Flying over the western states with mountainous terrain in a small aircraft is 40% riskier than flying over similar distances in flatter portions of the nation, according to a General Accounting Office study completed in response to a congressional request. The accident rate for small aircraft in the 11 mountainous western states is 2.1 accidents per 100,000 flight operations.(a) Let r = number of accidents for a given number of operations. Explain why the Poisson distribution would be a good choice for the probability distribution of r.Frequency of accidents is a rare occurrence. It is reasonable to assume the events are independent.Frequency of accidents is a rare occurrence. It is reasonable to assume the events are dependent. Frequency of accidents is a common occurrence. It is reasonable to assume the events are independent.Frequency of accidents is a common occurrence. It is reasonable to assume the events are dependent.(b) Find the probability of no accidents in 100,000 flight operations. (Use 4 decimal places.) (c) Find the probability of at least 6 accidents in 200,000 flight operations. (Use 4 decimal places.)
Flying over the western states with mountainous terrain in a small aircraft is 40% riskier than flying over similar distances in flatter portions of the nation, according to a General Accounting Office study completed in response to a congressional request. The accident rate for small aircraft in the 11 mountainous western states is 2.1 accidents per 100,000 flight operations.
(b) Find the probability of no accidents in 100,000 flight operations. (Use 4 decimal places.)
(c) Find the probability of at least 6 accidents in 200,000 flight operations. (Use 4 decimal places.)
(a) Poisson distribution:
It is a discrete probability distribution that models the number of events occurring within given time interval. In mass function of Poisson distribution formula, λ is the average number of events occur in an interval, r is the number of events occurring in the interval. The pmf of Poisson distribution is as follows.
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-2 е Pr r!
Some characteristics of Poisson distribution:
- The outcome of the experiment can be classified as success or failure.
- The average number of successes (λ) is known.
- The probability of success in each trial is very small.
- The events are independent of each other.
Poisson distribution as a good choice for the probability distribution of r:
It is given that the accident rate for small aircraft in the 11 mountainous western states is 2.1 accidents per 100,000 flight operations.
Here, λ is 2.1 and r denotes the number of accidents for a given number of operations.
Frequency of accidents (r) is a rare occurrence. And, it is reasonable to assume the events are independent.
Since this scenario follows the characteristics of Poisson distribution, this would be a good choice for the probability distribution of r.
(b) Calculation of probability of no accidents in 100,000 flight operations:
For no accidents occur in 100,000 flight operations, the value of r is 0.
The probability of no accidents in 100,000 flight operations is P(r = 0).
It is given that the accident rate is 2.1 accidents per 1...
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12.1) (2.1) P(r 0) 0! (0.1225) (1) 1 = 0.1225
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