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Model the branching process using probability generating functions (PGFs) and compute the probability of extinction after exactly two minutes by finding the probability that all lineages from the initial cell are extinct at time 2. Use the fact that each cell's offspring distribution is uniform over {0,1,2,3}, and compute the extinction probability recursively.
Pro tip: Clearly distinguish between extinction by time 2 and extinction exactly at time 2; here 'after exactly 2 minutes' means the population is extinct at time 2 but not necessarily at time 1. However, since extinction is absorbing, being extinct at time 2 implies it was extinct at time 1 if it was extinct at time 1, but the question likely means the population size is 0 at time 2. Clarify this with the interviewer.
Let X_n be the population size at minute n. Start with X_0 = 1. Each cell independently produces a random number of offspring according to the distribution: P(0)=P(1)=P(2)=P(3)=1/4.
The PGF for the number of offspring per cell is f(s) = (1 + s + s^2 + s^3)/4. This encodes the distribution of offspring.
The probability that a single cell's lineage is extinct after one minute is the probability it produces 0 offspring, which is f(0) = 1/4. So P(X_1 = 0) = 1/4.
The probability of extinction at time 2 is f(f(0)) = f(1/4). Compute f(1/4) = (1 + 1/4 + (1/4)^2 + (1/4)^3)/4 = (1 + 0.25 + 0.0625 + 0.015625)/4 = 1.328125/4 = 0.33203125. Thus P(X_2 = 0) = 0.33203125.
If the question means extinct exactly at time 2 (i.e., not extinct at time 1 but extinct at time 2), then compute P(X_2=0) - P(X_1=0) = 0.33203125 - 0.25 = 0.08203125. Clarify with the interviewer which interpretation is intended.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
First, clarify the stochastic process and parameters from the setup (e.g., branching process, birth-death chain). Then, set up equations for the probability of eventual extinction, typically using generating functions or first-step analysis. Solve for the smallest non-negative root of the equation, and interpret the result in context.
Pro tip: In branching processes, extinction is certain if the mean offspring number μ ≤ 1, but if μ > 1, the extinction probability is the unique solution in [0,1) of s = G(s), where G is the offspring generating function. Mentioning this theorem shows deep understanding.
Determine the type of stochastic process (e.g., Galton-Watson branching process, birth-death chain) and extract key parameters such as offspring distribution or transition rates.
Let q be the probability that the population eventually hits 0. For a branching process, q satisfies q = G(q), where G is the probability generating function of the offspring distribution.
Find the smallest non-negative solution to the equation. If the mean offspring μ ≤ 1, then q = 1; if μ > 1, q is the unique solution in [0,1).
Check that the solution makes sense (e.g., q=1 for subcritical/critical, q<1 for supercritical) and relate it back to the original problem context.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.