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Unformatted text preview: CS112 - Homework #1 Probability Axioms: Nonnegativity: P ( A ) 0 for every event A Additivity: if A and B are disjoint, then: P ( A B ) = P ( A ) + P ( B ) Normalization: P () = 1 1. Let A and B be independent events with A C denoting the complement of A . Prove that A C is independent of B . (Remember that P ( A C ) = 1- P ( A )) 2. (a) Let A 1 ,A 2 ,A 3 be mutually exclusive events. Show that P [ A 1 A 2 A 3 ] = P [ A 1 ]+ P [ A 2 ]+ P [ A 3 ] in the following two ways: i. Informally, using a Venn Diagram. ii. With a formal proof, using the Axioms of Probability. Note that in your textbook on page 9 are the probability axioms. The additivity axiom they use ( P [ A 1 A 2 ... ] = P [ A 1 ] + P [ A 2 ] + ... for disjoint A i ) is more general than the one we gave you here. A good method for this problem is to use the additivity axiom we provided to prove the more general property in the text. Then, it should be trivial to apply the general property to prove P [ A 1 A 2 A 3 ] = P [ A 1 ] + P [ A 2 ] + P [ A 3 ] ....
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- Spring '08