# What is the probability that the first manufactured product indicates out-of-control?

A process quality characteristic that is being monitored is thought to be Normally distributed with a true mean of 90 and a standard deviation of 10. The manufacturer devises a control chart with lower and upper control limits of LCL = 70; UCL = 110, respectively. If the measurement of a manufactured product lies outside these control limits, then the entire process is said to be \out-of-control” and shut down for further investigation.

(a) What is the probability that the first manufactured product indicates out-of-control?

(b) What is the probability that out-of-control is indicated in the 5th product? 10th product? 100th product? Hint: Consult the additional material on the Geometric probability distribution.

(c) What is the expected number of products manufactured until out-of-control is first indicated? Hint: Consult the additional material on the Geometric probability distribution.

(d) What is the probability that the control chart indicates out-of-control in one of the first three manufactured products?