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Unformatted text preview: GEM2900: Understanding Uncertainty & Statistical Thinking DSAP, NUS, Semester 1, 2008/2009 – 1 GEM2900: Understanding Uncertainty & Statistical Thinking Berwin Turlach [email protected] Department of Statistics and Applied Probability National University of Singapore How to calculate with probabilities (cont.) GEM2900: Understanding Uncertainty & Statistical Thinking DSAP, NUS, Semester 1, 2008/2009 – 114 Bayes’ Theorem : Let A 1 , A 2 , . . . , A n be a collection of n mutually exclusive and exhaustive events with P ( A i ) > for i = 1 , . . ., n . Then for any other event B for which P ( B ) > P ( A k  B ) = P ( A k ∩ B ) P ( B ) = P ( B  A k ) P ( A k ) ∑ n i =1 P ( B  A i ) P ( A i ) k = 1 , . . . , n In particular: P ( A  B ) = P ( A ∩ B ) P ( B ) = P ( B  A ) P ( A ) P ( B  A ) P ( A ) + P ( B  A c ) P ( A c ) Lindley (2006, Chapter 6), Olofsson (2007, Chapter 4) How to calculate with probabilities (cont.) GEM2900: Understanding Uncertainty & Statistical Thinking DSAP, NUS, Semester 1, 2008/2009 – 115 Example : How useful are screening tests? A rare disease has an incidence of . 2% ; this means that a person drawn at random from the population has probability . 002 of contracting the disease. A screening test is designed to detect the disease but it’s not a perfect indicator of the existence or absence of the disease. If a person has the disease the test will give a (true) positive result with probability . 997 . On a diseasefree person, the test gives a (false) positive result with probability . 015 . (a) a person is selected at random and tested. What is the probability that the test gives a positive result?...
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This note was uploaded on 03/19/2012 for the course SCIENCE 2900 taught by Professor Forgot during the Spring '08 term at National University of Singapore.
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