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

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Midterm Examination #1 Sta 113: Probability and Statistics in Engineering Thursday, 2007 Sep. 20, 1:15 – 2:30 pm This is a closed-book exam so do not refer to your notes, the text, or any other books (please put them on the floor). You may use a single sheet of notes or formulas and a calculator, but materials may not be shared. A formula sheet and two blank worksheets are attached to the exam. You must show your work to get partial credit. Even correct answers will not receive full credit without justification. Please give all numerical answers to at least two correct digits or as exact fractions reduced to lowest terms . Write your solutions as clearly as possible and make sure it’s easy to find your answers (circle them if necessary), since you will not receive credit for work that I cannot understand or find. Good Luck! If you find a question confusing please ask me to clarify it. Cheating on exams is a breach of trust with classmates and faculty, and will not be tolerated. After completing the exam please acknowledge the Duke Honor Code with your signature below: I have neither given nor received unauthorized aid on this exam . Signature: Print Name: 1. /15 2. /25 3. /25 4. /25 5. /10 Total: /100
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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
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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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