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*.

(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.)

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.)

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.

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...

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