= Positive common intersection from pair-intersection probabilities
{title2=$P(\bigcap_i A_i)\geq\sum_{i<j}P(A_i\cap A_j)-\binom n2+1$}
For $n\geq2$, the intersection of all pairwise intersections $A_i\cap A_j$ is $\bigcap_iA_i$. Apply the <union bound> to the complements of the pairwise intersections and then take the complement to obtain the displayed lower bound. A sum exceeding $\binom n2-1$ therefore forces positive common probability without any <independence> assumption.
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