Compute extinction probability in a branching process
Company: StackAdapt
Role: Data Scientist
Category: Statistics & Math
Difficulty: Hard
Interview Round: Technical Screen
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. \(P(\exists t: Z_t=0)\)?
Quick Answer: This question evaluates understanding of discrete-time branching processes (Galton–Watson), extinction probabilities, and probabilistic reasoning about independent offspring distributions in a stochastic process.