors as (p − 1)(p² + 2p − 1) = 0. The relevant root in (0,1) is p = √2 − 1 (the smallest nonnegative fixed point of a branching process's generating function).
Source: Standard Galton-Watson branching-process extinction problem, after Francis Galton and Henry William Watson (1875). Statement written for AxiomIQ.