Question

1. a service center receives an average of 0.9 customer complaints per hour. managements goal is to receive fewer than three complaints each hour. assume the number of complaints follows the poisson distribution. determine the probability that exactly four complaints will be received during the next eight hours. a) 0.844 b) 0.998 c) 0.084 d) 0.011 e) 0.156

Explanation:

Step1: Identify the Poisson - distribution formula

The Poisson - distribution formula is $P(X = k)=\frac{\lambda^{k}e^{-\lambda}}{k!}$, where $\lambda$ is the average number of events in the given time interval, $k$ is the actual number of events that occur, and $e\approx2.71828$. The service center receives an average of $\lambda = 0.9$ customer complaints per hour. For an eight - hour period, $\lambda_{total}=0.9\times8 = 7.2$.

Step2: Calculate the probability for $k = 4$

We want to find $P(X = 4)$ with $\lambda=7.2$. Substitute $\lambda = 7.2$ and $k = 4$ into the Poisson - distribution formula:
\[

$$\begin{align*} P(X = 4)&=\frac{(7.2)^{4}\times e^{- 7.2}}{4!}\\ &=\frac{2687.3856\times0.000746586}{24}\\ &=\frac{2.0077}{24}\\ &\approx0.084 \end{align*}$$

\]

Answer:

C) 0.084