There is a one amoeba in a pond. After every minute the amoeba may die, stay the same, split into two or split into three with equal probability. All its offspring, if it has any, will behave the same (and independent of other amoebas). What is the probability the amoeba population will die out?
The question has a straightforward solution by modeling the long-run probability using a cubic equation (answer is ~0.414), but I was curious about solving it by modeling it as a Markov Chain.
From my understanding, the population can be modeled as a Markov chain with multiple transient states, with the state of 0 population being its sole, self-absorbing recurrent state. Since this is the case, why is the long-run probability not equal to one? I was under the assumption that in such a Markov chain, in the long run we would always reach the single recurrent state.
I understand that the expectation of the time for this might be infinite, but I thought the probability of reaching this state was always 1?