Split-in-three or die extinction

Question

A colony starts from a single cell. Each minute, every cell independently either dies (probability $\tfrac12$ ) or splits into $3$ cells (probability $\tfrac12$ ).

What is the probability the colony eventually goes extinct?

Accepted Answer

Let p be the extinction probability starting from a single cell. If a cell splits into 3, all three lineages are independent and the colony dies only if every one of them dies, probability p^3. Conditioning on the first minute: p = \frac12\underbrace{(1)}{	ext{dies}} + \frac12\underbrace{(p^3)}{	ext{3 copies}}. Multiply by 2 and rearrange: p^3 - 2p + 1 = 0. Spot the always-present root p = 1 ("extinct forever" is a fixed point) and factor: p^3 - 2p + 1 = (p - 1)(p^2 + p - 1) = 0. The quadratic gives p = 	frac{-1 \pm \sqrt{5}}{2}. The extinction probability is the smallest root in [0, 1]: p…

Split-in-three or die extinction

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