You got a summer job at Bob’s metal shed as a machine operator doing metalwork. The very old machine you are in charge of cuts long metal tubes into shorter pieces. Customers specify a length l in increments of 1 cm and the number of pieces n. Bob guarantees that all n pieces will have length between l − 0.5 and l + 0.5.

You control the length of pieces produced by the machine with a mechanical hand-wheel in in-
crements of 1 cm. The machine will cut the tube into pieces of average length l. However the length
of the pieces has a variance of 1 . Assume that the actual length of produced pieces follows a normal 16
distribution.

Your task is to make sure that the order is correctly fulfilled, by measuring each of the pieces
produced (you can do this at the same speed as the machine produces pieces) and running the machine until n pieces of length between l − 0.5 and l + 0.5 have been produced. Produced pieces outside of the range l − 0.5 and l + 0.5 are called defectives. The length of the production run—that is, the total number of pieces produced—is denoted by m.



What is the probability that the machine produced a piece which can be delivered to the customer? Use the CDF for the standard normal Φ.