Compute extinction probability in a branching process
Company: StackAdapt
Role: Data Scientist
Category: Statistics & Math
Difficulty: hard
Interview Round: Technical Screen
# 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. \(P(\exists t: Z_t=0)\)?
### 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?
Overview: Compute extinction probability in a branching process evaluates statistical assumptions, formulas, estimation strategy, uncertainty, edge cases, and interpretation in a realistic interview setting. A strong answer states assumptions, handles edge cases, explains trade-offs, and shows how to validate the result clearly.