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ANS11 - EC2019 SAMPLING AND INFERENCE January 2011 1 Give...

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EC2019 SAMPLING AND INFERENCE, January 2011 1. Give an account of the axioms of Boolean algebra and, in the process, compare them with the axioms of arithmetic. Prove the rule of De Mor- gan that asserts that A c B c = ( A B ) c , where the su x c denotes complemetation. A student has a probability of 0.2 of forgetting the book that he must return, a probability of 0.35 of forgetting his library card and a probability of forgetting both of 0.3. (a) What is the probability that today he will have both the book and the library card in his possession (b) Given that he has the book, what is the probability that he will find the library card? Answer. We must prove that (i) ( A c B c ) ( A B ) = S and that (ii) ( A c B c ) ( A B ) = . (i) There is ( A c B c ) ( A B ) = { ( A c B c ) A } { ( A c B c ) B } . But { ( A c B c ) A } = { ([ A c A ] [ B c A ]) } = B c A and there is ( A c B c ) A = A c B , whence the union of the two is ( B c A ) ( A c B ) = S (ii) There is ( A c B c ) ( A B ) = { ( A c B c ) A } { ( A c B c ) B } = ∅ ∩ ∅ . (a) Let A denote the event of forgetting the library card and let B dentote the event of forgetting the book. The event of forgetting neither is A c B c = ( A B ) c . There is P ( A B ) = P ( A )+ P ( B ) P ( A B ) = 0 . 35+0 . 2 0 . 3 = 0 . 25 . Therefore P ( A c B c ) = P { ( A B ) c } = 1 P ( A B ) = 0 . 75 . (b) The probability of finding the library card given that he has found the book is P ( A c | B c ) = P ( A c B c ) /P ( B c ) = 0 . 75 / (1 0 . 2) = 15 / 16 . 2. Let H 1 , H 2 , . . . , H k denote an exhaustive set of mutually exclusive hy- potheses representing the possible causes of an event E . Show that P ( H i | E ) = P ( E | H i ) P ( H i ) P ( E ) , where P ( E ) = P i P ( E | H i ) P ( H i ). I always leave the house for work before my wife does. My wife takes the car on 3 out of the 5 workdays. Today, I have reckoned that there is an equal probabilty that either of us will be the first to return home.
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