= Janson inequality
{c}
{wiki}
For increasing containment events $A_i$ in a Bernoulli product space, let $X=\sum_i\mathbf1_{A_i}$, $\mu=\mathbb EX$, and
$$
\Delta=\sum_{i\ne j:\,A_i\sim A_j}\mathbb P(A_i\cap A_j),
$$
where the sum is ordered and dependency means overlapping coordinate supports. Then
$$
\mathbb P(X=0)\leq e^{-\mu+\Delta/2}.
$$
In particular it is at most $e^{-\mu/2}$ when $\Delta\leq\mu$, and at most $e^{-\mu^2/(2\Delta)}$ when $\Delta\geq\mu$.
= Janson inequalities
{synonym}
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