Assuming a suitable probability space $(\Omega, \mathcal{F}, P)$, the Frechet inequalities are given by:
$$\max \left( 0, \sum_{k=1}^n P(A_k) – (n-1) \right) \leq P\left( \bigcap_{k=1}^{n} A_k \right) \leq \min_k \{ P(A_k) \}$$
$$\max_k \{P(A_k)\} \leq P\left( \bigcup_{k=1}^{n} A_k \right) \leq \min \left(1, \sum_{k=1}^n P(A_k) \right)$$
Wikipedia defines "tight bounds":
An upper bound is said to be a tight upper bound, a least upper bound, or a supremum, if no smaller value is an upper bound. Similarly, a lower bound is said to be a tight lower bound, a greatest lower bound, or an infimum, if no greater value is a lower bound.
Are these bounds tight?
Best Answer
The definition from Wikipedia is about bounds for sets. Here you do not have a set but just an inequality which depends on may parameters ($n$, $A_1,\ldots,A_n$, $P$).
There are choices of the parameters for which the inequality becomes even an equality (for instance $A_1=\cdots=A_n$ for the first three inequalities, or $n=1$ for all inequalities); so in some sense the inequality is “tight”. But there are also other choices of parameters for which the inequality is not an equality, so in some sense simultaneously the inequality is not “tight”.