HW01 - A c and B c mutually exclusive? Give your reasoning....

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PROF HONG FALL 2011 ECONOMICS 619 PROBLEM SET # 1 A and B , ±nd formula for the probabilities of the following events in terms of the quantities P ( A ) , P ( B ) , and P ( A \ B ) : (a) either A or B or both; (b) either A or B but not both; (c) at least one of A or B ; (d) at most one of A or B ; S be a sample space. (a) Show that the collection B = f; ;S g is a sigma algebra. (b) Let B = f all subsets of S , including S itself g . Show that B is a sigma algebra. (c) Show that the intersection of two sigma algebra is a sigma algebra. 3. Consider two events A and B such that P ( A ) = 1 3 and P ( B ) = 1 2 : Determine the value of P ( B \ A c ) for each of the following conditions: (a) A and B are disjoint; (b) A B ; (c) P ( A \ B ) = 1 8 : 4. Let A and B be two events. Check if the following relations hold: (a) A [ B = A [ ( A c \ B )? (b) B = ( A \ B ) [ ( A c \ B )? Please give your reasoning clearly. 5. Suppose events A and B are mutually exclusive. (a) Can we say that
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Unformatted text preview: A c and B c mutually exclusive? Give your reasoning. (b) Please nd some example such that A c and B c are mutually exclusive as well? 6. Suppose P ( A ) = 1 3 and P ( B c ) = 1 4 : Is it possible that A and B are mutually exclusive? 7. If 50% of the families in a city subscribe to the morning newspaper, 65% of the families subscribe to the afternoon newspaper, and 85% of the families subscribe to at least one of the two newspapers, what is the proportion of the families subscribe to both newspapers? 8. Let A;B;C be three events in Sample space S: Express P ( A [ B [ C ) in terms of P ( A ) ;P ( B ) ;P ( C ) ;P ( A \ B ) ;P ( B \ C ) ;P ( C \ A ) and P ( A \ B \ C ) : Give your reasoning. 1...
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