A product is processed on two sequential machines, I and II$.$ Inspection takes place after a product unit is completed on a machine. There is a $5\%$ chance that the product will be scrapped before inspection. After inspection, there is a $3\%$ chance the product unit will be scrapped and a $7\%$ chance of its being returned to the same machine for reworking. Else, a unit passing inspection after both machines is accepted as non$-$defective and shipped to the customer.

a$)$ Define the states for the given process and construct the transition matrix.

b$)$ For a part starting at Machine I, determine the average number of visits to each station.

c$)$ If a batch of $1000$ units is started on Machine I, how many non$-$defective units will be produced.