Question on inclusion-exclusion principle when $n=2$

Question

Using the inclusion exclusion principle - http://www.proofwiki.org/wiki/Inclusion-Exclusion_Principle - if I set $n=2$ I get the following -

$$P(A_1 \cup A_2) = P(A_1) + P(A_2) - P(A_1 \cap A_2) + P(A_1 \cap A_2)$$

when the correct answer should be -

$$P(A_1 \cup A_2) = P(A_1) + P(A_2) - P(A_1 \cap A_2)$$

I have the term $P(A_1 \cap A_2)$ at the end the first equation due to the last part of the inclusion exclusion principle - $$(-1)^{n-1}P(\cap_i^n A_i)$$

It seems that I shouldn't be including that if I want to have the correct answer...but surely I have to include it as I can't just drop an arbitrary term from some formula...so what am I missing?

Answer 1

In both of the places where you wrote $A$ you probably mean $A_1$; I’m going to assume so.

When $n=2$ the last term is

$$(-1)^{n-1}P\left(\bigcap_{i=1}^nA_i\right)=(-1)^1P\left(\bigcap_{i=1}^2A_i\right)=-P(A_1\cap A_2)\;,$$

just as it should be. Somehow you added an extra term that is not present in the expression on the cited Proof Wiki page.