Extinction Probability of an Amoeba Population

This question considers a simple branching process model for a reproducing population, framed as the evolution of an amoeba colony in discrete time. At each step, an individual organism can die, stay as one, or produce multiple offspring, with all descendants behaving independently and identically. Candidates are asked to determine the probability that the entire population eventually disappears, starting from a single ancestor. The setup is a canonical example of a Galton–Watson process and is a common style of probability question in quantitative finance interviews, particularly for roles that value comfort with stochastic processes and long-term event probabilities.

The solution leans on conditional probability, recursion, and fixed-point reasoning. A key step is expressing the extinction probability in terms of itself by conditioning on the first reproduction event and using independence between offspring lineages. Candidates must correctly construct and solve a nonlinear equation that encodes the offspring distribution, then select the appropriate root based on probabilistic constraints. Interviewers look for clear identification of the recursive structure, comfort manipulating probability-generating ideas, and the ability to justify why one solution represents the true extinction probability while others are discarded.

What it tests

This problem class is governed by the recursive structure of branching processes, where the fate of the entire population can be described in terms of the independent fates of its descendants. The extinction probability is a fixed point of a function that encodes the offspring distribution: if each entity acts independently, the overall extinction probability satisfies a self-referential equation. This arises because after each branching event, the future evolution of each branch is probabilistically identical to the original process, so the total extinction probability is a function of itself. The reason this recursive equation works is that independence and identical behavior allow us to multiply probabilities for separate branches, and sum over all possible first-step outcomes weighted by their probabilities. The solution is always the smallest non-negative root of the resulting fixed-point equation, as larger roots correspond to trivial or impossible scenarios.