practice1d - Probability with Applications—Spring 2008...

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Unformatted text preview: Probability with Applications—Spring 2008 Problem Set 1 for ISYE 2027, Section B (January 30, 2008) Purpose. The first exam will focus on material largely covered in Chapter 2 of Walpole et al. This document contains two practice exams and a list of recommended exercises from the textbook. In the lecture periods prior to the exam, I will solve these practice exams at the board (along with additional exercises as time permits). Please see also the document “Study Guide 1,” which provides a theoretical overview of the material you are responsible for. Also important are your lecture notes and the document “Homework 1 Solutions.” Notes, books, and electronic devices may not be used during the exam. First practice exam. The following formulas are provided without context, as a memory aid: nPr = n ! / ( n- r )! nCr = ( nPr ) /r ! P ( S k i =1 E k ) = ∑ k i =1 P ( E k ) if E i ∩ E j = ∅ for all i 6 = j P ( A | B ) = P ( A ∩ B ) /P ( B ) if P ( B ) 6 = 0 P ( B ) = P ( B ∩ A ) + P ( B ∩ A ) 1. Consider events A and B in a sample space S . Are the following statements necessarily true? More specifically, are they guaranteed to be true for any A , B , and S we might consider? Answer “yes” or “no” for each of the three statements below. (Complements are taken relative to S .) (a) A ∪ B = A ∩ B (b) B ∩ C ⊂ ( A ∪ B ) ∩ ( A ∪ C ) (c) A ⊂ ( A ∩ C ) ∪ ( A ∩ B ) 2. Let S = { 1 , 2 , 3 , 4 , 5 , 6 } be the sample space for a die roll. How many distinct subsets of S could we define? (Don’t forget the subsets assumed in the second required property of probability measures.) 3. Relative to the sample space S = { 1 , 2 , 3 , 4 , 5 , 6 } , define the events A = { 1 , 2 , 3 , 4 , 5 } , B = { 1 , 2 , 3 } , and C = { 3 , 4 } . Assume complements are relative to S . Specify the following sets by listing their elements: (a) A ∩ B (b) B ∩ C (c) B ∪ C (d) ( A ∪ C ) (e) ( B ∩ C ) (f) A ∩ C (g) B ∪ C 4. Consider events A and B in a sample space S with probability measure P . If P ( A ) = 3 5 and P ( B ) = 2 3 , can A and B be disjoint sets? Answer “yes” or “no.” 5. How many license plates are possible if the first three places are occupied by letters (among5....
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This note was uploaded on 04/18/2008 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.

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practice1d - Probability with Applications—Spring 2008...

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