= Solution
Use the nearest-neighbour lattice <graph> with <graph vertex> set $\mathbb Z^d$: an unordered pair $\{x,y\}$ is an <edge> precisely when $\|x-y\|_1=1$. A bond configuration is $\omega\in\{0,1\}^{E_d}$; under the <product measure> $\mathbb P_p$, its <edge> coordinates are independent <Bernoulli distribution> with success <probability> $p$. <Edges> with coordinate one are open. The open cluster $C(0)$ is the <connected component of a graph> reachable from the origin through open <edges>.
Define the <percolation probability> and <percolation critical probability> by
$$
\boxed{\theta_d(p)=\mathbb P_p(|C(0)|=\infty),\qquad p_c(d)=\inf\{p\in[0,1]:\theta_d(p)>0\}.}
$$
Equivalently, $p_c(d)=\sup\{p:\theta_d(p)=0\}$. The <monotone coupling of Bernoulli percolation> proves the equivalence: assign independent <uniform distribution> $U_e$ to the <edges> and open $e$ at parameter $p$ when $U_e\leq p$. Connection events and $\theta_d$ then increase with $p$. For $d\geq1$, $\theta_d(0)=0$ and $\theta_d(1)=1$, so the defining set is nonempty. The definition does not settle what happens at $p=p_c(d)$.
By translation invariance all roots have the same <probability>. Since the <graph vertex> set is countable, $\theta_d(p)=0$ implies that no <graph vertex> lies in an <infinite percolation cluster> <almost surely>. Conversely a positive root <probability> gives positive <probability> of an <infinite percolation cluster>, and <translation ergodicity of Bernoulli percolation> then makes existence an almost-sure event. In question 2(b), $\theta$ means $\theta_2$.
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