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Chapter 5: Problem 86

A commuter airline receives an average of \(9.7\) complaints per day from its passengers. Using the Poisson formula, find the probability that on a certain day this airline will receive exactly 6 complaints.

Short Answer

Expert verified
The exact probability depends on the result of the calculation in step 6, but remember, probabilities range from 0 to 1. To convert to a percentage, multiply by 100.

Step by step solution

01

Understand the Poisson Distribution

The Poisson Distribution gives the probability of a certain number of events in a fixed interval of time or space. The Poisson formula for the probability of exactly x occurrences is \[ P(x; \lambda) = \frac{\lambda^x e^{-\lambda}}{x!} \] where \( \lambda \) is the average number of occurrences and \( e \) is the base of the natural logarithm.
02

Identify the Given Variables

In this case, \( \lambda = 9.7 \), which is the average number of complaints per day, and \( x = 6 \), the number of complaints we want to find the probability for.
03

Apply the Poisson Formula

By inputting the known values into the Poisson formula: \[ P(6; 9.7) = \frac{(9.7)^6 e^{-9.7}}{6!} \]
04

Calculate the Numerator

Calculate the numerator of the fraction, which is \( (9.7)^6 * e^{-9.7} \).
05

Calculate the Denominator

Calculate the denominator of the fraction, which is \( 6! \). It can be calculated by multiplying 6 by every positive integer less than it till 1.
06

Solve the Equation

Now by dividing the result of step 4 by the result of step 5, we get the required probability. It is critical to accurately carry out the arithmetic here to reach a correct solution.
07

Express the Answer

Express the answer in a probability format. That is, the answer should be a number between 0 and 1, which can also be expressed as a percentage.

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