Let $n \geq 3$ be a fixed integer and consider an arbitrary set of the events $\{A_i, 1 \leq i \leq n \}$ in the same probability space. Consider the following two sums of probabilities: \begin{align} S_1 ^{(n)} &= \sum_{1 \leq i \leq n} \mathbb{P}(A_i); \\ S_2 ^{(n)} &= \sum_{1 \leq i_1 < i_2 \leq n} \mathbb{P}(A_{i_1} \cap A_{i_2}). \end{align} The question is whether it holds that $S_1 ^{(n)} \geq S_2 ^{(n)}$ and $2 S_1 ^{(n)} \geq 3 S_2 ^{(n)}$?
Both inequalities hold for $n = 3$. But I could not have a proof based on induction according to this base case. Anyone has a idea? Thanks very much.