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Chapter 5: Problem 16

San Andreas Fault USA Today reported that Parkfield, California, is dubbed the world's earthquake capital because it sits on top of the notorious San Andreas fault. Since 1857 , Parkfield has had a major earthquake on the average of once every 22 years. (a) Explain why a Poisson probability distribution would be a good choice for \(r=\) number of earthquakes in a given time interval. (b) Compute the probability of at least one major earthquake in the next 22 years. Round \(\lambda\) to the nearest hundredth, and use a calculator.

Short Answer

Expert verified
A Poisson distribution is appropriate as earthquakes are independent events with a constant mean rate. The probability of at least one earthquake in the next 22 years is approximately 0.63.

Step by step solution

01

Understanding Poisson Distribution

A Poisson distribution is ideal for modeling the number of events happening within a fixed interval of time when these events occur with a known constant mean rate and independently of the time since the last event. In this scenario, major earthquakes are such events, occurring independently along the San Andreas fault with a known average rate.
02

Determine the Mean Rate (\(\lambda\))

Given that Parkfield has a major earthquake on average every 22 years, the average rate, \(\lambda\), is 1 earthquake per 22 years. Hence, \(\lambda = 1\).
03

Calculate Probability of At Least One Earthquake

We want to find the probability of at least one earthquake occurring, which is the complement of no earthquakes occurring. The probability of no earthquakes is given by the Poisson distribution formula for \(r = 0\): \(P(X=0) = \frac{e^{-\lambda} \cdot \lambda^0}{0!} = e^{-1}\).
04

Using the Complement Rule

To find the probability of at least one earthquake, use the complement rule: \(P(X \geq 1) = 1 - P(X = 0)\). Compute \(P(X \geq 1) = 1 - e^{-1}\).
05

Compute and Round Result

Using a calculator, compute \(e^{-1} \approx 0.3679\). Then, \(P(X \geq 1) = 1 - 0.3679 = 0.6321\). This is the probability of having at least one earthquake in the next 22 years.

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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 a branch of mathematics concerned with the analysis of random phenomena. It provides the foundation for understanding and calculating the likelihood of events happening.

In the context of earthquakes, probability theory helps us to model and predict earthquake occurrences. The Poisson distribution is a key tool in this area. It describes the probability of a given number of events happening in a fixed interval of time or space, assuming these events occur with a constant average rate and independently of each other.
  • Random Phenomena: Events or outcomes that cannot be precisely predicted.
  • Probability Distribution: A mathematical function that provides the probabilities of occurrence of different possible outcomes.
  • Poisson Process: A stochastic process that models the probability of a given number of events occurring in a fixed interval, commonly used in situations where events happen at a constant average rate.
Earthquake Statistics
Understanding earthquake statistics is crucial for predicting and preparing for these natural events. Earthquake statistics involve analyzing the frequency and magnitude of earthquakes over time.

Seismologists often use statistical models to estimate the probability of future earthquakes. Data such as the history of past earthquakes along a fault line is critical for developing these models. In the case of Parkfield, California, it is known to experience a major earthquake roughly once every 22 years.
  • Seismology: The scientific study of earthquakes and the propagation of elastic waves through the Earth.
  • Magnitude: A measure of the energy released during an earthquake.
  • Recurrence Interval: The average time between successive occurrences of an event, such as an earthquake.
San Andreas Fault
The San Andreas Fault is one of the most well-known fault lines in the world. It is a tectonic boundary between the Pacific Plate and the North American Plate.

This fault line is responsible for numerous earthquakes in California, including those in Parkfield. It runs approximately 800 miles through California and has been the site of significant seismic activity.
  • Fault Line: A fracture along which the Earth's crust has moved.
  • Tectonic Plates: Massive slabs of solid rock that make up Earth's crust and tectonic boundaries where most seismic activity occurs.
  • Seismic Activity: The frequency and intensity of earthquakes experienced over a period of time in a certain area.
Mean Rate Calculation
Calculating the mean rate, represented as \( \lambda \), is essential in applying the Poisson distribution. The mean rate is the average number of times an event occurs in a specified interval.

For Parkfield, with an average of one major earthquake every 22 years, the mean rate is \( \lambda = 1 \) event per 22 years.
  • Mean Rate (\( \lambda \)): The average number of occurrences of an event in a given time period.
  • Poisson Formula: \( P(X=k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!} \) where \( P(X=k) \) is the probability of \( k \) events in the interval.
  • Complement Rule: Used to find the probability of at least one event happening; \( P(X \geq 1) = 1 - P(X=0) \).

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Most popular questions from this chapter

: Syringes The quality-control inspector of a production plant will reject a batch of syringes if two or more defective syringes are found in a random sample of eight syringes taken from the batch. Suppose the batch contains \(1 \%\) defective syringes. (a) Make a histogram showing the probabilities of \(r=0,1,2,3,4,5,6,7\), and 8 defective syringes in a random sample of eight syringes. (b) Find \(\mu .\) What is the expected number of defective syringes the inspector will find? (c) What is the probability that the batch will be accepted? (d) Find \(\sigma\).

Caribbean Cruise The college student senate is sponsoring a spring break Caribbean cruise raffle. The proceeds are to be donated to the Samaritan Center for the Homeless. A local travel agency donated the cruise, valued at \(\$ 2000 .\) The students sold 2852 raffle tickets at \(\$ 5\) per ticket. (a) Kevin bought six tickets. What is the probability that Kevin will win the spring break cruise to the Caribbean? What is the probability that Kevin will not win the cruise? (b) Expected earnings can be found by multiplying the value of the cruise by the probability that Kevin will win. What are Kevin's expected earnings? Is this more or less than the amount Kevin paid for the six tickets? How much did Kevin effectively contribute to the Samaritan Center for the Homeless?

Pyramid Lake is located in Nevada on the Paiute Indian Reservation. This lake is famous for large cutthroat trout. The mean number of trout (large and small) caught from a boat is \(0.667\) fish per hour (Reference: Creel Chronicle, Vol. 3, No. 2). Suppose you rent a boat and go fishing for 8 hours. Let \(r\) be a random variable that represents the number of fish you catch in the 8 -hour period. (a) Explain why a Poisson probability distribution is appropriate for \(r\). What is \(\lambda\) for the 8 -hour fishing trip? Round \(\lambda\) to the nearest tenth so that you can use Table 4 of Appendix II for Poisson probabilities. (b) If you have already caught three trout, what is the probability you will catch a total of seven or more trout? Compute \(P(r \geq 7 \mid r \geq 3)\). See Hint below. (c) If you have already caught four trout, what is the probability you will catch a total of less than nine trout? Compute \(P(r<9 \mid r \geq 4)\). See Hint below. (d) List at least three other areas besides fishing to which you think conditional Poisson probabilities can be applied. Hint for solution: Review item 6 , conditional probability, in the summary of basic probability rules at the end of Section \(4.2\). Note that $$ P(A \mid B)=\frac{P(A \text { and } B)}{P(B)} $$

On the leeward side of the island of Oahu, in the small village of Nanakuli, about \(80 \%\) of the residents are of Hawaiian ancestry (Source: The Honolulu Advertiser). Let \(n=1,2,3, \ldots\) represent the number of people you must meet until you encounter the first person of Hawaiian ancestry in the village of Nanakuli. (a) Write out a formula for the probability distribution of the random variable \(n\). (b) Compute the probabilities that \(n=1, n=2\), and \(n=3\). (c) Compute the probability that \(n \geq 4\). (d) In Waikiki, it is estimated that about \(4 \%\) of the residents are of Hawaiian ancestry. Repeat parts (a), (b), and (c) for Waikiki.

Poisson Approximation to Binomial: Comparisons (a) For \(n=100, p=0.02\), and \(r=2\), compute \(P(r)\) using the formula for the binomial distribution and your calculator: $$ P(r)=C_{n, t} p^{r}(1-p)^{n-r} $$ (b) For \(n=100, p=0.02\), and \(r=2\), estimate \(P(r)\) using the Poisson approximation to the binomial. (c) Compare the results of parts (a) and (b). Does it appear that the Poisson distribution with \(\lambda=n p\) provides a good approximation for \(P(r=2) ?\) (d) Repeat parts (a) to \((c)\) for \(r=3\)
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