It is a well-known fact(oid) that armadillos always have identical quadruplets (four offspring). Each of the 4 little armadillos has a 1/3 chance of becoming a doctor, a lawyer or a scientist, independently of its 3 siblings. A doctor armadillo will reproduce further with probability 2/3, a lawyer with probability 1/2 and a scientist with probability 1/4, again, independently of everything else. If it reproduces at all, an armadillo reproduces only once in its life, and then leaves the armadillo scene. (For the purposes of this problem assume that armadillos reproduce asexually.) Let us call the armadillos who have offspring fertile.
What is the distribution of the number of fertile offspring? Write down its generating function.
What is the generating function for the number of great-grandchildren an armadillo willhave? What is its expectation? (Note: do not expand powers of sums)
Let the armadillo population be modeled by a branching process, and let's suppose that itstarts from exactly one individual at time 0. Is it certain that the population will go extinct sooner or later?