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# m1sol - Midterm Examination 1 Sta 113 Probability and...

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Name: Sta 113 Problem 1 : Let A,B,C denote events in a probability space with P ( A ) ,P ( B ) ,P ( C ) > 0. Decide if the following statements are true or false for any A,B,C and explain WHY. i P ( A ) P ( B ) = P ( A | B ) P ( B ) implies that the events A and B are dis- joint. False. This statement is an implication of independence not being disjoint. if the events were disjoint the P ( A ) P ( B ) = 0. ii If A c and B c are disjoint then P ( A B ) = P ( A ) + P ( B ) + P ( A B ) . False. The true statement is P ( A B ) = P ( A ) + P ( B ) - P ( A B ) . iii If A,B,C are independent then P ( A B C ) = P ( A B | C ) P ( C ) = P ( A C | B ) P ( B ) = P ( A B C ) = P ( B C | A ) P ( A ) = P ( B C ) P ( A ) . I took true or false based on arguments since this question was not completely speciﬁed. iv P ( A | B C ) > P ( A B C ) P ( B | C ) P ( C ) False. It is an equality by the deﬁnition of conditional probability P ( A | B C ) = P ( A B C ) P ( B | C ) P ( C ) Fall 2007 Page 1 of 10 Sep. 20, 2006
Sta 113 v P ( A ) = 1 - P ( B ) = . 001 P ( C | A ) = P ( C c | B ) = . 95 Is it true that P ( A | C ) > 10 × P ( B | C c ) ? False. P ( A | C ) = P ( C | A ) P ( A ) P ( C | A ) P ( A ) + P ( C | B ) P ( B ) = . 95 × . 001 . 95 × . 001 + . 05 × . 999 = . 0186 . P ( B | C c ) = P ( C c | B ) P ( B ) P ( C c | B ) P ( B ) + P ( C c | A ) P ( A ) = . 95 × . 999 . 95 × . 999 + . 05 × . 001 1 . Fall 2007

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m1sol - Midterm Examination 1 Sta 113 Probability and...

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