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Chapter 5: Problem 53

A university police department receives an average of \(3.7\) reports per week of lost student ID cards. a. Find the probability that at most 1 such report will be received during a given week by this police department. Use the Poisson probability distribution formula. b. Using the Poisson probabilities table, find the probability that during a given week the number of such reports received by this police department is i. 1 to 4 ii. at least 6 iii. at most 3

Short Answer

Expert verified
a. The probability that the police department will receive at most 1 report in a week can be calculated by adding the Poisson probabilities for 0 and 1 report.\n b. i. The probability of receiving 1 to 4 reports in a week can be found by adding the Poisson probabilities for 1, 2, 3, and 4 reports. ii. The probability of receiving at least 6 reports is found by subtracting the sum of the probabilities of 0 to 5 reports from 1. iii. The probability of receiving at most 3 reports is found by adding the probabilities for 0, 1, 2, and 3 reports.

Step by step solution

01

Calculating the probability of at most 1 report using Poisson formula

In this case, 'at most 1' means 0 or 1 report. To calculate this, add the two probabilities together. \n For 0 reports: Use the Poisson formula by plugging \(位=3.7\) (average reports per week) and \(x=0\) (number of successes) into it. \(P(0; 3.7) = \frac{3.7^0\times e^{-3.7}}{0!}\)\n For 1 report: Apply the same process with \(x=1\). \(P(1; 3.7) = \frac{3.7^1\times e^{-3.7}}{1!}\)\n Add the two probabilities to find the probability of at most 1 report being received during a given week.
02

Using Poisson probabilities table to calculate the probabilities

a. For 1 to 4 reports: To calculate this, add the Poisson probabilities for 1,2,3, and 4 reports. This can be looked up directly in a Poisson probabilities table.\n b. For at least 6 reports: The probability of at least 6 reports is the same as 1 minus the probability of at most 5 reports. So, calculate or look up the probabilities for 0 to 5 reports, add them together, and subtract the result from 1.\n c. For at most 3 reports: This simply requires adding the probabilities for 0, 1, 2, and 3 reports, which again can be looked up in the table.

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Most popular questions from this chapter

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