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HW1_solution - Econ 3190 Fall 2011 HW#1 Solution 1 Denote...

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Econ 3190 Fall 2011 HW #1 Solution 1. Denote event in (a), (b), (c) and (d) as event E. (a) P ( E ) = P ( A B ) = P ( A ) + P ( B ) - P ( A B ) (b) P ( E ) = P ( A B ) - P ( A B ) = P ( A ) + P ( B ) - 2 P ( A B ) (c) P ( E ) = P ( A B ) = P ( A ) + P ( B ) - P ( A B ) (d) P ( E ) = 1 - P ( A B ) 2. Following De Morgan’s Laws (a) A ( A c B ) = ( A A c ) ( A B ) = S ( A B ) = ( A B ) (b) ( A B ) ( A c B ) = ( A A c ) B = B 3. (a) Not necessarily. We know A B = φ, but can we conclude A c B c = φ as well? We have to consider 2 cases: i. A and B are collectively exhaustive, that is to say A B = S. From above, A c B c = ( A B ) c = S c = φ. As such, if A and B are collectively exhaustive and A and B are mutually exclusive, A c and B c are mutually exclusive as well. ii. A and B are NOT collectively exhaustive, that is to say A B S. And A c B c = ( A B ) c 6 = S c = ˙ φ . Hence, in this case, A c and B c are NOT mutually exclusive. (b) If A and B are collectively exhaustive and A and B are mutually exclusive, A c and B c are mutually exclusive as well.

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HW1_solution - Econ 3190 Fall 2011 HW#1 Solution 1 Denote...

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