According to the 1986 Statistical Abstracts of the United States, in the ?ve- year period from 1977-1981, failures among banks insured by the Federal Deposit Insurance Corporation (FDIC) averaged approximately 8.5 per year. Speci?cally, 10 failures were reported in 1980. If FDIC-insured bank failures are considered rare events,

(i) Postulate an appropriate model for the random variable X representing the total number of FDIC-insured bank failures per year, and use this model to compute the probability of observing the number of failures reported in 1980.

(ii) What is the "most likely" number of failures in any one year, if this number, x∗, is so designated because f (x∗) is the highest probability of all possible values of x? Determine the probability of having more failures in one year than this "most likely" number of failures.

(iii) An FDIC "quality control inspector" suggested that the occurrence of 13 or more failures in one year should be considered cause for alarm. What is the proba- bility of such an event occurring and why do you think that such an event should truly be a cause for alarm?