. A branching process has initially one individual. Use the law of total probability in the form Pr (extinction) = ^ Pr (extinction |/c descendants) Pr{k descendants)

k to deduce that the extinction probability x is a solution of x = P(x).

Let n = 0, 1,2,...} be a branching process with Xq = 1 and with the number of offspring per individual 0,1,2 with probabilities p,q,r, respectively, where p + -f r = 1 and p,q,r> 0. Show that if + 2r > 1, the probability of extinction is

^ I - q - - q ) ^ -4 p r 2r