A laboratory keeps a population of aphids. The probability of an aphid passing a day uneventfully is $q<1$. Given that a day is not uneventful, there is probability $r$ that the aphid will have one offspring, probability $s$ that it will have two offspring and probability $t$ that it will die, where $r+s+t=1$. Offspring are ready to reproduce the next day. The fates of different aphids are independent, as are the events of different days. The laboratory starts out with one aphid.

Let $X_{1}$ be the number of aphids at the end of the first day. What is the expected value of $X_{1}$ ? Determine an expression for the probability generating function of $X_{1}$.

Show that the probability of extinction does not depend on $q$, and that if $2 r+3 s \leqslant 1$ then the aphids will certainly die out. Find the probability of extinction if $r=1 / 5, s=2 / 5$ and $t=2 / 5$.

[Standard results on branching processes may be used without proof, provided that they are clearly stated.]