If e1 ej is a partition of the sample space and x

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Unformatted text preview: obability, P (R1 ) = P (R1 |E1 )P (E1 ) + P (R1 |E2 )P (E2 ) + P (R1 |E3 )P (E3 ) 11 21 31 1 = + + =. 63 63 63 3 (b) We need to find P (R2 |R1 ), and we’ll begin by finding P (R1 R2 ). By the law of total probability, P (R1 R2 ) = P (R1 R2 |E1 )P (E1 ) + P (R1 R2 |E2 )P (E2 ) + P (R1 R2 |E3 )P (E3 ) 1 6 = 2 1 + 3 2 2 6 1 + 3 3 6 2 1 7 =. 3 54 (1 7 Therefore, P (R2 |R1 ) = PPRRR)2 ) = 18 . (Note: we have essentially derived/used Bayes formula.) (1 (c) We need to find P (E3 |R1 R2 R3 ), and will do so by finding the numerator and denominator in the definition of P (E3 |R1 R2 R3 ). The numerator is given by P (E3 R1 R2 R3 ) = P (E3 )P (R1 R2 R3 |E3 ) = 1 1 33 = 24 . Using the law of total probability for the denominator yields 36 P (R1 R2 R3 ) = P (R1 R2 R3 |E1 )P (E1 ) + P (R1 R2 R3 |E2 )P (E2 ) + P (R1 R2 R3 |E3 )P (E3 ) = 1 6 3 Therefore, P (E3 |R1 R2 R3 ) = formula.) 1 + 3 2 6 3 1 + 3 P (E3 R1 R2 R3 ) P (R1 R2 R3 ) = 3 6 18 24 3 1 1 =. 3 18 3 = 4 . (Note: we have essentially derived/used Bayes...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Tech.

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