# Finding the probability that the population becomes extinct for the first time in the third generation.

I am trying to solve the following problem and am wondering firstly if my solution is correct and alternatively if there is a shorter way to compute this.

In a branching process the number offspring per individual has a binomial distribution with parameters 2, p. Starting with a single individual, what is the probability that the population becomes extinct for the first time in the third generation.

The PGF of the binomial distribution is $G(s) = (ps + q)^n$

The probability that the population becomes extinct for the first time in the third generation can be evaluated by the function $G(G(G(0))) - G(G(0))$ and more precisely it is meant to be the smallest non-negative solution to the above equation.

$$G(G(G(s))) - G(G(s)) = (p(p(ps + q)^2+ q)^2)^2 - (p(ps + q)^2+ q)^2$$ Solving and substituting $n = 2$, $q= 1-p$ and $s=0$ I got $$G(G(G(0))) - G(G(0)) = p^2(p^3 - 2p^2 + 1)^4 - (p(p - 1)^2 - p + 1)^2$$ which expands out

$$= p^{14} - 8p^{13} + 24p^{12} - 28p^{11} - 8p^{10} + 48p^9 - 26p^8 - 24p^7 + 23p^6 + 8p^5 - 12p^4 - 2p^3 + 5p^2 - 1$$

According to Matlab other than 1 most of the other roots are either greater than, negative or complex.

I am unsure how to interpret these results and i suspect there to be a mistake in my solution.

Any help would be much appreciated.

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Your solution is almost corrrect. I get different coefficients for $p^8$ and below. Apparently, you left out a $+q$ in the outermost parentheses of the first summand. Plotting the graph of the correct expression $p^{14}-8p^{13} \pm\ldots$ for $0\le p\le1$ shows a nice curve that starts at 0, ends at 0 and has a single max inbetween. Why are you looking for roots? That would be values of $p$ such that extinction in exactly 3 generations is impossible and this is obviously the case only for $p=0$ or $p=1$.