(Solved):
6. An engineering inspector works at an automobile factory where he is assigned to check most of t ...
6. An engineering inspector works at an automobile factory where he is assigned to check most of the installed transducers. Suppose 3 in every 100 installed transducers checked by all inspectors do not work properly. In addition, suppose 90% of installed transducers are assigned to Mark for checking. Assuming independence between installed transducers, what is the probability that 9 installed transducers are to be selected to find the first transducer that does not work properly and was assigned to Mark for checking? (a) 0.0433 (b) 0.0235 (c) 0.0217 (d) 0.195 (e) 0.212
Here is the asnwerThis problem can be solved using the negative binomial distribution. Let X be the number of transducers that Mark inspects before he finds the first one that does not work properly. Then X follows a negative binomial distribution with parameters r = 1 (since we only need to find one defective transducer) and p = 0.03 (the probability of a transducer being defective).Now, we need to find the probability that the first defective transducer that Mark finds is among the first 9 transducers he inspects. Let Y be the number of defective transducers among the first 9 that Mark inspects. Then Y follows a binomial distribution with parameters n = 9 and p = 0.03.We want to calculate The problem requires us to find the probability that the first defective transducer that Mark finds is among the first 9 transducers he inspects. We are given that 90% of the installed transducers are assigned to Mark for checking. We can use the negative binomial distribution to model the situation where we are counting the number of successes until the first failure (in this case, finding the first defective transducer). We are also given that the probability of a transducer being defective is 0.03.Let X be the number of transducers that Mark inspects before he finds the first defective transducer. We know that X follows a negative binomial distribution with parameters r=1 (since we only need to find one defective transducer) and p=0.03 (the probability of a transducer being defective).