The exponential distribution can be used to solve Poisson-type problems in which the intervals are not time. The Airline Quality Rating Study published by the U.S. Department of Transportation reported that in a recent year, Airtran led the nation in fewest occurrences of mishandled baggage, with a mean rate of 4.06 per 1,000 passengers. Assume mishandled baggage occurrences are Poisson distributed. Using the exponential distribution to analyze this problem, determine the average number of passengers between occurrences. Suppose baggage has just been mishandled.

a. What is the probability that at least 500 passengers will have their baggage handled properly before the next mishandling occurs?

b. What is the probability that the number will be fewer than 200 passengers?