Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 209 / 2 / a3 / Solution

Both $\{\#C_N\ge \theta \#H_N/2\}$ and $\{0\in C_N\}$ are increasing events. The first has probability at least $\theta /2$ by part (a2), while the second has probability at least $\theta$ because an infinite origin cluster reaches $h_N$. The Harris-FKG inequality gives