Consider an amoeba. At each iteration, it can split into 2, 3 amoebas, stay the same, or die. These 4 events occur with equal probability.
Let $E$ denote the event that all current amoeba dies, and $F_1, F_2, F_3, F_4$ denote the above 4 events, then we have
\begin{align} P(E) = P(E|F_1)P(F_1) + P(E|F_2)P(F_2) + P(E|F_3)P(F_3) + P(E|F_4)P(F_4) \\ P(F_i) = \frac{1}{4} \\ P(E|F_1) = 1 \\ P(E|F_2) = P(E)^2 \\ P(E|F_3) = P(E)^3 \\ P(E|F_4) = P(E) \\ P(E) = \frac{1}{4}[1 + P(E) + P(E)^2 + P(E)^3] \\ \end{align}
Solving this cubic, we find $P(E) = 1, 1 \pm \sqrt{2}$. We throw out the negative root, and we are left with $P(E) = 1, \sqrt{2} - 1$. In the book I am reading, it restricts the probabilities to $P(E) < 1$.
But intuitively, I do not understand why $P(E) = 1$ isn't possible. Can someone intuitively explain this?
In addition, using $P(E) = \sqrt{2} - 1$, does this mean that the expectation of the number of iterations that the amoeba population dies out is infinity?
One thought that occurred to me is that the expected number of amoeba after one iteration is $0.25(1 + 2 + 3 + 0) = 1.5$, and by induction we see that this continues to grow.
On average one amoeba produces $1.5$ the next round, so if there are a large number of them the probability of extinction is very small. This shows you can't have extinction with probability $1$.