Binary-splitting bacterium extinction

Question

A single bacterium reproduces once per generation. Each generation it either dies (probability $\tfrac13$ ) or divides into exactly $2$ daughter cells (probability $\tfrac23$ ), independently.

What is the probability the lineage eventually goes extinct?

Accepted Answer

Let q be the extinction probability starting from one bacterium. Condition on the first generation. If it dies (probability 	frac13) the line is gone; if it divides (probability 	frac23), both daughters found independent lineages and the line dies only if both die, probability q^2: q = \frac13 + \frac23 q^2. Multiply by 3 and rearrange into a quadratic: 2q^2 - 3q + 1 = 0 \quad\Longrightarrow\quad (2q - 1)(q - 1) = 0. The roots are q = 	frac12 and q = 1. The extinction probability is the smaller root in [0, 1]: q = \frac12. Why not q = 1? The mean offspring is \mu = 	frac13(0) + 	frac23(2…

Binary-splitting bacterium extinction

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