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Unformatted text preview: ECEN 303: Assignment 2 Problems: 1. Show that the probability of the union of two events can be generalized to three events as follows: Pr( A ∪ B ∪ C ) = Pr( A ) + Pr( B ) + Pr( C )- Pr( A ∩ B )- Pr( A ∩ C )- Pr( B ∩ C ) + Pr( A ∩ B ∩ C ) . Let D = B ∪ C , then we can write Pr( A ∪ B ∪ C ) = Pr( A ∪ D ). Using previously derived results, we have Pr( A ∪ D ) = Pr( A ) + Pr( D )- Pr( A ∩ D ) . Applying the same result to Pr( D ), we get Pr( D ) = Pr( B ∪ C ) = Pr( B ) + Pr( C )- Pr( B ∩ C ) . Now, using elementary set operation, we have A ∩ D = A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ), from which we gather that Pr( A ∩ D ) = Pr(( A ∩ B ) ∪ ( A ∩ C )) = Pr( A ∩ B ) + Pr( A ∩ C )- Pr(( A ∩ B ) ∩ ( A ∩ C )) = Pr( A ∩ B ) + Pr( A ∩ C )- Pr( A ∩ B ∩ C ) . Putting these pieces together, we obtain Pr( A ∪ B ∪ C ) = Pr( A ) + Pr( B ) + Pr( C )- Pr( A ∩ B )- Pr( A ∩ C )- Pr( B ∩ C ) + Pr( A ∩ B ∩ C ) . 2. A retail establishment accepts either the American Express or the VISA credit card. A total of 24 percent of its customers carry an American Express card, 61 percent carry a VISA card, and 11 percent carry both. What percentage of its customers carry a credit card that the establishment will accept? Let A denote the set of customers that carry an American Express card; and let V be the collection of customers that possess a VISA card. Then, from above, we have Pr( A ) = 0 . 24, Pr( V ) = 0 . 61, and Pr( A ∩ V ) = 0 . 11. The probability that a customer carries a credit card accepted by this establishment becomes Pr( A ∪ V ) = Pr( A ) + Pr( V )- Pr( A ∩ V ) = 0 . 74 ....
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This note was uploaded on 02/11/2010 for the course ECEN 303 taught by Professor Chamberlain during the Fall '07 term at Texas A&M.
- Fall '07