Extinction chance for a colony that starts with three cells

A bacterial colony evolves in generations: every living cell, independently of the others, either dies without offspring or produces a random positive number of new cells. Starting from a single cell, the colony eventually dies out with probability $\tfrac13$. What is the probability of eventual extinction for a colony that starts with three cells?

$\tfrac1{27}$. The three founding cells generate independent family trees, each a copy of the one-cell process. The colony dies out only if all three lines do, which has probability $(\tfrac13)^3=\tfrac1{27}$.