HW1 solution

# HW1 solution - Econ 3190 Fall 2010 HW#1 Solution 1 Denote...

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Econ 3190 Fall 2010 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 ) (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 = but can we conclude A c \ B c = 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 = 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 = _ . Hence, in this case, A c and B c are NOT mutually exclusive.

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HW1 solution - Econ 3190 Fall 2010 HW#1 Solution 1 Denote...

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