A sales firm receives an average of four calls per hour on its toll-free number. For any given hour, find the probability that it will receive exactly nine calls. Use the Poisson distribution.
A)0.0132
B)0.0001
C)146.3700
D)0.0003

Question

Quizplus · Accepted Answer

This problem involves using the Poisson distribution, which gives the probability of a certain number of events occurring in a fixed interval of time, given the average rate of events occurring. In this case, the average rate is four calls per hour, and we want to find the probability of nine calls in one hour. Using the Poisson formula, P(X=9) = (e^-4)*(4^9)/9! = 0.0132. Therefore, the correct answer is choice A. This problem involves finding the probability of a rare event (receiving exactly 9 calls in an hour) using the Poisson distribution. The Poisson distribution models rare events that occur randomly over time or space, such as accidents, defects, or phone calls. The Poisson distribution has one parameter λ (lambda), which represents the average rate or intensity of the events. This parameter is also equal to the mean and variance of the distribution. Therefore, if the average rate is 4 calls per hour, we can set λ=4 and use the Poisson formula:

P(X=9) = (e^(-λ) * λ^9) / 9! = (e^(-4) * 4^9) / 362880 ≈ 0.0132

where X is the random variable representing the number of calls in an hour, and e is the base of natural logarithms. The probability of getting exactly 9 calls in an hour is very low (less than 2%), but not impossible. Therefore, the best choice is A) 0.0132. The other choices are either too high (C) or too low (B, D) to be plausible probabilities.

A Sales Firm Receives an Average of Four Calls Per