Fundamentals of Probability and Statistics for Engineers

186 Fundamentals of Probability and Statistics for Engineers6.10 A manufacturing firm receives a lot of 100 parts, of which 5 are defective. Supposethat the firm accepts all 100 parts if and only if no defective ones are found in asample of 10 parts randomly selected for inspection. Determine the probability thatthis lot will be accepted.6.11 A shipment of 10 boxes of meat contains 2 boxes of contaminated goods. Aninspector randomly selects 4 boxes; let Z be the number of boxes of contaminatedmeat among the selected 4 boxes.(a) What is the pmf of Z?(b) What is the probability that at least one of the four boxes is contaminated?(c) How many boxes must be selected so that the probability of having at last onecontaminated box is larger than 0.75?6.12 In a sequence of Bernoulli trials with probability p of success, determine theprobability that r successes will occur before s failures.6.13 Cars arrive independently at an intersection. Assuming that, on average, 25% ofthe cars turn left and that the left-turn lane has a capacity for 5 cars, what is theprobability that capacity is reached in the left-turn lane when 10 cars are delayed bya red signal?6.14 Suppose that n independent steps must be taken in the sterilization procedure fora biological experiment, each of which has a probability p of success. If a failurein any of the n steps would cause contamination, what is the probability ofcontamination if n ˆ 10 and p ˆ 0:99?6.15 An experiment is repeated in a civil engineering laboratory. The outcomes of theseexperiments are considered independent, and the probability of an experimentbeing successful is 0.7.(a) What is the probability that no more than 6 attempts are necessary to produce3 successful experiments?(b) What is the average number of failures before 3 successful experiments occur?(c) Suppose one needs 3 consecutive successful experiments. What is the probabilitythat exactly 6 attempts are necessary?6.16 The definition of the 100-year flood is given in Example 6.7.(a) Determine the probability that exactly one flood equaling or exceeding the100-year flood will occur in a 100-year period.(b) Determine the probability that one or more floods equaling or exceeding the100-year flood will occur in a 100-year period.6.17 A shipment of electronic parts is sampled by testing items sequentially until the firstdefective part is found. If 10 or more parts are tested before the first defective partis found, the shipment is accepted as meeting specifications.(a) Determine the probability that the shipment will be accepted if it contains 10%defective parts.(b) How many items need to be sampled if it is desired that a shipment with 25%defective parts be rejected with probability of at least 0.75?6.18 Cars enter an interchange from the south. On average, 40% want to go west, 10%east, and 50% straight on (north). Of 8 cars entering the interchange:(a) Determine the joint probability mass function (jpmf) of X 1 (cars westbound),X 2 (cars eastbound), and X 3 (cars going straight on).(b) Determine the probability that half will go west and half will go east.(c) Determine the probability that more than half will go west.TLFeBOOK