If p is the probability that a single amoeba's descendants will die
out eventually, the probability that N amoebas' descendents will all
die out eventually must be `p ^{N}`, since each amoeba is independent of
every other amoeba. Also, the probability that a single amoeba's
descendants will die out must be independent of time when averaged
over all the possibilities. At

The generating function for the sequence `P(n,i)`, which gives the
probability of i amoebas after n minutes, is `f ^{n(x)}`,
where

Then `f _{n}(0)` gives the probability of 0 amoebas after

`f _{n}(x) = ( 1 + f_{n}(x) + (f_{n}(x))^{2} + (f_{n}(x))^{3} )/4`

so that if `f _{n+1}(0) -> f^{n}(0)` we can solve the equation.

The generating function also gives an expression for the expectation
value of the number of amoebas after n minutes. This is ` ^{d}/_{dx}(f_{n}(x))`
evaluated at