Explanation (a) Using Venn diagram to verify the given extension of the addition rule. P ( A ∪ B ∪ C ) = P ( A ) + P ( B ) + P ( C ) − P ( A ∩ B ) − P ( A ∩ C ) − P ( B ∩ C ) + P ( A ∩ B ∩ C ) L.H.S: R.H.S: From L.H.S and R.H.S, the given extension of the addition rule is verified. (b) Using the axioms of probability and the propositions to verify the given extension of the addition rule. Distributive laws is given below: ( A ∪ B ) ∩ C = ( A ∩ C ) ∪ ( B ∩ C ) ( A ∩ B ) ∪ C = ( A ∪ C ) ∩ ( B ∪ C ) General addition rule for two events: P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B ) Using the general addition rule, then P ( A ∪ B ∪ C ) = P ( ( A ∪ B ) ∪ C ) = P ( A ∪ B ) + P ( C ) − P ( ( A ∪ B ) ∩ C ) On simplification, P ( A ∪ B ∪ C ) = P ( ( A ∪ B ) ︸ Using addition rule ∪ C ) = { P ( A ) + P ( B ) − P ( A ∩ B ) + P ( C ) − P ( ( A ∪ B )

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Chapter 1.8, Problem 6P

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Verify the given 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.

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