probability or statistics - Inclusion-Exclusion Principle Implementation

I would like to know how I can write a Mathematica code for the Inclusion–exclusion principle.

The formulas governing it are:

$ P(\bigcup_{i=1}^n A_i)=\sum_{k=1}^n (-1)^{k-1} \sum_{\substack{I\subset\{1,2,...,n\}\\|I|=k} }P(\bigcap_{i\in I} A_i) $

where $P(A_i)$ denotes the probability of the event $A_i$. Similarly for cardinality of union we have:

$ |\bigcup_{i=1}^n A_i|=\sum_{k=1}^n |A_k|- \sum_{1\leq i_1

or in closed form

$|\bigcup_{i=1}^n A_i|=\sum_{\substack{\emptyset\neq I\subset\{1,2,...,n\}\\|I|=k} }(-1)^{|I|-1} |\bigcap_{i\in I} A_i|$