I know the first two BC lemmas.
However, I was looking at this generalization of it and I don't understand the intuition behind it.
In the 2nd BC lemma, we require independence of events. The generalization is used to replace the condition of pairwise independence.
Pairwise independence can be replaced with the weaker condition of $P(A_k A_j) \leq P(A_k)P(A_j)$ for ever $k$ and $j$ such that $k\neq j$
What does this mean in simple terms? I want to know how I can explain it without having to use math. What results ensue from not requiring independence and replacing it with the weaker condition?
This is known as the "pairwise negative correlation" property. This is found for example in the paper Balanced Matroids by Feder, Mihail, available here.
What this means intuitively is the appearance of one element in the pairwise condition means that the appearance of the other element is less likely.