Selecting the best wafer-slicing machine. Silicon wafer slicing is a critical step in the production of semiconductor devices (e.g., diodes, solar cells, transistors). Yuanpei (China) University researchers used control charts to aid in selecting the best silicon wafer-slicing machine (Computers & Industrial Engineering, Vol. 52, 2007). Samples of n = 2 wafers were sliced each hour for 67 consecutive hours and bow measurements (a measure of precision) were recorded. The resulting x-chart for one of the machines tested revealed that the cutting process was out of control on the 19th, 40th, and 59th hours. For each of these 3 hours, the mean bow measurement fell above the upper control limit. Assume the mean bow measurements are normally distributed.

a. If the process is in control, what is the probability that a mean bow measurement for a randomly selected hour will fall above the upper control limit?

b. If the process is in control, what is the probability that 3 of 67 mean bow measurements fall above the upper control limit?