Compute extinction probability in a branching process

Q: Compute extinction probability in a branching process

This question evaluates understanding of discrete-time branching processes (Galton–Watson), extinction probabilities, and probabilistic reasoning about independent offspring distributions in a stochastic process.

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Question

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.

What is the probability the population is extinct at $t=2$ minutes , i.e. $P(Z_2=0)$ ?
What is the probability the population eventually goes extinct (at some finite time), i.e. $P(\exists t: Z_t=0)$ ?

Compute extinction probability in a branching process

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Compute extinction probability in a branching process

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