hw3sol - Math 361 X1 Homework 3 Solutions Spring 2003...

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Unformatted text preview: Math 361 X1 Homework 3 Solutions Spring 2003 Graded problems: 2, 4, 6, 7; 3 points each, 12 points total score. Instructions. The first and second problems serve to review material that has come up earlier; the first one is similar to the lottery problem discussed in class, and the second boils down to a set-theoretic exercise once you have defined appropriate events and expressed the probabilities sought in terms of these events. The remaining problems fall into the repeated success/failure trial framework, and you should do these problems within that framework, as in the class examples. Be sure to state what you mean by “success” , specify the parameters p and n , and rephrase the event in question within this model (i.e., in terms of successes and/or failures). One of these problems requires an (easy) “assembly level” probability computation, but for the majority the formulas for one of the two special cases discussed in class can be applied. Problem 1. [1.Rev.19] Sampling with/without replacement. A box contains 5 tickets numbered 1,2,3,4,5. Two tickets are drawn at random. Find the probability that the numbers on the two tickets differ by two or more if (a) the draws are made with replacement; (b) the draws are made without replacement. Solution. (a) If the samples are made with replacement, the problem is equivalent to that of rolling a 5 sided die twice and asking for the probability that the two numbers differ by at most 2. We can take as outcomes the tuples ( n 1 ,n 2 ) with n i = 1 , 2 ,..., 5, where n 1 denotes the number on the first ticket and n 2 the number on the second ticket, and as outcome space Ω = { ( n 1 ,n 2 ) : n i = 1 , 2 ,..., 5 } , with equally likely probabilities. The event we are interested in, “the two tickets differ by at least 2,” corresponds to the subset A = { ( n 1 ,n 2 ) : n i = 1 , 2 ,..., 5; | n 1- n 2 | ≥ 2 } , which written out consists of the 6 elements (1 , 3) , (1 , 4) , (1 , 5) , (2 , 4) , (2 , 5) , (3 , 5) plus the same tuples with the order switched , so #( A ) = 12. Since #(Ω) = 5 2 = 25, it follows that P ( A ) = 12 / 25. (b) If the samples are made without replacement, the condition n 1 6 = n 2 has to be added to the definition of Ω, while the set A stays the same. We now have #(Ω) = 5 · 4 = 20, so P ( A ) = 12 / 20 = 3 / 5. Problem 2. [1.6:4] Winning probabilities at slot machines. A typical slot machine in a Nevada casino has three wheels, each marked with twenty symbols at equal spacings around the wheel. The machine is engineered so that on each play the three wheels spin independently, and each wheel is equally likely to show any one of the twenty symbols when it stops spinning. On the central wheel, nine out of the twenty symbols are bells, while there is only one bell on the left wheel and one bell on the right wheel. The machine pays out the jackpot only if the wheels come to rest with each wheel showing a bell. (a) Calculate the probability of hitting the jackpot. (b) Calculate the probability ofshowing a bell....
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This note was uploaded on 04/03/2008 for the course STAT 134 taught by Professor Aldous during the Spring '03 term at Berkeley.

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hw3sol - Math 361 X1 Homework 3 Solutions Spring 2003...

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