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34                     Fundamentals of Probability and Statistics for Engineers

           2.29 A machine part may be selected from any of three manufacturers with probabilities
               p 1 ˆ 0:25, p 2 ˆ 0:50, and p 3 ˆ 0:25.  The probabilities that it will function properly
               during a specified period of time are 0.2, 0.3, and 0.4, respectively, for the three
               manufacturers. Determine the probability that a randomly chosen machine part
               will function properly for the specified time period.
           2.30 Consider the possible failure of a transportation system to meet demand during
               rush hour.
               (a) Determine the probability that the system will fail if the probabilities shown in
                  Table 2.3 are known.
                       Table 2.3  Probabilities of demand levels and of system
                        failures for the given demand level, for Problem 2.30
                                                             j
                       Demand level   P(level)   P(system failure level)
                       Low            0.6        0
                       Medium         0.3        0.1
                       High           0.1        0.5

               (b) If system failure was observed, find the probability that a ‘medium’ demand
                  level was its cause.
           2.31 A cancer diagnostic test is 95% accurate both on those who have cancer and on
               those who do not. If 0.005 of the population actually does have cancer, compute
               the probability that a particular individual has cancer, given that the test indicates
               he or she has cancer.
           2.32 A quality control record panel of transistors gives the results shown in Table 2.4
               when classified by manufacturer and quality.
                 Let one transistor be selected at random. What is the probability of it being:
               (a) From manufacturer A and with acceptable quality?
               (b) Acceptable given that it is from manufacturer C?
               (c) From manufacturer B given that it is marginal?
                        Table 2.4  Quality control results, for Problem 2.32
                   Manufacturer                 Quality
                                Acceptable  Marginal  Unacceptable  Total

                   A            128        10       2            140
                   B             97         5       3            105
                   C            110         5       5            120

           2.33 Verify Equation (2.26) for three events.
           2.34 In an elementary study of synchronized traffic lights, consider a simple four-light
               system. Suppose that each light is red for 30 seconds of a 50-second cycle, and suppose
                                     P…S j‡1 jS j †ˆ 0:15
               and
                                     P…S j‡1 jS j †ˆ 0:40








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