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Chapter 5: Problem 19

Meteorology: Winter Conditions Much of Trail Ridge Road in Rocky Mountain National Park is over 12,000 feet high. Although it is a beautiful drive in summer months, in winter the road is closed because of severe weather conditions. Winter Wind Studies in Rocky Mountain National Park by Glidden (published by Rocky Mountain Nature Association) states that sustained gale- force winds (over 32 miles per hour and often over 90 miles per hour) occur on the average of once every 60 hours at a Trail Ridge Road weather station. (a) Let \(r=\) frequency with which gale-force winds occur in a given time interval. Explain why the Poisson probability distribution would be a good choice for the random variable \(r\). (b) For an interval of 108 hours, what are the probabilities that \(r=2,3\), and \(4 ?\) What is the probability that \(r<2 ?\) (c) For an interval of 180 hours, what are the probabilities that \(r=3,4\), and \(5 ?\) What is the probability that \(r<3 ?\)

Short Answer

Expert verified
(a) The Poisson distribution is appropriate due to independent events and constant rate. (b) \( P(r=2) \approx 0.267, P(r=3) \approx 0.160, P(r=4) \approx 0.072, P(r<2) \approx 0.571. (c) \( P(r=3) \approx 0.224, P(r=4) \approx 0.168, P(r=5) \approx 0.101, P(r<3) \approx 0.423.

Step by step solution

01

Understanding Poisson Distribution

The Poisson probability distribution is useful for modeling the number of events that happen over a fixed interval of time or space when the events occur independently at a constant average rate. In the context of the problem, gale-force winds occur independently with an average frequency of once every 60 hours.
02

Calculate Average Frequency (位)

For a Poisson distribution, the parameter \( \lambda \) is the average rate of occurrence. For a 108-hour interval, \( \lambda = \frac{108}{60} = 1.8 \) events. For a 180-hour interval, \( \lambda = \frac{180}{60} = 3 \) events.
03

Compute Poisson Probabilities for 108-Hour Interval

Using the Poisson probability formula: \( P(r) = \frac{e^{-\lambda} \lambda^r}{r!} \). For \( \lambda = 1.8 \):Compute \( P(r=2) = \frac{e^{-1.8} \times 1.8^2}{2!} \), \( P(r=3) = \frac{e^{-1.8} \times 1.8^3}{3!} \), and \( P(r=4) = \frac{e^{-1.8} \times 1.8^4}{4!} \).
04

Probability That r < 2 in 108 Hours

Calculate \( P(r < 2) = P(r=0) + P(r=1) \) for \( \lambda = 1.8 \) using the Poisson formula. Specifically, calculate \( P(r=0) = \frac{e^{-1.8} \times 1.8^0}{0!} \) and \( P(r=1) = \frac{e^{-1.8} \times 1.8^1}{1!} \). Sum these probabilities.
05

Compute Poisson Probabilities for 180-Hour Interval

For \( \lambda = 3 \): Compute \( P(r=3) = \frac{e^{-3} \times 3^3}{3!} \), \( P(r=4) = \frac{e^{-3} \times 3^4}{4!} \), and \( P(r=5) = \frac{e^{-3} \times 3^5}{5!} \) using the Poisson formula.
06

Probability That r < 3 in 180 Hours

Calculate \( P(r < 3) = P(r=0) + P(r=1) + P(r=2) \) for \( \lambda = 3 \). Use the Poisson formula to find each: \( P(r=0) = \frac{e^{-3} \times 3^0}{0!} \), \( P(r=1) = \frac{e^{-3} \times 3^1}{1!} \), and \( P(r=2) = \frac{e^{-3} \times 3^2}{2!} \). Sum these probabilities.

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Key Concepts

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

Probability Theory
Probability theory is the mathematical framework that studies random events and helps quantify the likelihood of various outcomes. It's an essential part of understanding natural phenomena and everyday events, such as predicting weather patterns or rolling dice. The core idea is to measure how likely an event is to occur. This is expressed using probabilities, which are numbers between 0 and 1. A probability of 0 means the event will not happen, while a probability of 1 means it will certainly happen.
- **Applications**: Probability theory is used in various fields including finance, insurance, and meteorology. - **Tools**: Common tools include probability distributions, which model the expected frequencies of different outcomes.
Meteorologists, for instance, use probability theory to predict weather events. This involves calculations that factor in numerous data points to assess the likelihood of events such as rain, wind, or storms. The Poisson probability distribution is an example often applied when analyzing frequency of events over time or space, like the occurrences of high winds in the exercise.
Meteorology
Meteorology is the science of the atmosphere and is primarily concerned with weather processes and forecasting. It involves studying the various forces and elements like wind, heat, and pressure that affect weather and climate. Meteorologists collect and analyze atmospheric data to make predictions about the weather.
- **Wind Studies**: In the scenario given in the exercise, meteorologists have studied wind patterns to understand the frequency of gale-force winds. - **Data Collection**: Meteorologists rely on data collected from weather stations situated in different terrains to model the atmosphere's dynamics.
Weather stations like the one on Trail Ridge Road gather continuous data, providing crucial insights into atmospheric conditions. Understanding these patterns allows meteorologists to calculate probabilities of severe weather conditions, which is vital to public safety and planning.
Random Variables
A random variable is a concept in probability theory used to quantify random events. It assigns numerical values to the outcomes of random phenomena. In the context of the exercise, the random variable represents the frequency with which gale-force winds occur.
- **Discrete Random Variables**: These are variables that have countable outcomes. The number of gale-force wind events in a given period is an example. - **Poisson Distribution**: This distribution models the number of times an event occurs in a fixed period or space, given a constant average rate and independent occurrences.
The Poisson distribution is ideal for our example because it effectively models situations where events occur randomly yet with a specific average frequency. Using it, we can calculate the probability of different numbers of events (e.g., wind occurrences) happening over a set interval, like the 108-hour and 180-hour periods discussed. Understanding random variables and their distributions helps in making informed predictions about the natural world.

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Most popular questions from this chapter

Engineering: Cracks Henry Petroski is a professor of civil engineering at Duke University. In his book To Engineer Is Human: The Role of Failure in Successful Design, Professor Petroski says that up to \(95 \%\) of all structural failures, including those of bridges, airplanes, and other commonplace products of technology, are believed to be the result of crack growth. In most cases, the cracks grow slowly. It is only when the cracks reach intolerable proportions and still go undetected that catastrophe can occur. In a cement retaining wall, occasional hairline cracks are normal and nothing to worry about. If these cracks are spread out and not too close together, the wall is considered safe. However, if a number of cracks group together in a small region, there may be real trouble. Suppose a given cement retaining wall is considered safe if hairline cracks are evenly spread out and occur on the average of \(4.2\) cracks per 30 -foot section of wall. (a) Explain why a Poisson probability distribution would be a good choice for the random variable \(r=\) number of hairline cracks for a given length of retaining wall. (b) In a 50 -foot section of safe wall, what is the probability of three (evenly spread-out) hairline cracks? What is the probability of three or more (evenly spread-out) hairline cracks? (c) Answer part (b) for a 20 -foot section of wall. (d) Answer part (b) for a 2 -foot section of wall. Round \(\lambda\) to the nearest tenth. (e) Consider your answers to parts (b), (c), and (d). If you had three hairline cracks evenly spread out over a 50 -foot section of wall, should this be cause for concern? The probability is low. Could this mean that you are lucky to have so few cracks? On a 20 -foot section of wall [part (c)], the probability of three cracks is higher. Does this mean that this distribution of cracks is closer to what we should expect? For part (d), the probability is very small. Could this mean you are not so lucky and have something to worry about? Explain your answers.

Critical Tbinking Consider a binomial distribution of 200 trials with expected value 80 and standard deviation of about \(6.9\). Use the criterion that it is unusual to have data values more than \(2.5\) standard deviations above the mean or \(2.5\) standard deviations below the mean to answer the following questions. (a) Would it be unusual to have more than 120 successes out of 200 trials? Explain. (b) Would it be unusual to have fewer than 40 successes out of 200 trials? Explain. (c) Would it be unusual to have from 70 to 90 successes out of 200 trials? Explain.

Statistical Literacy Consider two binomial distributions, with \(n\) trials each. The first distribution has a higher probability of success on each trial than the second. How does the expected value of the first distribution compare to that of the second?

Sociology: Ethics The one-time fling! Have you ever purchased an article of clothing (dress, sports jacket, etc.), worn the item once to a party, and then returned the purchase? This is called a one-time fling. About \(10 \%\) of all adults deliberately do a one-time fling and feel no guilt about it (Source: Are You Normal?, by Bernice Kanner, St. Martin's Press). In a group of seven adult friends, what is the probability that (a) no one has done a one-time fling? (b) at least one person has done a one-time fling? (c) no more than two people have done a one-time fling?

Ecology: Wolves The following is based on information taken from The Wolf in the Southwest: The Making of an Endangered Species, edited by David Brown (University of Arizona Press). Before 1918 , approximately \(55 \%\) of the wolves in the New Mexico and Arizona region were male, and \(45 \%\) were female. However, cattle ranchers in this area have made a determined effort to exterminate wolves. From 1918 to the present, approximately \(70 \%\) of wolves in the region are male, and \(30 \%\) are female. Biologists suspect that male wolves are more likely than females to return to an area where the population has been greatly reduced. (a) Before 1918 , in a random sample of 12 wolves spotted in the region, what was the probability that 6 or more were male? What was the probability that 6 or more were female? What was the probability that fewer than 4 were female? (b) Answer part (a) for the period from 1918 to the present.
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