I just started statistics and need to use the general addition rule. I know what it looks like for $3$ events:

$$P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(A \cap C) - P(B \cap C) - (2 * P(A \cap B \cap C)) + P(A'\cup B' \cup C').$$

But I'm confused to exactly how it extends to 4 events?


  • $\begingroup$ To find $P(A\cup B\cup C\cup D)$: You first add all individual probabilities, then subtract all probabilities of events taken two at a time ($P(A\cap B)$, e.g.; there are other terms here...), then add all probabilities of events taken three at a time ($P(A\cap B\cap D)$, e.g.; note there are three other terms here), and finally subtract the probability of the intersection of all events. $\endgroup$ – David Mitra Sep 22 '12 at 19:23

There's this thing called the inclusion-exclusion identity, for any number $n$, this is the formula.

enter image description here

for a proof of how this is works, you can check out this link at wikipedia


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