= Solution
By the <union bound>,
$$
\mathbb P_p(0\leftrightarrow\partial B(n))
\leq\sum_{z\in\partial B(n)}
\mathbb P_p(0\leftrightarrow z).
$$
Hence some $z\in\partial B(n)$ has connection probability at least the left side divided by $|\partial B(n)|$. Some coordinate of $z$ equals $n$ or $-n$. A coordinate permutation and reflection of $\mathbb Z^d$ sends that face to the face $x_1=n$ and preserves the <bond percolation> law. Its image $x$ therefore satisfies
$$
\boxed{\mathbb P_p(0\leftrightarrow x)
\geq\frac{\mathbb P_p(0\leftrightarrow\partial B(n))}{|\partial B(n)|}.}
$$
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