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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.
 Spring '08
 Zahrn
 The Land

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