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Airplane safety has been improving over the years. From 2000 to \(2010,\) the average number of global airline deaths per year was over 1000 , even when excluding the nearly 3000 deaths in the United States on September 11,2001 . The number of global airline deaths declined in 2011 , again in \(2012,\) and then hit a low of only 265 in \(2013 .\) In \(2013,\) there were a total of 825 million passengers globally. Sources: en.wikipedia.org and www.transtats.bts.gov1. a. Can you consider the 2013 data as a long run or short run of trials? Explain. b. Estimate the probability of dying on a flight in \(2013 .\) (Note, the probability of dying from a 1000 -mile automobile trip is about 1 in 42,000 by contrast.) c. Raul is considering flying on an airplane. He noticed that over the past two months, there have been no fatal airplane crashes around the world. This raises his concern about flying because the airlines are "due for an accident." Comment on his reasoning.

Step by step solution

01

Understanding Long Run vs Short Run

Long run means a large number of trials have occurred, allowing statistical trends to stabilize. For the 2013 data, with flights involving 825 million passengers, this large number of trials suggests a long-run scenario as it provides a stable dataset for statistical analysis.

02

Calculate Probability of Dying on a Flight

To find the probability of dying in a flight in 2013, divide the number of deaths by the number of passengers. The probability is calculated as: \[ P(\text{dying in 2013}) = \frac{265}{825,000,000} \approx 3.21 \times 10^{-7} \] This indicates an extremely low probability of a person dying on a flight.

03

Evaluate Raul's Concerns About Accidents Being "Due"

Raul's reasoning is based on the gambler's fallacy, the belief that a random event is "due" after a series of non-occurrences. However, airplane crashes are independent events, not influenced by past events, so no crash is "due." The statistical probability of events doesn't change based on recent history.

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Key Concepts

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

Long Run vs Short Run

When discussing probability in statistics, it's important to differentiate between long run and short run perspectives. The long run involves a large number of trials, which helps identify stable patterns or statistical trends over time. For example, the data on airline safety in 2013 involved 825 million passengers worldwide. This large dataset is considered a long run. The extensive number of trials allows us to rely on the resulting statistics as they reflect a stable pattern—showing the safety of air travel over a significant period.

On the other hand, a short run might involve only a handful of flights. The smaller data size can result in more variance and less stable trends. In such cases, conclusions might not be as reliable since they could be skewed by a few unusual outcomes. Thus, the long run approach gives a clearer, more dependable picture of probabilities in scenarios with large datasets.

Probability Calculation

Calculating probability involves finding the likelihood of an event occurring, typically expressed as a fraction or decimal. For airplane safety statistics in 2013, we want to determine the probability of dying on a flight. To calculate this, we divide the number of fatalities by the total number of passengers.

In 2013, there were 265 deaths among 825 million passengers. Therefore, the probability is given by:
\[ P(\text{dying in 2013}) = \frac{265}{825,000,000} \approx 3.21 \times 10^{-7} \]

This minuscule number shows that the likelihood of dying on a flight is incredibly low.

• It's crucial to remember that probabilities between 0 and 1 show impossibility and certainty, respectively.
• The lower the probability, the less likely the event is.

Comparatively, the probability of dying from a 1000-mile automobile trip is much higher, indicating air travel's relative safety.

Independent Events

One of the key concepts in probability is understanding independent events. These are events where the outcome of one does not affect the outcome of another. In the context of airplane travel, consider Raul's concern about crashes being "due." This is known as the gambler's fallacy, where one falsely believes that a random event is more likely after a series of non-events.

However, airplane crashes are independent events; their probability doesn't change regardless of recent accident history. Each flight operates independently, and the safety measures are consistent. No matter how many flights operate without incident, it doesn't increase or decrease the chance of a crash occurring.

• Past occurrences do not affect the probability of future events.
• Understanding the concept of independent events helps in evaluating risks logically.

Thus, flying remains one of the safest ways to travel, independent of recent trends in accident data.