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# hw1-sol - AMS 311(Fall 2009 Joe Mitchell Homework Set 1 –...

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AMS 311 (Fall, 2009) Joe Mitchell Homework Set # 1 – Solution Notes (1). (13 points) Problem 5, Ross, Chapter 2. (same problem in 7th or 8th edition) (a). Since each x i is either 0 or 1 (2 choices), the total number of choices for the outcome vector, ( x 1 ,x 2 ,x 3 ,x 4 ,x 5 ), is 2 5 . Thus, | S | = 2 5 = 32. (b). W has 15 elements, namely: W = { (1,1,0,0,0), (1,1,0,0,1), (1,1,0,1,0), (1,1,0,1,1), (1,1,1,0,0), (1,1,1,0,1), (1,1,1,1,0), (1,1,1,1,1), (0,0,1,1,0), (0,0,1,1,1), (0,1,1,1,0), (0,1,1,1,1), (1,0,1,1,0), (1,0,1,1,1), (1,0,1,0,1) } . (c). Since A consists of outcomes having x 4 = x 5 = 0, we see that there are 2 choices for each of x 1 ,x 2 ,x 3 ; thus, A has 2 · 2 · 2 = 8 elements. (You should be able to list them.) (d). A W = { (1 , 1 , 0 , 0 , 0) , (1 , 1 , 1 , 0 , 0) } , since these are the only two elements of W having x 4 = x 5 = 0. (2). (16 points) Among 33 students in a class, 17 of them earned A’s on the midterm exam, 14 earned A’s on the Fnal exam, and 11 did not earn A’s on either examination. What is the probability that a randomly selected student from this class earned an A on both exams? What is the sample space and event, in your notation? The “experiment” is the selection of one student from the class of 33; we can record the “out- come” as a pair ( m,f ), where m is the grade the student made on the midterm and f is the grade the student made on the ±nal. The sample space is the set S of 33 students. Let E be the event that the selected student makes an “A” on the midterm; we know that P ( E ) = 17 / 33. Let F be the event that the student earns an “A” on the ±nal; we know that P ( F ) = 14 / 33. We are also told that P ( E c F c ) = 11 / 33 = 1 / 3. We want to compute the probability

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hw1-sol - AMS 311(Fall 2009 Joe Mitchell Homework Set 1 –...

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