Extinction of the population - Branching process with separate generations > Consider a branching process with separate generations and it is assumed that each individual of each generation, independently of the others, produces a number of individuals of the next generation according to a binomial distribution with parameter $3$ and $1/2$. > > At the initial generation, there are two individuals. Did you certain extinction of the population? Otherwise, what is the probability of extinction of the population? In this problem, what is meant by a branching process with separate generations?

The offspring distribution has generating function $$P(s) = \frac18 + \frac38 s + \frac38 s^2 + \frac18 s^3 = \frac18(1+s)^3, $$ and mean $$\mu = P'(1) = \frac32>1, $$ so if the initial generation had one individual, the probability of extinction is the unique solution to $P(\pi)=\pi$ with $0<\pi<1$, in which case $\pi=2-\sqrt3$. Now, since the initial generation has two individuals, we can treat these as two separate processes, and extinction occurs exactly when both processes expire. Hence the extinction probability is $$(2-\sqrt 3)^2=7-4\sqrt3\approx 0.0718. $$