What is the addition law for probability with multiple elements?

[christang_1023]

The additional law with two elements can be expressed $$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$, while the law with three elements can be $$P(A\cup B\cup C)=P(A)+P(B)+P(C)-P(A\cap B)-P(B\cap C)-P(A\cap C)+P(A\cap B\cap C)$$
Now I wonder if there is the more general form of addition law, which applies to any n elements.

 

[S.G. Janssens]

Now I wonder if there is the more general form of addition law, which applies to any n elements.

Yes, there is. It is knows as the principle of inclusion-exclusion, usually discussed in probability books and also in various places online.

 

[StoneTemplePython]

unless you really like combinatorial manipulations, the easiest derivation that I'm aware of is

i) $P\Big( A_1 \bigcup A_2 \bigcup A_3 \bigcup... \bigcup A_n\Big) = 1- P\Big( A_1^C \bigcap A_2^C \bigcap A_3^C \bigcap ... \bigcap A_n^C\Big)$
i.e. recall that the union of events is equivalent to the complement of the intersection of complements

ii) then use indicator random variables to model the RHS, which gives
$P\Big( A_1 \bigcup A_2 \bigcup A_3 \bigcup ... \bigcup A_n\Big) = 1 - E\Big[\big(1-\mathbb I_{A_1}\big)\big(1-\mathbb I_{A_2}\big)\big(1-\mathbb I_{A_3}\big)...\big(1-\mathbb I_{A_n}\big)\Big]$

if you expand the polynomial (i.e. apply elementary symmetric functions of the indicators and pay attention to the sign), and then use linearity of the expectations operator, you recover Inclusion-Exclusion for probability.