The general disjunction rule for events $A_1$ and $A_2$ is $$P(A_1 vee A_2) = P(A_1) + P(A_2) – P(A_1 wedge A_2).$$

What about when there are $n$ events? What is $P(bigvee_i^n A_i)$ where $A_i$ is the $i$th event?

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#### Best Answer

The general formula is $$ P(A_1 cup cdots cup A_n) = sum_{k=1}^n (-1)^{k+1} sum_{i in C_{k,n}} P(A_{i_1}cap cdots cap A_{i_k}) $$ where $C_{k,n}$ is the set of all ordered $k$-uples $i_1 < cdots < i_k$ of ${1,dots,n}$.

You can prove it by induction, it’s not conceptually difficult but painful enough to write.

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