A production process contains three stages in series. An entering item starts in the first manufacturing stage. The output from each stage is inspected. An item of acceptable quality is passed on to the next stage, a defective item is scrapped, and an item of marginal quality is reworked at the current stage. An item at stage 1 has a 0.08 probability of being defective, a 0.12 probability of being of marginal quality, and a 0.80 probability of being of acceptable quality. An item at stage 2 has a 0.06 probability of being defective, a 0.09 probability of being of marginal quality, and a 0.85 probability of being acceptable. The probabilities that an item at stage 3 will be defective, marginal in quality, or acceptable, are 0.04, 0.06, and 0.90, respectively. All items of acceptable quality produced by stage 3 are sold. 

(a) Model the production process as a five-state absorbing multichain with two absorbing states. Choose one absorbing state to represent scrapping a defective item, and the other to represent selling an item of acceptable quality produced by stage 3. Choose the transient states to represent the stages of the production process. Construct the transition probability matrix. Represent it in canonical form. 

(b) Find the probability that an item will be scrapped, given that it is in stage 3. 

(c) Find the probability that an item will be sold without being reworked.

(d) Find the probability that an item will be in stage 2 after three inspections.

(e) Find the probability that an item will be scrapped.

(f) Find the probability that an item will be sold.

(g) Find the mean number of inspections that an item will receive in stage 3. 

(h) Given that an item is in stage 2, find the mean number of inspections that it will receive in stage 3.