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Chapter 5: Problem 80

The article "A Crash Course in Probability" from The Economist included the following information: The chance of being involved in an airplane crash when flying on an Airbus 330 from London to New York City on Virgin Atlantic Airlines is 1 in \(5,371,369 .\) This was interpreted as meaning that you "would expect to go down if you took this flight every day for 14,716 years." The article also states that a person could "expect to fly on the route for 14,716 years before plummeting into the Atlantic." Comment on why these statements are misleading.

Short Answer

Expert verified
The statements in the article are misleading because they incorrectly assume that probabilities can be linearly summed over time and imply that a crash becomes inevitable after a certain number of flights. However, the risk of each flight is independent and the overall probability of experiencing a crash at some point during those 14,716 years is not a simple linear calculation. Furthermore, it is not accurate to suggest that crashes become a certainty after a specific number of flights.

Step by step solution

01

Understanding the Probability

The given probability of being involved in an airplane crash when flying on an Airbus 330 from London to New York City on Virgin Atlantic Airlines is 1 in 5,371,369. This means that if there were 5,371,369 flights, we should expect 1 of these flights to be involved in a crash. It is important to understand that this probability is for a single flight and not for multiple flights taken over time.
02

Misleading Statements

The article implies that by taking this flight every day for 14,716 years, a person would "expect to go down" or have a crash. To derive this number, the article assumes that it is safe to fly every day for 14,716 years. The basis of this interpretation is the misconception that individual flight probabilities can be summed linearly over time, which is incorrect. In reality, the risk of each flight is independent, meaning that the overall probability of experiencing a crash at some point during those 14,716 years is not a simple linear calculation. Furthermore, the article suggests that a person could "expect to fly on the route for 14,716 years before plummeting into the Atlantic." This statement is also misleading because it implies that flying for that period guarantees a crash, which is not the case. Once again, each flight is an independent event. While the probability of experiencing a crash over thousands of flights taken would indeed be higher than the probability of a crash on a single flight, it is not accurate to suggest that crashes become a certainty after a certain number of flights. In conclusion, both statements are misleading because they make incorrect assumptions about the nature of probabilities, suggesting the inevitability of experiencing a crash after a certain number of flights. In reality, each flight's risk is independent, and the overall probability of experiencing a crash at some point in time is more complex than simple linear assumptions.

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Key Concepts

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

Risk Assessment
When considering risk, particularly in activities as serious as flying, it's crucial to accurately interpret probabilities. The probability of being involved in a crash on any specific flight is a measure of risk. However, it's a common mistake to misinterpret this number as a guarantee over time.

In the original exercise, the probability was given as 1 in 5,371,369. This doesn't mean that if you fly 5,371,369 times, you're destined to experience an accident. Instead, it highlights the very low likelihood of a crash on each individual flight.

Effective risk assessment involves understanding what these probabilities convey about safety. It means considering each flight's risk independently rather than assuming aggregated probabilities over multiple journeys determine an eventual outcome. This ensures we clearly comprehend how safe or risky an activity truly is without falling into the trap of misleading interpretations.
Independent Events
Each flight operates as an independent event in statistical terms. This concept is essential to grasp why the statements from the article are misleading. In probability theory, independent events mean that the outcome of one flight does not affect the outcomes of following flights.

Given probability as 1 in 5,371,369 on this specific route, every flight has this same risk level, unaffected by previous travel experiences. Even if you traveled this route every day, every single journey carries its distinct risk, which speaks to independence.

This independence is why the notion of flying for 14,716 years leading to a crash is a misconception. Probabilities don't accumulate in a straightforward manner across independent events. Instead, each flight, or independent event, begins with a clean slate in terms of risk.
Probability Calculation
Calculating probability effectively means doing more than simply adding. Each calculation must respect the nature of the events involved, particularly their independence. The chance provided in the exercise, 1 in 5,371,369, must be understood as a probability for a single occurrence, not a cumulative risk over repeated events.

In probability calculations, particularly for independent events, multiplicative rules often apply rather than additive ones. For instance, if two flights are taken, the probability of both having accidents would involve multiplying individual probabilities, not adding them.

Understanding probability calculations ensures an accurate interpretation of risk. It prevents erroneous conclusions that suggest certainty of adverse outcomes after numerous repetitions of a low-probability event. Thus, the probability doesn't guarantee a crash even after numerous flights; it simply remains constant for each independent flight.

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Most popular questions from this chapter

A small college has 2700 students enrolled. Consider the chance experiment of selecting a student at random. For each of the following pairs of events, indicate whether or not you think they are mutually exclusive and explain your reasoning. a. the event that the selected student is a senior and the event that the selected student is majoring in computer science. b. the event that the selected student is female and the event that the selected student is majoring in computer science. c. the event that the selected student's college residence is more than 10 miles from campus and the event that the selected student lives in a college dormitory. d. the event that the selected student is female and the event that the selected student is on the college football team.

Suppose you want to estimate the probability that a randomly selected customer at a particular grocery store will pay by credit card. Over the past 3 months, 80,500 purchases were made, and 37,100 of them were paid for by credit card. What is the estimated probability that a randomly selected customer will pay by credit card?

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The article "Scrambled Statistics: What Are the Chances of Finding Multi-Yolk Eggs?" (Significance [August 2016]: 11) gives the probability of a double-yolk egg as 0.001 . a. Give a relative frequency interpretation of this probability. b. If 5000 eggs were randomly selected, about how many double-yolk eggs would you expect to find?
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