Chapter 8.3, Problem 26E

Quality Control A manufacturer of precise measuring devices carefully tests each device manufactured. Seventy percent of all newly manufactured devices are approved, 20% are sent back to be recalibrated, and 10% are determined to be beyond repair and are destroyed. Eighty percent of the devices that have been recalibrated are approved, 10% are sent back to be recalibrated again, and 10% are destroyed. Therefore, the four states of a devices are New, Approved, Recalibrated, and Destroyed.

a. Set up the $4\times 4$ absorbing stochastic matrix with columns and rows labeled, A, D, N, and R that describes this transition.

b. Find the stable matrix.

c. Determine the probability that a New device will eventually be Approved.

d. Determine the expected number of times a New device will be tested.