There are many many approximations that I know on the probability of union of events, $P(A\cup B \cup C...)$ available in terms of $S_1 = P(A) + P(B) + P(C)....$ and $S_2 = P(A \cap B) + P(B \cap C) + ...$ etc. I would ideally not like $S_3, S_4....$I found these in papers from the 60s and 70s.
I do not know where to find bounds on the errors in these approximations. Where should I look? I need to find a bound on the error if I approximate the union probability with, say, Bonferroni's bound of $S_1-S_2$. Or some other bound.