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Chapter 5: Problem 44

An average of 15 aircraft accidents occur each year (The World Almanac and Book of Facts, 2004 ). a. Compute the mean number of aircraft accidents per month. b. Compute the probability of no accidents during a month. c. Compute the probability of exactly one accident during a month. d. Compute the probability of more than one accident during a month.

Short Answer

Expert verified
Mean accidents per month: 1.25. Probability of no accidents: 0.2865. Probability of exactly one accident: 0.3581. Probability of more than one accident: 0.3554.

Step by step solution

01

Identify the Mean per Month

The first step is to determine the average number of aircraft accidents per month, given that there are 15 accidents per year. Since there are 12 months in a year, the mean number of accidents per month, \( \lambda \), can be calculated as follows:\[ \lambda = \frac{15 \text{ accidents per year}}{12 \text{ months}} \]Calculate the value:\[ \lambda = 1.25 \text{ accidents per month} \]
02

Probability of No Accidents in a Month

To find the probability of no accidents in a given month, we'll use the Poisson probability formula:\[ P(X = k) = \frac{e^{-\lambda} \cdot \lambda^{k}}{k!} \]Where:- \( \lambda = 1.25 \)- \( k = 0 \) (no accidents)- \( e \approx 2.71828 \)Substitute \( k = 0 \):\[ P(X = 0) = \frac{e^{-1.25} \cdot 1.25^{0}}{0!} = e^{-1.25} \]Calculate \( e^{-1.25} \):\[ P(X = 0) \approx 0.2865 \]
03

Probability of Exactly One Accident in a Month

Next, compute the probability of exactly one accident in a month, using the Poisson formula with \( k = 1 \):\[ P(X = 1) = \frac{e^{-1.25} \cdot 1.25^{1}}{1!} \]Calculate the value:\[ P(X = 1) = e^{-1.25} \cdot 1.25 \approx 0.3581 \]
04

Probability of More Than One Accident in a Month

To determine the probability of more than one accident in a month, we need to consider all scenarios where \( k \geq 2 \). Instead of calculating each case separately, use the cumulative probabilities for \( k = 0 \) and \( k = 1 \), and subtract from 1:\[ P(X > 1) = 1 - P(X = 0) - P(X = 1) \]Substitute the values computed earlier:\[ P(X > 1) = 1 - 0.2865 - 0.3581 \]Calculate to find:\[ P(X > 1) \approx 0.3554 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Calculation
Probability calculation is a method to determine the likelihood of a particular event happening. In this context, we use the Poisson Distribution to calculate probabilities. It is ideal for counting the number of events that happen over a fixed interval of time or space, like aircraft accidents in a month.

The Poisson formula is used:
  • \( P(X = k) = \frac{e^{-\lambda} \cdot \lambda^{k}}{k!} \)
Where:
  • \( \lambda \) is the average number of events (accidents here) that occur in the given interval.
  • \( k \) is the number of events for which we want to find the probability.
  • \( e \) is a constant approximately equal to 2.71828.
This formula allows us to compute the probability of no accidents (\( k = 0 \)), exactly one accident (\( k = 1 \)), and more than one accident (\( k \geq 2 \)) in a given month. Understanding how to apply this formula provides a powerful tool to evaluate event occurrences over time.
Aircraft Accidents
An aircraft accident is a significant event characterized by unexpected and unintended incidents involving aircraft. In this problem, the idea is to predict the probability of such accidents occurring within specified time intervals, using historical data.

When we talk about aircraft accidents:
  • They are treated as rare events, suggesting that such events are suited to be modelled by the Poisson distribution.
  • This helps aviation regulatory authorities to strategize safety measures effectively based on probable frequencies.
Using statistics, such as the average number of accidents in a year, allows for a calculated approach to prepare for accidents, rather than a reactive one. This can help improve safety and reduce risk over time.
Mean Calculation
In statistics, the mean (average) is central to many analyses. Here, we are focused on calculating the average number of aircraft accidents per month, based on annual data.

The formula to calculate the mean per month is simple:
  • \( \lambda = \frac{\text{Total accidents per year}}{\text{Number of months in a year}} \)
Applying this to the problem:
  • The total number of aircraft accidents is 15 per year.
  • Dividing by 12 months, the mean, \( \lambda \), becomes 1.25 accidents per month.
This mean value is critical because it forms the basis of the Poisson distribution used for calculating probabilities. It represents expected outcomes and serves as a benchmark for evaluating monthly accident occurrences.

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Most popular questions from this chapter

The following data were collected by counting the number of operating rooms in use at Tampa General Hospital over a 20 -day period: \(\mathrm{On} 3\) of the days only one operating room was used, on 5 of the days two were used, on 8 of the days three were used, and on 4 days all four of the hospital's operating rooms were used. a. Use the relative frequency approach to construct a probability distribution for the number of operating rooms in use on any given day. b. Draw a graph of the probability distribution. c. Show that your probability distribution satisfies the required conditions for a valid discrete probability distribution.

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