KöMaL 2022 A832

Let us assume that the number of offsprings for every man can be $0, 1,\dots$ or $n$ with probabilities $p_0,p_1,\dots,p_n$ independently from each other, where $p_0+p_1+\dots+p_n=1$ and $p_n \ne 0$. (This is the so called Galton-Watson process.) Which positive integer $n$ and probabilities $p_0,p_1,\dots,p_n$ will maximize the probability that the offsprings of a given man go extinct in exactly the tenth generation?