= Solution
In <bond percolation> on an <undirected graph> $G=(V,E)$, a configuration is $\omega\in\{0,1\}^E$. The <edge> $e$ is open when $\omega_e=1$ and closed when $\omega_e=0$. Under $\mathbb P_p$, the coordinates are <independent and identically distributed random variables> with the <Bernoulli distribution> of parameter $p$. For a countable <edge> set, use the <product sigma-algebra> and the <product measure>
$$
\mathbb P_p=\bigotimes_{e\in E}\operatorname{Bernoulli}(p).
$$
The open <edges> form a random subgraph with the original <graph vertices>. Its <connected components of a graph> are the <percolation clusters>. Write $x\leftrightarrow y$ when an open <graph path> joins $x$ to $y$; a zero-length <graph path> is allowed, so $x\leftrightarrow x$ always holds. \b[The randomness is in the independently open bonds; all vertices remain present.]
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