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Unformatted text preview: University of Illinois Fall 2009 ECE 313: Hour Exam I Monday October 12, 2009 7:00 p.m. — 8:00 p.m. 100 Noyes Laboratory 1. [15 points] Let A , B , and C denote three events defined on a sample space Ω, and suppose that P ( A ) = 0 . 6 ,P ( B ) = P ( C ) = 0 . 3, and P ( B c ∩ C ) = P ( A ∩ B c ∩ C c ) = 0 . 2. Solution: The Karnaugh maps shown below are very useful in visualizing the problem. . 3 0.2 . 3 0.2 B C A B C A A c A c B c B c 0.2 0.1 (a) [5 points] Find P ( B ∩ C ). Solution: P ( B ∩ C ) = P ( C ) P ( B c ∩ C ) = 0 . 3 . 2 = 0 . 1. Note that P ( B ∪ C ) = P ( B ) + P ( B c ∩ C ) = 0 . 5. (b) [5 points] Find P ( B ∩ C c ). Solution: P ( B ∩ C c ) = P ( B ) P ( B ∩ C ) = 0 . 3 . 1 = 0 . 2. (c) [5 points] Find P (( A ∪ B ∪ C ) c ). Solution: P (( A ∪ B ∪ C ) c ) = 1 P ( A ∪ B ∪ C ) = 1 [ P ( B ∪ C ) + P ( A ∩ B c ∩ C c )] = 1 [0 . 5 + 0 . 2] = 0 . 3. Note that P (( A ∪ B ∪ C ) c ) = P ( A c ∩ B c ∩ C c ), and also that P ( B c ∩ C c ) = 1 P ( B ∪ C ) = 0 . 5. 2. [10 points] A and B are events defined on a sample space Ω. Assume P ( A ) ,P ( B ) > 0. Mark each of the two statements below as TRUE or FALSE. No justification is needed. TRUE FALSE 2 2 If P ( A  B ) = P ( B  A ), then P ( A ) = P ( B ). 2 2 P ( A  B ) P ( B ) + P ( A c  B ) P ( B ) = P ( B ) . Solution: The statement “If P ( A  B ) = P ( B  A ), then P ( A ) = P ( B )” seems to be true at first glance, but is in fact FALSE in general. (The statement is true if we are also told thatfirst glance, but is in fact FALSE in general....
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This note was uploaded on 10/25/2010 for the course ECE 313 taught by Professor Milenkovic,o during the Spring '08 term at University of Illinois, Urbana Champaign.
 Spring '08
 Milenkovic,O

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