lecture2 - Die example revisited Problem: Consider the...

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Unformatted text preview: Die example revisited Problem: Consider the experiment of tossing a fair die , and let A = { observe an even number } , B = { observe a number less than or equal to 4 } . Are A and B independent events? P ( A ) = 1 / 2 ,P ( B ) = 2 / 3 ,P ( A B ) = 1 / 3. What is the conditional probability P ( A | B )? (Of course, you can also check P ( B | A )). 1 Bayes Rule and its application, p. 167 Given k mutually exclusive and exhaustive events, B 1 ,B 2 ,...,B k such that P ( B 1 )+ P ( B 2 )+ + P ( B k ) = 1, and given an observed event A , it follows that P ( B i | A ) = P ( B i A ) /P ( A ) = P ( B i ) P ( A | B i ) P ( B 1 ) P ( A | B 1 ) + + P ( B k ) P ( A | B k ) . The statement above is called the Bayes The- orem. 2 Problem Setup: An unmanned monitoring system uses high-tech equipment and micro- processors to detect intruders. One such sys- tem has been used outdoors at a weapons mu- nitions plant. The system is designed to detect intruders with a probability of .90, however, its performance may vary with the weather. Nat- urally design engineers want to test how reli- able the system is. Suppose after a long se- quence of tests, the following information has been available: Given that the intruder was in- deed detected by the system, the weather was clear 75% of the time, cloudy 20% of the time, and raining 5% of the time. When the system failed to detect the intruder, 60% of the days...
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lecture2 - Die example revisited Problem: Consider the...

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