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521exam1key

521exam1key - ECN 521 KEY TO EXAM 1 SPRING 2011 S GENSEMER...

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ECN 521 - KEY TO EXAM 1 - SPRING 2011 - S. GENSEMER 1. (a) False. For example, the events A and B in Problem 2 are independent; they are not mutually exclusive. (b) True. By the axiom of probability assigning a probability of 1 to the sample space and by the axiom of additivity when the events are mutually exclusive, 1 = P( S ) = P( A [ A 0 ) = P( A ) + P( A 0 ) ) P( A 0 ) = 1 P( A ) : (c) True. If A and B are independent events, then P( A j B ) = P( A \ B ) P( B ) = P( A ) ± P( B ) P( B ) = P( A ) : 2. (a) In the following, we use the addition rule (where events are not necessarily mutually exclusive) and we use the fact that A and B are independent events. P( A [ B ) = P( A ) + P( B ) P( A \ B ) = P( A ) + P( B ) P( A ) ± P( B ) ) P( A [ B ) = : 80 + : 60 : 80 ± : 60 = : 92 (b) P( A j A [ B ) = P [ A \ ( A [ B )] P( A [ B ) = P ( A ) P( A [ B ) = : 80 : 92 = 20 23 ² : 87 ; where we have used the fact that A \ ( A [ B ) = A:

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521exam1key - ECN 521 KEY TO EXAM 1 SPRING 2011 S GENSEMER...

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