Confusing probability riddle

Question

There exists a Snail lineage . in each generation every snail, independent from the other: Dies at probability 1/3, Have a single offspring and then dies at probability 1/3 Have 2 offspring and then dies at probability 1/3

the lineage goes extinct if from the single offspring in generation 0 ,there are no remaining offspring in some generation i . what is the probability that the lineage goes extinct?

this is so confusing to me i cant even wrap my head around it to start, please help.

Answer 1

Let's go through each of the three cases for the first generation:

1. With probability $\frac 1 3$ the first snake dies without any offspring, which makes the lineage go extinct.
2. With probability $\frac 1 3$ the first snake dies after having a single offspring. We are then left with one generation consisting of one snake, which is the same scenario we started with. Hence, if we denote by $P$ the probability that our lineage goes extinct (what we're trying to solve for), in this case the lineage will go extinct with probability $P$.
3. With probability $\frac 1 3$ the first snake has two offsprings and then dies. In order for this lineage to go extinct, both of the offspring's lineages need to go extinct. Each of these events happens with probability $P$, which gives a probability $P^2$ of both events happening at once.

Hence, in total, we get that $P = \frac 1 3 + \frac 1 3P + \frac 1 3 P^2$. Solving for $P$, we get \begin{alignat*}{2} &&3P &= 1 + P + P^2\\ &\iff&\qquad P^2 - 2P + 1 &= 0\\ &\iff&\qquad (P - 1)^2 &= 0\\ &\iff&\qquad P &= 1 &\end{alignat*} so for time tending to infinity the lineage is certain to go extinct.

(There are a few issues with this, since we assume that such a probability $P$ exists in the first place, but an approach like this is nice for getting a feeling for the problem)

Answer 2

Hint: the probability $p(n)$ that there are no descendants of a given individual after $n$ generations is $\frac{1}{3} + \frac{1}{3}p(n-1) + \frac{1}{3}p(n-1)^2$.