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Unformatted text preview: University of Illinois Fall 2009 ECE 313: Problem Set 2: Solutions Axioms of Probability; Countably Infinite Sample Spaces 1. [Maximum and minimum values for probabilities] (a) P ( A B ) = P ( A ) + P ( B ) P ( A B ) P ( A ) + P ( B ). Thus, P ( A B ) . 8 with equality occurring when P ( A B ) = 0. Similarly, P ( A C ) = P ( A ) + P ( C ) P ( A C ) P ( A ) + P ( C ) But, P ( A ) + P ( C ) > 1 and so we conclude that P ( A C ) 1 with equality occurring when P ( A C ) = 0 . 3. (b) Since A A B and B A B , we know (Ross, Proposition 4.2 in Chapter 2) that P ( A ) P ( A B ), P ( B ) P ( A B ) and so max { P ( A ) ,P ( B ) } = 0 . 6 P ( A B ). Thus, P ( A B ) . 6 with equality occurring when B A . Similarly, max { P ( A ) ,P ( C ) } = 0 . 7 P ( A C ) with equality occurring when A C . (c) Since A B A and A B B , we have that P ( A B ) P ( A ), P ( A B ) P ( B ) and so P ( A B ) min { P ( A ) ,P ( B ) } = 0 . 2. Thus, P ( A B ) . 2 with equality occurring when B A . Similarly, P ( A C ) min { P ( A ) ,P ( C ) } = 0 . 6 with equality occurring when A C . (d) The smallest possible value of P ( A B ) is 0 when A and B are mutually exclusive. On the other hand, the smallest possible value of P ( A C ) is 0.3 in which case P ( A C ) = 1 as noted in part (a)....
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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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