Suppose the number of earthquakes occurring in an area approximately follows a Poisson distribution with an average rate of 2 earthquakes every year. Show your work when solving the following problems. (a) Find the probability that there will be 1 to 3 (inclusive) earthquakes during the next year in this area. (b) Find the probability that there will be exactly 5 earthquakes during the next 3-year- period. (c) Consider 10 randomly selected years during last century. What is the probability that there will be at least 3 of those years with no earthquake?
Suppose the number of earthquakes occurring in an area approximately follows a Poisson distribution with an average rate of 2 earthquakes every year. Show your work when solving the following problems. (a) Find the probability that there will be 1 to 3 (inclusive) earthquakes during the next year in this area. (b) Find the probability that there will be exactly 5 earthquakes during the next 3-year- period. (c) Consider 10 randomly selected years during last century. What is the probability that there will be at least 3 of those years with no earthquake?
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![**Title: Poisson Distribution and Earthquake Probability Analysis**
---
**Introduction:**
This exercise explores the use of the Poisson distribution to model the occurrence of earthquakes in a specific area. Given an average rate of 2 earthquakes per year, we will calculate various probabilities.
---
**Problem (a):**
*Find the probability that there will be 1 to 3 (inclusive) earthquakes during the next year in this area.*
**Solution Approach:**
1. Identify the lambda (λ) for one year, which is 2.
2. Use the Poisson probability formula:
\[
P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}
\]
3. Compute the probabilities for k = 1, 2, and 3.
4. Sum the probabilities: \( P(1) + P(2) + P(3) \).
---
**Problem (b):**
*Find the probability that there will be exactly 5 earthquakes during the next 3-year-period.*
**Solution Approach:**
1. For a 3-year period, the average rate (λ) becomes 2 * 3 = 6.
2. Use the Poisson probability formula for k = 5:
\[
P(X = 5) = \frac{e^{-6} \times 6^5}{5!}
\]
---
**Problem (c):**
*Consider 10 randomly selected years during last century. What is the probability that there will be at least 3 of those years with no earthquake?*
**Solution Approach:**
1. Calculate the probability of 0 earthquakes in a year: \( P(0) \) using λ = 2:
\[
P(X = 0) = \frac{e^{-2} \times 2^0}{0!}
\]
2. Use the binomial distribution to find the probability of at least 3 years with 0 earthquakes out of 10 years.
3. Calculate for k = 3 to 10 using the formula:
\[
P(Y = k) = \binom{10}{k} (P(0))^k (1 - P(0))^{10-k}
\]
4. Sum probabilities for k = 3 to 10.
---
**Conclusion:**
This problem set illustrates how the Poisson distribution can](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1941a39f-dbc9-477a-8528-f1bffe3dd4bc%2F7d9a6566-be04-4573-8201-51350fbe0e75%2F7pk4t1_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Poisson Distribution and Earthquake Probability Analysis**
---
**Introduction:**
This exercise explores the use of the Poisson distribution to model the occurrence of earthquakes in a specific area. Given an average rate of 2 earthquakes per year, we will calculate various probabilities.
---
**Problem (a):**
*Find the probability that there will be 1 to 3 (inclusive) earthquakes during the next year in this area.*
**Solution Approach:**
1. Identify the lambda (λ) for one year, which is 2.
2. Use the Poisson probability formula:
\[
P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}
\]
3. Compute the probabilities for k = 1, 2, and 3.
4. Sum the probabilities: \( P(1) + P(2) + P(3) \).
---
**Problem (b):**
*Find the probability that there will be exactly 5 earthquakes during the next 3-year-period.*
**Solution Approach:**
1. For a 3-year period, the average rate (λ) becomes 2 * 3 = 6.
2. Use the Poisson probability formula for k = 5:
\[
P(X = 5) = \frac{e^{-6} \times 6^5}{5!}
\]
---
**Problem (c):**
*Consider 10 randomly selected years during last century. What is the probability that there will be at least 3 of those years with no earthquake?*
**Solution Approach:**
1. Calculate the probability of 0 earthquakes in a year: \( P(0) \) using λ = 2:
\[
P(X = 0) = \frac{e^{-2} \times 2^0}{0!}
\]
2. Use the binomial distribution to find the probability of at least 3 years with 0 earthquakes out of 10 years.
3. Calculate for k = 3 to 10 using the formula:
\[
P(Y = k) = \binom{10}{k} (P(0))^k (1 - P(0))^{10-k}
\]
4. Sum probabilities for k = 3 to 10.
---
**Conclusion:**
This problem set illustrates how the Poisson distribution can
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