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practice1d - Probability with Applications-Spring 2008...

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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 0 ) 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 0 B = A B 0 (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 0 (b) B C (c) B C (d) ( A C ) 0 (e) ( B C ) 0 (f) A 0 C 0 (g) B 0 C 0
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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 (among the 26 letters A–Z) and the last three by numbers (among the ten digits 0–9)? For example,
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