Solution
= Solution
Both $\{\#C_N\geq\theta\#H_N/2\}$ 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
$$
\mathbb P(A_N)\geq\frac{\theta}{2}\theta=\frac{\theta^2}{2}.
$$