Midterm2 - Mathematics Department, UCLA T. Richthammer...

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Unformatted text preview: Mathematics Department, UCLA T. Richthammer winter 09, midterm 2 Mar 02, 2009 Midterm 2: Math 170A Probability, Sec. 1 1. (5 pts) In a probability course there are 5 excellent students, 10 good students and 5 regular students. Suppose that the chance to get a good result on the second midterm is 100 % for an excellent student, 75 % for a good student and 50 % for a regular student. Is a student with a good result on the exam most likely excellent, good or regular? (Describe an appropriate model and calculate the three probabilities.) Answer: A good model for this situation is a sequential model with S = { e, g, r } { g, b } , where the first projection X 1 describes whether a student is excellent, good or regular, and the second projection X 2 describes whether a student has a good or a bad exam. P (( x 1 , x 2 )) = p x 1 p x 2 | x 1 , where p e = 1 4 , p g = 1 2 , p r = 1 4 , p g | e = 1, p g | g = 3 4 and p g | r = 1 2 . Using Bayes formula we get P ( X 1 = e | X 2 = g ) = p e p g | e p e p g | e + p g p g | g + p r p g | r = 1 4 1 1 4 1+ 1 2 3 4 + 1 4 1 2 = 2 2+3+1 = 2 6 , and similarly we get P ( X 1 = g | X 2 = g ) = 3 2+3+1 = 3 6 and P ( X 1 = r | X 2 = g ) = 1 2+3+1 = 1 6 . So most likely the student is good....
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Midterm2 - Mathematics Department, UCLA T. Richthammer...

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