Background to the problem: First Maryland Bank has recently acquired a very large bank in Texas, and is in the process of merging the operations of the two institutions. The new bank will be called "First American Bank" and, befitting its new and grand name, the CEO, Donald McLean, wants the new bank to be the most efficient in the country. Among the many tasks involved in merging the operations of the two banks is the processing of customers' checks. Currently, each bank has its own central check processing facility. The question now is whether to continue to maintain the two facilities, close one or the other down, or eventually close both down and build a larger, more efficient facility to take over the combined check processing.
One problem which confounds this decision is that most of the acquired bank's customers are in the South, while most of First Maryland Bank customers are on the mid-Atlantic coast. A preliminary decision tree analysis of this problem has indicated that the number of checks processed per week at each of the two facilities is critical to which decision is best.
One of the dedicated employees at First Maryland is the operations manager, Gregoire Currey. Fortunately, Currey has maintained 6 years of records on the number of checks processed each week at the First Maryland check processing facility. He has found that over the 6 years just prior to the merger, the number of checks processed each week at Piedmont's facility is normally distributed with a mean of 90,000 checks per week and a standard deviation of 18,000 checks per week.
The question: Suppose Currey randomly chooses 9 weeks from this pre-merger period. What is the probability that more than 90,000 checks were processed in 7 or more of these weeks?
This uses normal and binomial probability distributions to find probabilities regarding the number of bank checks processed.