Amoeba Question: A population of amoeba starts with 1. After 1 period, that amoeba can divide into 1, 2, 3, or 0 (it can die) with equal probability. What is the probability that the entire population will die out eventually?
The answer is generally given as the solution to :
$P=\frac{1}{4} + \frac{1}{4}P + \frac{1}{4}P^2 + \frac{1}{4}P^3$
Isn't it possible the amoeba, maybe let's say split into 2 and again split into 2 and then the 4 die? Basically
$P=\frac{1}{4} + \frac{1}{4}P + \frac{1}{4}P^2 + \frac{1}{4}P^3 + \frac{1}{16}P^4 + ...( \text{and other such possibilites}) $
Do we not go into deeper depth since it doesn't change the answer much or am I approaching the solution the wrong way?