Probability

$P\left(\bigcap_{i=1}^{15} A_i\right) \leq \sum_{i=1}^{15} P\left(A_i\right)-15$

$P\left(\bigcap_{i=1}^{15} A_i\right) \geq \sum_{i=1}^{15} P\left(A_i\right)-14$

$P\left(\bigcup_{i=1}^{15} A_i\right) \geq \sum_{i=1}^{15} P\left(A_i\right)$

\(P\left(\bigcup_{i=1}^{15} A_i\right) < \sum_{i=1}^{15} P\left(A_i\right)-\sum_{1 \leq i < j<15} P\left(A_i \cap A_j\right)\)