Problem 11: The Civil Air Patrol Cap Owns Airplanes Used For

Problem 11 The Civil Air Patrol Cap Owns Airplanes Used For Search

The Civil Air Patrol (CAP) owns airplanes used for search and rescue missions, and over the past 10 years, has recorded data on property damages to their airplanes in different regions of Pennsylvania. For the Northeast (NE) region, data includes the number of accidents per airplane per year along with their probabilities. For the Southwest (SW) region, similar data is available, including the mean and variance of accidents per airplane. The assignment involves calculating the probability distribution, expected values, variances, risk assessments, and comparisons between regions, as well as applying probability concepts to scenarios involving loss estimates, risk control strategies, insurance risk management, and other related problems.

Paper For Above instruction

The task involves multiple components centered around analyzing the risks faced by the Civil Air Patrol (CAP) regarding their airplanes used for search and rescue missions across two major regions—Northeast (NE) and Southwest (SW)—in Pennsylvania. These components include probability distribution estimation, risk measurement, risk comparison, financial loss assessment, and evaluation of risk control measures.

Analysis of Accident Data and Probabilities in the Northeast Region

Firstly, an understanding of the random variable involved is essential. The data scenario reflects the number of accidents per airplane per year, which is a classic example of a discrete random variable. This variable measures the count of accidents associated with an individual airplane over a fixed time period. The probability distribution of this variable is determined based on the provided data, including the probability that an airplane in the NE region experiences zero, one, or two accidents within a year. Specifically, the probabilities are calculated by dividing the number of airplanes with a given accident count by the total number of airplanes in the region. For example, if 481 airplanes have zero accidents, then the probability for zero accidents is 481/500 = 96.2%. Similarly, 13 airplanes with one accident give a probability of 2.6%, and 6 airplanes with two accidents give 1.2%. These probabilities sum to 100%, confirming the distribution's consistency.

Next, the expected number of accidents per airplane in the NE region is calculated using the probability distribution. The formula involves summing the product of each possible accident count and its probability. The resulting expected value (mean) indicates that, on average, each airplane encounters approximately 0.0500 accidents annually. The units of measurement are "accidents per airplane per year," reflecting the average accident frequency.

The total expected accidents for all airplanes in the NE region are then derived by multiplying the per-airplane expectation by the total number of airplanes. For 500 airplanes, this yields an expected total of 25 accidents for the entire NE fleet annually. This enables the CAP to anticipate and allocate resources accordingly.

To verify the data's integrity, the CAP has calculated the variance of the accident distribution, which measures the dispersion or variability around the mean. Using the given variance of 0.0715, one can validate the calculation by explicitly computing the variance—by summing the squared deviations from the mean weighted by their probabilities—and confirming it matches the reported value. This step ensures the accuracy of subsequent risk assessments.

Comparison Between NE and SW Regions

For the SW region, the data indicates a mean of 1.1 accidents per airplane and a variance of 0.81 for a fleet of 1,000 planes. Calculating the standard deviation reveals the extent of variability in accidents per airplane, and the coefficient of variation (standard deviation divided by mean) allows a risk comparison between the two regions. The NE region exhibits a higher coefficient of variation (approximately 5.35) than the SW region (approximately 5.21), implying that the NE region faces relatively greater variability in accidents per airplane. This suggests the NE region is subject to higher risk, primarily because higher variability indicates less predictability and greater uncertainty.

Financial Loss Analysis and Risk Exposure

Assuming a fixed loss of $25,000 when accidents occur, the expected loss per airplane in the NE region is derived by multiplying the accident probability (0.05) by the loss amount, resulting in an expected loss of $1,250 per airplane. Extending this to the entire fleet of 500 planes yields a total expected annual loss of $625,000. Additional data from reported accidents allow for a more refined calculation of the average severity of losses, which, when multiplied by accident probability, provides the expected monetary loss that helps in insurance premium formulations.

In terms of risk management, the CAP must decide on appropriate insurance premiums based on these expected losses and variability. The variance and standard deviation inform risk managers about the uncertainty level—higher variability translates into higher premiums needed to cover potential losses.

Scenario Analysis: Risk Control Strategies

The article on “Superstorm Alters Companies’ Risk Focus” highlights resilience planning as a risk control measure. Resilience planning involves preparing comprehensive strategies to minimize the impact of natural disasters, which is an example of a loss reduction technique aimed at decreasing the severity and likelihood of losses. This proactive approach enhances a company’s ability to recover quickly and continue operations, thereby reducing overall risk exposure.

In the case of toy magnet manufacturing, the implementation of avoidance as a risk control method—by ceasing production and sales of Buckyballs—eliminates the specific risk (children swallowing magnets). Avoidance is an effective risk control strategy where the risk source is eliminated entirely, preventing potential losses altogether.

Risk Management through Structural Risk Separation

The scenarios involving inventory management exemplify risk separation strategies. In Option A, consolidating all inventory in one warehouse yields a risk of total loss if a fire occurs, with a 10% chance of losing the entire $6,000 inventory. The probability distribution is straightforward: 90% of no loss and 10% of total loss.

Option B, on the other hand, involves dividing the inventory into two warehouses, each with a 10% fire risk. The probabilities of no fire, fire at one warehouse, or both warehouses burning are derived using the rules of probability, assuming independence. The probability distribution becomes more nuanced, with an 81% chance of no loss, 9% chance of loss from either warehouse, and 1% chance of total loss with both warehouses burning. The increased dispersion (higher variance and standard deviation) under Option B indicates a higher risk level, despite the risk being spread out geographically, illustrating the importance of strategic risk separation based on probabilistic analyses.

Analysis and Conclusions

In comparing the regions, the NE faces a higher relative risk due to increased variability (higher coefficient of variation), signaling a greater unpredictability in accident costs. Financial risk assessments reveal that anticipatory measures—such as adequate insurance coverage and resilient planning—are vital for risk mitigation. Similarly, the scenarios with inventory and loss management demonstrate how risk separation and avoidance can reduce systemic vulnerabilities.

Fundamentally, understanding the probabilistic nature of accidents and losses allows for more precise risk quantification, enabling organizations like CAP to allocate resources efficiently, set appropriate insurance premiums, and develop effective risk control measures. These practices are essential for effective risk management in aviation operations, disaster preparedness, manufacturing, and other high-risk sectors.

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