Part II: Practical case Consider a production line producing some commodity with a high failure rate $($maybe because of variance in materials, or dificulty of production$),$ To finish the product $2$ different steps need to be completed, step A and step $B.$ The overall structure of the production line is described in the figure below; in steps $A$ or $B$ the picce goes through the respective procedure. The control steps are to determine it the quality is good cnough to continue to the next steps if quality is qood enough, the piece is finished. Otherwise, the piece is sent to be foued in the attempl fix steps we try to fix the piece, it the issue is fixable or small enough, then the piece is sent back to quality control otherwise the piece is trashed. The values on the edges represernt the probabilities to go from one stale to another. a$).($Create the slochastic matrix describing the Markv chain coeresponding to this production line. b$)$ What is the probability that a product starling in step A ends up in the trash? What is the prothability that it is tinished? c$)$ What is the average number of is the average cost of producing $10$ finished products? $($Products that are lrashed incur no extra cost$).$ e$)$ Acsume no slarted products are trashed. Instead, a more expensive fix costing $4l$ is attempted on those defective pleces. If each finished plece could be sold ior $25lon,$ would it still be profitable to finish these products? $($Mou can assurne that the extra time spent on these products is neglectable$).$