Question:
Let $A_1,A_2,...,A_n \subset \Omega $ a sequence of outcome such that $P(A_i) \geq c $ for all $i$.
Prove that it exists $i \neq j$ s.t. $P(A_i \cap A_j) \geq \frac{nc^2-c}{n-1}$
I have tried different ways including the following one but I did not succeed to conclude.
1-First note that: $\int (\sum_{1 \leq i \leq n} I_{A_i})^2dP= \sum_{i \neq j} \int I_{A_i}I_{A_j}dP + \sum_{i = j} \int I_{A_i}^2dP= \sum_{i \neq j} P(A_i \cap A_j) + \sum_{i = j}P(A_i)$
2-Now we know that $\sum_{i = j}P(A_i)$ is the sum of $n$ element, thus $\sum_{i = j}P(A_i) \geq nc$
3-We know too that $\sum_{i \neq j} P(A_i \cap A_j)$ is the sum of $n(n-1)$ elements.
More over $\forall i \neq j$ by absurd let suppose that $P(A_i \cap A_j) < \frac{nc^2-c}{n-1}$.
But after I don't succeed to continue. Can someone help me please?
Thank for your help.