# Compute extinction probability in a branching process You start with **1 cell at time 0**. Every **minute**, **each existing cell independently** transitions with **equal probability (1/4)** into one of four outcomes: - **Dies** (produces 0 cells) - **Stays the same** (produces 1 cell) - **Divides into 2 cells** (produces 2 cells) - **Divides into 3 cells** (produces 3 cells) Let \(Z_t\) be the number of cells after \(t\) minutes. 1. What is the probability the population is **extinct at \(t=2\) minutes**, i.e. \(P(Z_2=0)\)? 2. What is the probability the population **eventually goes extinct** (at some finite time), i.e. \(P(\exists t: Z_t=0)\)? ### Constraints & Assumptions - Preserve the scope, facts, inputs, and requested outputs from the prompt above. - If the prompt leaves a detail unspecified, state a reasonable assumption before relying on it. - Keep the answer interview-ready: concise enough to present, but concrete enough to implement or evaluate. ### Clarifying Questions to Ask - Clarify the random variables, distributional assumptions, independence assumptions, and desired output. - Show enough derivation for the interviewer to follow the reasoning. - Explain how you would validate the result with simulation or sensitivity checks. ### What a Strong Answer Covers - A correct setup with definitions, formulas, and boundary conditions. - A step-by-step derivation or estimation plan. - Interpretation of the result, including uncertainty and practical limitations. - Checks for assumptions, edge cases, and numerical stability. ### Follow-up Questions - How would the result change if the assumptions were relaxed? - Can you verify the answer with a simulation? - What is the most likely source of estimation error?