a) Find the probability of exactly 32 particles arrive in a particular one minute period. b) Find the probability of exactly one particle arrives in a particular one second period. c) Find the probability that at least one particle arrives in a particular one second period.
a) Find the probability of exactly 32 particles arrive in a particular one minute period. b) Find the probability of exactly one particle arrives in a particular one second period. c) Find the probability that at least one particle arrives in a particular one second period.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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![**Title: Calculating Probabilities using Poisson Distribution**
**Introduction:**
A Geiger counter is used to count the number of alpha particles from radioactive material. On average, 27 particles are detected per minute. This scenario assumes that the arrival of particles at the counter follows a Poisson distribution.
**Problem Statements:**
a) **Calculate the Probability of 32 Particles in One Minute:**
Find the probability that exactly 32 particles arrive in a specified one-minute period.
[Input Field for Answer]
b) **Calculate the Probability of 1 Particle in One Second:**
Determine the probability that exactly one particle arrives in a specific one-second period.
[Input Field for Answer]
c) **Calculate the Probability of At Least 1 Particle in One Second:**
Calculate the probability that at least one particle arrives during a particular one-second period.
[Input Field for Answer]
d) **Calculate the Probability of At Least 2 Particles in Four Seconds:**
Find the probability that at least two particles arrive within a specific four-second period.
[Input Field for Answer]
**Instructions:**
- Use the properties of the Poisson distribution to solve each problem.
- Remember to adjust the average rate (λ) based on the time period (one minute vs. one second).
**Note:** Each part of the problem requires careful interpretation of the Poisson distribution formula to ensure accuracy in the probability calculations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F80864b91-d3a3-4910-ac0e-8d4cb607fd82%2F68f2c0be-f77d-4b57-8cdf-b105cbe6fa65%2F8od38g_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Calculating Probabilities using Poisson Distribution**
**Introduction:**
A Geiger counter is used to count the number of alpha particles from radioactive material. On average, 27 particles are detected per minute. This scenario assumes that the arrival of particles at the counter follows a Poisson distribution.
**Problem Statements:**
a) **Calculate the Probability of 32 Particles in One Minute:**
Find the probability that exactly 32 particles arrive in a specified one-minute period.
[Input Field for Answer]
b) **Calculate the Probability of 1 Particle in One Second:**
Determine the probability that exactly one particle arrives in a specific one-second period.
[Input Field for Answer]
c) **Calculate the Probability of At Least 1 Particle in One Second:**
Calculate the probability that at least one particle arrives during a particular one-second period.
[Input Field for Answer]
d) **Calculate the Probability of At Least 2 Particles in Four Seconds:**
Find the probability that at least two particles arrive within a specific four-second period.
[Input Field for Answer]
**Instructions:**
- Use the properties of the Poisson distribution to solve each problem.
- Remember to adjust the average rate (λ) based on the time period (one minute vs. one second).
**Note:** Each part of the problem requires careful interpretation of the Poisson distribution formula to ensure accuracy in the probability calculations.
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