Principle of Inclusion-Exclusion

We have seen the addition rule (OR Rule) of probabilities. It says
P (A OR B) = P(A) + P(B) – P (A AND B)

$P (A \cup B) = P(A) + P(B) - P (A \cap B)$

The above equation is a special case of the Principle of Inclusion-Exclusion (PIE). For the union of three events that are not necessarily disjoint:

$\\ P (A \cup B \cup C) = P(A) + P(B) + P(C) \\ - P (A \cap B) - P (A \cap C)- P (B \cap C) \\ + P (A \cap B \cap C)$

It goes by the sequence of adding and subtracting combinations of interactions.

P(A OR B OR C) =
Add all individual probabilities
subtract all pairs of probabilities (AND rule)
add all triplets of probabilities (AND rule), etc.

Let’s do an example to illustrate.

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