There is a colony consisting of 100 cells. Every minute, a cell dies with probability $\frac{1}{3}$; otherwise it splits into two identical copies. What is the probability that the colony never goes extinct?
The answer is $1-\left(\frac{1}{2}\right)^{100}$.

Let $p$ be the probability that a colony consisting of just one cell will survive. Then

$$
p=\frac{1}{3} \cdot 0+\frac{2}{3}\left(1-(1-p)^{2}\right)
$$

owing to the fact that when the cell splits in two, the probability both of them go extinct is $(1-p)^{2}$. Solving for $p$, we obtain $p=\frac{1}{2}$.

The colony initially has 100 cells; if we treat these as 100 distinct colonies, we obtain the claimed answer.