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(a) The probability that fewer than 7 field mice are found on a given acre is (Round to four decimal places as needed.) (b) The probability that fewer than 7 field mice are found on 3 of the next 4 acres inspected is (Round to four decimal places as needed.)

(a) The probability that fewer than 7 field mice are found on a given acre is (Round to four decimal places as needed.) (b) The probability that fewer than 7 field mice are found on 3 of the next 4 acres inspected is (Round to four decimal places as needed.)

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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### Estimating the Probability of Finding Field Mice in a Wheat Field The average number of field mice per acre in a 6-acre wheat field is estimated to be 12. - (a) Find the probability that fewer than 7 field mice are found on a given acre. - (b) Find the probability that fewer than 7 field mice are found on 3 of the next 4 acres inspected. #### Instructions: Click here to view the table of Poisson probability sums. --- #### Questions and Answers: (a) The probability that fewer than 7 field mice are found on a given acre is [____]. *(Round to four decimal places as needed.)* (b) The probability that fewer than 7 field mice are found on 3 of the next 4 acres inspected is [____]. *(Round to four decimal places as needed.)* --- #### Poisson Distribution Explanation: The Poisson distribution is used to model the number of events occurring within a given time interval or spatial area when the events are independent and the rate at which events occur is constant. To solve these types of problems, we often use Poisson probability tables or formulas. The formula for the Poisson probability is given by: \[ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \] Where: - \( \lambda \) is the average rate (mean) - \( k \) is the number of occurrences - \( e \) is the base of the natural logarithm (approximately equal to 2.71828) - \( k! \) denotes the factorial of \( k \) In this context, you will refer to a Poisson probability table which provides cumulative probabilities for various values of \( \lambda \) and \( k \). This can greatly simplify finding the probability of fewer than a certain number of events occurring. For part (b), solving usually involves the binomial probability formula or related tables after determining the individual Poisson probabilities. --- Please refer to the provided Poisson probability sums table for exact values needed to complete the calculations. Note: Ensure your final answers are rounded accurately to four decimal places as required.
Transcribed Image Text:### Estimating the Probability of Finding Field Mice in a Wheat Field The average number of field mice per acre in a 6-acre wheat field is estimated to be 12. - (a) Find the probability that fewer than 7 field mice are found on a given acre. - (b) Find the probability that fewer than 7 field mice are found on 3 of the next 4 acres inspected. #### Instructions: Click here to view the table of Poisson probability sums. --- #### Questions and Answers: (a) The probability that fewer than 7 field mice are found on a given acre is [____]. *(Round to four decimal places as needed.)* (b) The probability that fewer than 7 field mice are found on 3 of the next 4 acres inspected is [____]. *(Round to four decimal places as needed.)* --- #### Poisson Distribution Explanation: The Poisson distribution is used to model the number of events occurring within a given time interval or spatial area when the events are independent and the rate at which events occur is constant. To solve these types of problems, we often use Poisson probability tables or formulas. The formula for the Poisson probability is given by: \[ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \] Where: - \( \lambda \) is the average rate (mean) - \( k \) is the number of occurrences - \( e \) is the base of the natural logarithm (approximately equal to 2.71828) - \( k! \) denotes the factorial of \( k \) In this context, you will refer to a Poisson probability table which provides cumulative probabilities for various values of \( \lambda \) and \( k \). This can greatly simplify finding the probability of fewer than a certain number of events occurring. For part (b), solving usually involves the binomial probability formula or related tables after determining the individual Poisson probabilities. --- Please refer to the provided Poisson probability sums table for exact values needed to complete the calculations. Note: Ensure your final answers are rounded accurately to four decimal places as required.
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