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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.)

Question

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.)
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Step 1

(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.

-2
е
Pr
r!
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-2 е Pr r!

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Step 2

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.

Step 3

(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...

12.1) (2.1)
P(r 0)
0!
(0.1225) (1)
1
= 0.1225
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12.1) (2.1) P(r 0) 0! (0.1225) (1) 1 = 0.1225

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