Probability and recurrence One day, one alien has come to the earth. Every day, each alien does one of four things, each with a probability of $\tfrac{1}{4}$:


*

*destroying himself

*splitting into 2 aliens

*splitting into 3 aliens

*doing nothing.


A newly created alien also does one of these 4 things every day starting on the next day, but different aliens do not necessarily do the same thing and act independently. Find the probability that there is eventually no alien on the earth.

Attempt: Let $p_n$ be the probability that there is no alien on the earth in the n-th day. Then the answer is $p=\sum_{i=1}^{\infty}p_i.$ We have $p_1=1/4$, $p_2= (1/4)^2+(1/4)^3+(1/4)^4$. Then naturally I think that $p$ is just a geometric series, but then an analysis on the 3rd day shows that this is a wrong guess.
Then I tried to find a recurrence relation of $p_n$ but in vein. I have also tried to find the complement $1-p$ but also failed.
 A: Let $P$ be the probability of the lineage of one alien dying out. Since we start with $1$ alien, $P$ is precisely the probability that there is eventually no alien left.
$P$ obeys $P=\frac{1}{4}(1+P+P^2+P^3)$, and moreover $P$ is the smallest positive root of this equation. Hence, $P=\sqrt{2}-1$.
Edit To see that $P=\frac{1}{4}(1+P+P^2+P^3)$, let $E$ be the event that a particular alien's line dies out eventually. Then, 
\begin{align}P&=\mathbb{P}(E)\\&=\mathbb{P}(E | \text{Destroys himself})\mathbb{P}(\text{Destroys himself}) + \mathbb{P}(E | \text{Splits into 2})\mathbb{P}(\text{Splits into 2})+\mathbb{P}(E | \text{Splits into 3})\mathbb{P}(\text{Splits into 3})+\mathbb{P}(E | \text{Does nothing})\mathbb{P}(\text{Does nothing})\\&=1\cdot\frac{1}{4}+P^2\cdot\frac{1}{4}+P^3\cdot\frac{1}{4}+P\cdot\frac{1}{4}\end{align}
where we note that $\mathbb{P}(E |\text{Splits into } n)=P^n$, since $E$ occurs iff each of the $n$ descendant lines die out independently (and each with probability $P$).
The fact that $P$ is the smallest positive root of the equation is a basic result of the theory of branching processes.
