Can someone please explain in detail how we get the equations for Pjk? Also what will the transition probability matrix look like?

(IN DETAIL PLS)

Many thanks

4.2.3 A Continuous Sampling Plan Consider a production line where each item has probability p of being defective. Assume that the condition of a particular item (defective or nondefective) does not depend on the conditions of other items. The following sampling plan is used. Initially every item is sampled as it is produced; this procedure continues until i consecutive nondefective items are found. Then, the sampling plan calls for sampling only one out of every r items at random until a defective one is found. When this happens the plan calls for reverting to 100% sampling until i consecutive nondefective items are found. The process continues in the same way. State Ex(k = 0,1,...,.-1) denotes that k consecutive nondefective items have been found in the 100% sampling portion of the plan, while state E; denotes that the plan is in the second stage (sampling one out of r). Time m is considered to fol- low the mth item, whether sampled or not. Then, the sequence of states is a Markov chain with Pik = Pr{in state Ex after m +1 items in state E; after m items) P for k=0,0