Computing conditional extinction probabilities for pollen cells

[psie]

Homework Statement

The growth dynamics of pollen cells can be modeled by binary splitting as follows: After one unit of time, a cell either splits into two or dies. The new cells develop according to the same law independently of each other. The probabilities of dying and splitting are 0.46 and 0.54, respectively.

(a) Determine the maximal initial size of the population in order for the probability of extinction to be at least 0.3.

(b) What is the probability that the population is extinct after two generations if the initial population is the maximal number obtained in (a)?

Relevant Equations

My previous post collects some useful assumptions about the Galton-Watson process: https://www.physicsforums.com/threads/understanding-extinction-probability-in-simple-gwp.1064663/

I'm studying branching processes, and I am bit confused about this exercise. The assumption of the processes I am studying (Galton-Watson processes) is that $X(0)=1$, i.e. the population starts with one single ancestor. Then $X(1)$ denotes the number of children obtained by the ancestor. The answer I get for (a) is 7, and this is what the book gets. The way I computed this was using the fact that the extinction probability $\eta$ is the smallest nonnegative root to $t=g(t)$, where $g(t)=q+pt^2$, so $\eta=q/p<1$ (here $q=1-p$). Now, from my previous post above, we've got $$P(\text{extinction}\mid A_k)=\eta^k,$$where $A_k=\{\text{the founding member produces } k\text{ children}\}, k\geq 0$. So we want to solve for $k$ in $$\eta^k\geq0.3,$$which gives approximately 7.5, and we round off to 7, since the inequality does not hold if $k=8$.

What confuses me about part (a) and how I solved it is that we are studying binary splitting, so the ancestor can only get 2 children. I am not sure if the number 7 relates to the ancestor's children or not. If 7 relates to the number of ancestors, then I'm not sure my formulas hold because they were derived under the assumption that $X(0)=1$. Also, I am confused about the probability one wants to find in (b). I can compute $P(X(n)=0)$ for $n\geq 1$, but I don't think this is the probability they are after. The answer here is 0.0206.

 

[psie]

Both parts are solved.

(a) Let there be $n$ cells to start with; each cell defines a Galton-Watson process and the extinction probability for each branch is $\eta$. So the probability for the entire population to go extinct is $\eta^n$.

(b) The probability we are looking for in (b) is that each branch goes extinct by generation $2$. From my previous post, we have $q_n=P(X(n)=0)$ and $q_n=\sum_{k=0}^\infty p_kq_{n-1}^k=p_0+p_2q_{n-1}^2$, where $q_0=0$. So $q_2\approx 0.5743$. For the entire population to go extinct by generation $2$, this probability is just $(0.5743)^7\approx 0.0206$.