= Negative correlation of increasing and decreasing events
{title2=$\mathbb P(E\cap G)\leq\mathbb P(E)\mathbb P(G)$}
Under a <product measure> on a finite <Boolean lattice>, an <increasing event> $E$ and a <decreasing event> $G$ obey the displayed inequality. The complement $G^c$ is an <increasing event>, so <Harris' inequality> gives $\mathbb P(E\cap G^c)\geq\mathbb P(E)\mathbb P(G^c)$. Subtract from $\mathbb P(E)$ to obtain the result. This implication includes degenerate <Bernoulli distribution> parameters $0$ and $1$.
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