Verify the following extension of the addition rule (a) by...

Verify the following extension of the addition rule (a) by an appropriate Venn diagram and (b) by a formal argument using the axioms of probability and the propositions in this chapter. $$\begin{aligned} P(A \cup B \cup C)=& P(A)+P(B)+P(C)-P(A \cap B) \\ &-P(A \cap C)-P(B \cap C)+P(A \cap B \cap C) \end{aligned}$$

Verify the following extension of the addition rule (a) by an appropriate Venn diagram and (b) by a formal argument using the axioms of probability and the propositions in this chapter.
$$\begin{aligned}
P(A \cup B \cup C)=& P(A)+P(B)+P(C)-P(A \cap B) \\
&-P(A \cap C)-P(B \cap C)+P(A \cap B \cap C)
\end{aligned}$$

Key Concepts

Venn Diagram

A Venn diagram is a graphical tool that represents sets as regions in a plane and their relationships through overlaps. In the context of probability, it visually displays events and their intersections, providing an intuitive way to illustrate and verify the addition rule and the inclusion-exclusion principle by clearly marking the regions corresponding to single events, overlaps between pairs, and the common overlap of all events.

Probability Axioms

These are the foundational rules that define any probability measure—non-negativity, normalization (the probability of the sample space is 1), and countable additivity. They provide the basis for combining probabilities of multiple events and ensure that operations like unions and intersections behave consistently within the probabilistic framework.

Inclusion-Exclusion Principle

This principle is used to compute the probability of the union of multiple events by adding the probabilities of individual events, subtracting the probabilities of their pairwise intersections, and then adding back the probability of the intersection of all events. It corrects for overcounting when events overlap, ensuring that each outcome is counted exactly once.

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