Suppose I have the following events $A_0, A_1, ... A_N$. These events are not independent. I want to know the probability of the event $A_0 \cap A_1 \cap ... \cap A_N$ occurring. However, I don't have the information I need to do this. All I know is the probability of each individual event (i.e. $P(A_0)$, $P(A_1)$, etc). Instead, I would like to create a lower bound on the probability.
From here I found the following inequality
$$P(B \cap C) \geq max(0, P(B) + P(C) - 1)$$
Using this inequality I can write (I believe, correct me if I am wrong)
$$P(A_0 \cap A_1 \cap ... \cap A_N) \geq max(0, P(A_0) + P(A_1 \cap ... \cap A_N) - 1)$$
$$P(A_1 \cap ... \cap A_N) \leq min(P(A_1), ... , P(A_N))$$
I then can observe that
$$P(A_0 \cap A_1 \cap ... \cap A_N) \geq max(0, P(A_0) + min(P(A_1), ... , P(A_N)) - 1)$$
WLOG, if I were to sort $A_0, A_1, ... A_N$ such that $P(A_0)$ has the highest probability and $P(A_N)$ the lowest I would then have a "decent" bound given the information I have available (I say decent since most of the time there will be strong dependencies between events, i.e. event $A_0$ can only occur if $A_1$ has occurred. However, this statement is not true in general.).
My question(s) are the following.
(1) Is my reasoning correct?
(2) Is there a better way to do what I want?