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Unformatted text preview: Homework Problems, 550.311 Spring 2009 Collected for 550.311; many are taken from textbooks by Jay Devore, Neil Weiss, Bernard Rosner, John Rice, Sheldon Ross, Cincich, Levine & Stephan, Walpole, Myers, Myers, & Ye. Probability basics Problem 1.1: Later in the semesters homework you will show that if a 1 inch needle is ran- domly dropped on a floor lined with parallel lines spaced 2 inches apart, then the probability of the needle touching a line is exactly 1 . Assuming this is true, assuming the only arithmetic operation you may do is to count, and assuming you have access to such a needle and such a floor, how might you compute an integer numerator and an integer denominator of a fraction approximating ? Problem 1.2: (Devore ed5, p58) An engineering construction firm is currently working on power plants at three different sites. Let A i denote the event that the plant at site i is completed by the contract date. Use the operations of union, intersection, and complementation to describe each of the following events in terms of A 1 , A 2 , and A 3 , draw a Venn diagram, and shade the region corresponding to each one: a) At least one plant is completed by the contract date. b) All plants are completed by the contract date. c) Only the plant at site 1 is completed by the contract date. d) Exactly one plant is completed by the contract date. e) Either the plant at site 1 or both of the other two plants are completed by the contract date. Problem 1.3: (Rice p25) Show from the probability axioms: For any events A , B it holds that P( A B ) P( A )+P( B )- 1. (You may use the results derived in class without reproving them.) Problem 1.4: Show from the probability axioms: For any events E 1 ,E 2 ,E 3 ,... (even if they are not disjoint) it holds that P( i E i ) i P( E i ). [Hint: Consider events of form A i := E i \ ( i- 1 j =1 E j ).] Combinatorial probability Problem 2.1: (Devore ed5, p74) A class has 20 nonsmokers, 15 light smokers, and 10 heavy smokers. Six of these will be randomly selected to participate in a study (each set of 6 is equiprob- able). a) What is the probability that all 6 are heavy smokers? b) What is the probability that all 6 have the same status (non/light/heavy smoking)? 1 Problem 2.2: (Ross p54) Suppose 8 rooks (castles) are randomly laid on a chessboard. What is the probability that they are non-attacking, that is, that no two are in the same row or column? Problem 2.3: (Rice p25) The first 3 digits of a university telephone exchange are 452. If all sequences of the remaining four digits are equally likely, what is the probability that a randomly selected university number contains 7 distinct digits? Problem 2.4: (Rice p26) A deck of 52 cards is shuffled thoroughly. What is the probability that all 4 aces are next to each other?...
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This note was uploaded on 03/24/2009 for the course MATH 550.311 taught by Professor Fishkind during the Spring '09 term at Johns Hopkins.
- Spring '09