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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?

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    The set $A$ is not in the union on the left, so should not appear on the right side.2012-11-30

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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.

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    @csss: You’re not reading the formula correctly. When $n=2$ the only terms are $\sum_{i=1}^2P(A_i)$ and -\sum_{1\le i. That second term **is** the term $(-1)^{2-1}P\left(\bigcap_{i=1}^2A_i\right)$.2012-11-30