A regional director responsible for business development in the state of Pennsylvania is concerned about the number of small business failures. If the mean number of small business failures per month is 8.8, what is the probability that exactly 4 small businesses will fail during a given month (to 4 decimals)? Assume that the probability of a failure is the same for any two months and that the occurrence or nonoccurrence of a failure in any month is independent of failures in any other month.
Since the probability of failure is an independent event, this follows a Poisson distribution, where the mean is 8.8 and the given x = 4: [tex]P(k) = \frac{e^{-\lambda}(\lambda)^{k}}{k!} \\ P(4) = \frac{e^{-8.8}(8.8)^{4}}{4!} = 0.0377[/tex]
