5. For the output of a factory, the error of the first kind (false positive) is that an item is good given that it is tested to be defective. The error of the second kind (false negative) is that an item is defective given that it is tested to be good. Previously, the factory relied on a simple test such that a defective item passes with probability 0.15 while a good item passes with probability 0.95. Assume the fraction of defective items produced is 0.1. (a) Compute the probability of errors of the first kind and of the second kind for the simple test. [4] (b) Consider a monitoring system that runs n independent simple tests and requires the [6] item to pass at least m of them. Find the smallest m, n 6 {1, 2, 3, 4,5} such that a defective item passes this system with probability at most 0.1 and a good item passes with probability at least 0.99. Hint: n = 3. (c) Find the probability of errors of the first kind and of the second kind for the monitoring [4] system constructed in part (b) rounded to 3 decimal places. Compare with the corresponding errors of the simple test computed in part (a)