squiz2 - the probability that there is a connection between...

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Solutions to quiz # 2 (January 19) 1. Let I , II and III be the events that we choose the frst, second and the third box respectively, and let R be the event that we choose a red marble. Then P ( III | R ) = P ( III R ) P ( R ) . Now, P ( III R ) = P ( III ) · P ( R | III ) = 1 3 · 1 = 1 3 and P ( R ) = P ( R | I ) P ( I ) + P ( R | II ) P ( II ) + P ( R | III ) P ( III ) = 1 5 · 1 3 + 1 2 · 1 3 + 1 · 1 3 = 17 30 . Hence P ( III | R ) = 10 / 17. Answer: the probability in question is 10 / 17. 2. Let us call C the point with the wires numbered 1 , 2 , 5 and 8 coming out oF it. There is a connection between A and B iF and only iF there is a connection between A and C and there is a connection between C and B . The connection between A and C is either via wires 1 and 4 or via wires 3 and 2 or both. Using the inclusion-exclusion principle we conclude that
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Unformatted text preview: the probability that there is a connection between A and C is (1-p ) 2 +(1-p ) 2-(1-p ) 4 . Similarly, the probability that there is a connection between C and B is 2(1-p ) 2-(1-p ) 4 . ThereFore, the probability that there is a connection between A and B is ( 2(1-p ) 2-(1-p ) 4 ) 2 = 1-8 p 2 + 8 p 3 + 14 p 4-32 p 5 + 24 p 6-8 p 7 + p 8 . Answer: the probability in question is ( 2(1-p ) 2-(1-p ) 4 ) 2 . 1...
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This note was uploaded on 02/06/2012 for the course STAT 525 taught by Professor Alexanderbarvinok during the Winter '12 term at University of Michigan-Dearborn.

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