PracticeSolutions - Dr. Edwin Romeijn Practice Problem Set...

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Dr. Edwin Romeijn ESI 6321 Applied Probability Methods in Engineering Practice Problem Set - Solutions Class of 2009 - 7 - ESI 6321 APPLIED PROBABILITY METHODS IN ENGINEERING OUTREACH ENGINEERING MANAGEMENT Class of 2009 PRACTICE PROBLEM SET - SOLUTIONS Exercise 2.3 (a) P(At least 4 years experience) = P(4 years experience) + P(5 or more years experience) = 15/100 + 35/100 = 50/100. (b) P(At least 4 years experience | At least 3 years experience) = P(At least 4 years experience)/P(At least 3 years experience) = (50/100)/(80/100) = 50/80. Exercise 2.9 Let H + denote the event that the person is infected with HIV and H that the person is not infected. Furthermore, let R denote the event that the test is positive and N that the test is negative. Then it is given that P(H + ) = 550,000/250,000,000 = 0.0022, P(R | H + ) = 0.99, and P(N | H ) = 0.99. (a) We want to compute P(H + | R). Notice that P(R) = P(R | H + ) x P(H + ) + P(R | H ) x P(H ) = = 0.99 x 0.0022 + 0.01 x (1-0.0022) = 0.002178 + 0.009978 = 0.012156. Therefore, P(H + | R) = P(R | H + ) x P(H + )/P(R) = 0.002178/0.012156 = 0.18. (b) Let D denote the event that the person is a drug user. We want to compute P(H + | R and D). Note that P(H + and D) = 275,000/250,000,000 = 0.0011 and P(H and D) = (10,000,000 – 275,000)/250,000,000 = 0.0389. We assume that P(R | H + and D) = P(R | H + ) = 0.99 and P(N | H and D) = P(N | H ) = 0.99. That is, drug using doesn’t influence the test accuracy. Then P(R D) = P(R | H + and D) x P(H + and D) + P(R | H and D) x P(H and D) = = P(R | H + and D) x P(H + and D) + (1 - P(N | H and D)) x P(H and D) = = (0.99) x (0.0011) + (1 - 0.99) x (0.0389) = 0.001478. Therefore, P(H + | R and D) = P(H + and R and D)/P(R and D) = = P(R | H + and D) x P(H + and D)/P(H + and D) = (0.99) x (0.0011)/(0.001478) = 0.74.
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Dr. Edwin Romeijn ESI 6321 Applied Probability Methods in Engineering Practice Problem Set - Solutions Class of 2009 - 8 - Exercise 2.14 (a) Let the random variable A denote the time to complete Task A, B denote the time to complete Task B, and
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PracticeSolutions - Dr. Edwin Romeijn Practice Problem Set...

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