4. (Conditional Probabilities, Finite Stochastic Processes; 25pts). Three machines A, B and C produce respectively 48%, 27% and 25% of the total number of items of a factory. The percentages of defective output of these machines are 3%, 4% and 8%, respectively. (i). If an item is selected at random, find the probability that the item is defective. (ii). If an item is selected at random and is found to be defective, find the probability that the item was produced by machine A. Finding a Transition Matrix In Exercises 17-24, find the transition matrix from B to B'. 20. B = {(1, 1), (1, 0)}, B' = {(1, 0), (0, 1)} Finding Transition and Coordinate Matrices In Exercises 37-40, (a) find the transition matrix from B to B', (b) find the transition matrix from B' to B, (c) verify that the two transition matrices are inverses of each other, and (d) find the coordinate matrix [x], given the coordinate matrix [x]. - 38. B = {(2, 2), (6, 3)}, B' = {(1, 1), (32, 31)}, [X = [-1] Problem 2: Please check whether each of the following processes is Recurrent Nonnull or not. (1) A birth and death process with rates 1 An =1+ Vn 0, n=1 Vn 1 n+1 (2) A birth and death process with rates 2 n = 1- Vn 0, n+3 Min=1 Vn 1