In an article on August 15, 1998, edition of the Journal News (Hamilton, Ohio),2 the Associated Press reported that the Cleveland Air Route Traffic Control Center, the busiest in the nation for guiding planes on cross-country routes, had experienced an unusually high number of errors since the end of July. An error occurs when controllers direct flights either within five miles of each other horizontally, or within 2,000 feet vertically at a height of 18,000 feet or more (the standard is 1,000 feet vertically at heights less than 18,000 feet). 

The controllers’ union blamed the errors on a staff shortage, whereas the Federal Aviation Administration (FAA) claimed that the cause was improved error reporting and an unusual number of thunderstorms. Suppose that an air traffic control center has been averaging 20.8 errors per year and that the center experiences 3 errors in a week. 

The FAA must decide whether this occurrence is unusual enough to warrant an investigation as to the causes of the (possible) increase in errors. To investigate this possibility, we will find the probability distribution of x, the number of errors in a week, when we assume that the center is still averaging 20.8 errors per year. 

Calculate the probability of no errors in a week and the probability of three errors in a week.