An improved bonferroni-type inequality

Let $A_1, A_2, \ldots, A_n$ be a sequence of events on a given probability space and let $m_n$ be the number of those A's which occur. Put $S_{0,n} = 1$ and $$ S_{k,n} = \Sigma P(A_i_1 \cap A_i_2 \cap \cdots \cap A_i_k), (a \leq k)$$ where the summation is over all subscripts satisfying $1 \let i_1 < i_2 < \cdots < i_k \leq n$.