Past exam of the mathematics course of the University of Cambridge / 2026 / ia / Paper 2 / 12F / iii / Solution

The extinction probability is the smallest fixed point of the convex generating function $F_1$ in $[0,1]$. Since $F_1\prime (1)=\mu \le 1$ and $p_0,p_2>0$ make strict convexity relevant, the only fixed point is $1$.