ps11-12sol - Massachusetts Institute of Technology...

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Unformatted text preview: Massachusetts Institute of Technology 6.042J/18.062J, Fall 02 : Mathematics for Computer Science November 27 Prof. Albert Meyer and Dr. Radhika Nagpal revised November 19, 2002, 35 minutes Solutions to Problem Set 11-12 Problem 1. Prove that (a) Pr { A | B } = Pr A B iff A and B are independent. Solution. Proof. ( ): Assuming Pr { A | B } = Pr A B , Pr { A } = Pr { A | B } Pr { B } + Pr A B Pr B (Law of Total Probability) = Pr { A | B } Pr { B } + Pr { A | B } Pr B (assumption) = Pr { A | B } ( Pr { B } + Pr B ) (distributivity) = Pr { A | B } 1 = Pr { A | B } (Complement Rule) , so A is independent of B by definition. ( ): Assuming A and B are independent, we know that A and B are also independent, so Pr { A | B } = Pr { A } (independence of A and B ) = Pr A B (independence of A and B ) . For the record, heres the proof that A and B are independent: Pr A B = Pr A B Pr B (definition of conditional probability) = Pr { A- B } Pr B (definition of set difference) = Pr { A } - Pr { A B } Pr B (Difference Rule) = Pr { A } - Pr { A } Pr { B } Pr B (independence of A and B ) = Pr { A } (1- Pr { B } ) Pr B (distributivity) = Pr { A } Pr B Pr B (Complement Rule) = Pr { A } . Copyright 2002, Prof. Albert R. Meyer & Dr. Radhika Nagpal. All rights reserved. 2 Solutions to Problem Set 11-12 (b) If A,B,C are mutually independent events, then A and B C are independent. Solution. Proof. To show independence of the event A and the event B C , it suffices to show that the probability of both events happening is equal the the product of the probabilities that each event happens. That is, we need only show that the Pr { A ( B C ) } = Pr { A } Pr { B C } (1) But Pr { A ( B C ) } = Pr { ( A B ) ( A C ) } (distributivity of over ) = Pr { A B } + Pr { A C } - Pr { ( A B ) ( A C ) } (Inclusion-exclusion) = Pr { A B } + Pr { A C } - Pr { A B C } (since A A = A ) = Pr { A } Pr { B } + Pr { A } Pr { C } - Pr { A } Pr { B } Pr { C } (mutual independence) = Pr { A } ( Pr { B } + Pr { C } - Pr { B } Pr { C } ) (distributivity of multiplication) = Pr { A } Pr { B C } (Inclusion-exclusion) , which proves ( 1 ). Problem 2. There is a coursenot 6.042, naturallyin which 10% of the assigned problems contain errors . If you pick a random problem and send an email to your TA and your Lecturer asking whether the problem has an error, then the TAs reply will be correct 80% of the time . This 80% accuracy holds regardless of whether or not a problem has an error. Likewise, the Lecturers reply will be correct with only 75% accuracy . 1 Furthermore, the TA and Lecturers tend to be confused by different kinds of problems. The net result of this is that the correctness of the lecturers answer and the TAs answer are independent of each other, regardless of whether there is an error ....
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This note was uploaded on 01/15/2012 for the course ECON 101 taught by Professor N/a during the Fall '11 term at Middlesex CC.

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ps11-12sol - Massachusetts Institute of Technology...

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