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We most often hear of the power of earthquakes given in terms of the Richter scale, but this tells only the power of the earthquake at its epicenter. Of more immediate importance is how an earthquake affects the location where we are. Seismologists measure this in terms of ground movement, and for technical reasons they find the acceleration of ground movement most useful. For the purpose of this problem, a major earthquake is one that produces a ground acceleration of at least \(5 \%\) of \(g\), where \(g\) is the acceleration 4 due to gravity near the surface of the Earth. In California, the probability \(p(n)\) of one's home being affected by exactly \(n\) major earthquakes over a 10 -year period is given approximately \({ }^{5}\) by $$ p(n)=0.379 \times \frac{0.9695^{n}}{n !} $$ See Exercise 7 for an explanation of \(n !\). a. What is the probability of a California home being affected by exactly 3 major earthquakes over a 10 -year period? b. What is the limiting value of \(p(n)\) ? Explain in practical terms what this means. c. What is the probability of a California home being affected by no major earthquakes over a 10 year period? d. What is the probability of a California home being affected by at least one major earthquake over a 10-year period? Hint: It is a certainty that an event either will or will not occur, and the probability assigned to a certainty is 1. Expressed in a formula, this is Probability of an event occurring \+ Probability of an event not occurring \(=1\). This, in conjunction with part c, may be helpful for part d.

Step by step solution

01

Calculate Probability for Exactly 3 Earthquakes

We need to calculate \( p(3) \) using the given formula: \[ p(n) = 0.379 \times \frac{0.9695^n}{n!} \]Plugging in \( n = 3 \), the probability becomes:\[ p(3) = 0.379 \times \frac{0.9695^3}{3!} \]Calculating further:\( 0.9695^3 = 0.91 \) approximately, \( 3! = 6 \),So:\[ p(3) = 0.379 \times \frac{0.91}{6} \approx 0.0574 \].

02

Determine the Limiting Value of p(n)

The limiting value of \( p(n) \) as \( n \) approaches infinity can be understood by recognizing that the probability of being affected by an ever-increasing number of events becomes negligible. Mathematically, as \( n \to \infty \), the term \( \frac{0.9695^n}{n!} \) tends towards zero because the factorial \( n! \) grows much faster than \( 0.9695^n \). Thus, \( \lim_{n \to \infty} p(n) = 0 \). In practical terms, this means there is virtually no situation where an extremely high number of earthquakes could occur within a single 10-year period.

03

Calculate Probability of No Major Earthquakes

To find the probability of experiencing no major earthquakes (\( n = 0 \)), use the formula:\[ p(0) = 0.379 \times \frac{0.9695^0}{0!} \]Since \( 0! = 1 \) and \( 0.9695^0 = 1 \), it simplifies to:\[ p(0) = 0.379 \times 1 = 0.379 \].Hence, the probability of a California home being affected by no major earthquakes over a 10-year period is \( 0.379 \).

04

Probability of At Least One Earthquake

To determine the probability of experiencing at least one earthquake, we use the formula:\[ 1 = p(0) + P(\text{at least one earthquake}) \]With part c showing \( p(0) = 0.379 \), the formula becomes:\[ p(\text{at least one}) = 1 - p(0) = 1 - 0.379 = 0.621 \].Thus, the probability of experiencing at least one major earthquake over a 10-year period is \( 0.621 \).

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Key Concepts

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

Probability

Probability is the mathematical measure of how likely an event is to occur. It ranges from 0 to 1, where 0 signifies impossibility and 1 represents certainty. When dealing with earthquakes, we use probability to predict the number of events over a given period.

For example, in the problem given, we're determining the likelihood of being affected by a certain number of earthquakes in a 10-year span. The formula used is:

• \(p(n) = 0.379 \times \frac{0.9695^n}{n!}\)
• Here, \(n!\) represents the factorial of \(n\), which is the product of all positive integers up to \(n\).

These probabilities help us understand how often certain numbers of earthquakes can be expected in an area, which is essential for planning and risk management.

Richter Scale

The Richter Scale is a mathematical scale that measures the magnitude or the energy released by an earthquake. It's a logarithmic scale, meaning each whole number increment represents a tenfold increase in amplitude and roughly 31.6 times more energy.

This scale specifically measures the energy at the earthquake's epicenter, which is the point on the Earth's surface directly above where the quake originated. However, it doesn't directly tell us how the earthquake will affect a particular location other than the epicenter.

Thus, seismologists also measure ground acceleration to better understand impacts at various distances from the epicenter.
This is crucial for assessing safety and structural impacts in earthquake-prone areas.

Earthquakes

An earthquake is a natural phenomenon characterized by the shaking of the Earth due to underground movements along faults. Earthquakes release energy stored in the Earth's crust from geological processes. While consistently small earthquakes happen regularly, large quakes can cause significant damage.

The impact of earthquakes isn't limited to the point of origin or epicenter; instead, it spans out to different locations depending on factors like distance, ground composition, and structural elements.
Understanding these events helps communities prepare and respond effectively by reinforcing buildings and infrastructure, and implementing early warning systems.

• Seismologists analyze both the immediate and secondary effects to offer additional protection against potential disasters.

Ground Acceleration

Ground acceleration refers to the rate of change in velocity of ground motion as a result of an earthquake. It gives a clearer picture of how much a structure or area will physically move during a quake.

Seismologists often express ground acceleration as a percentage of \(g\), the acceleration due to gravity. In the given problem, an acceleration of at least \(5\%\) of \(g\) defines a major earthquake.

This metric is crucial because it directly impacts how we construct buildings and infrastructure in earthquake-prone areas. Calculating probable ground accelerations helps engineers design and reinforce structures to withstand potential shocks, thereby reducing risks to human life and property.

• By incorporating these calculations, seismic standards are established ensuring structural safety and resilience.