Inequality on probability of intersection of n events
$$P\left(\bigcap_{i=1}^n A_i\right) = 1- P\left(\bigcup_{i=1}^n A_i^c\right) \\ \geq 1-\left(\sum_{i=1}^n P(A_i^c)\right) =1-\left(\sum_{i=1}^n (1-P(A_i))\right) = \left(\sum_{i=1}^n P(A_i)\right) -(n-1)$$
with equality when the $A_i^c$ are mutually disjoint, or at least any intersections have zero probability since then $P\left(\bigcup_i A_i^c\right) = \sum_i P(A_i^c) $