A population at each step dies, or doubles or triples all with equal probability. What is the probability that the population dies?
I solved it as: If the probability of dying is $x$ , then after 1 time step either the population doubles , triples or dies (each with equal likelihood). Now a population can only die by doubling and dying $x^2$ , tripling and dying $x^3$ or 1 (since its dead). Hence
$ x = \frac{x^2 + x^3 + 1}{3} $
This has 3 solutions $ 1, \sqrt{2} -1 , -\sqrt{2} -1$. By constraints one can eliminate the negative answer, but what can be the reason to eliminate the answer 1. Could it be possible that the population is guaranteed to die ($x$ =1)
If at every step the population faces a chance of $1/3$ of dying, the chance that will never happen is $\displaystyle \lim_{n\to\infty}\left(\frac23\right)^n=0$. So it will die with probability $1$, indeed.