Find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine if the events are unusual. If convenient, use the appropriate probability table or technology to find the probabilities.

A newspaper finds that the mean number of typographical errors per page is eight. Find the probability that (a) exactly five typographical errors are found on a page, (b) at most five typographical errors are found on a page, and (c) more than five typographical errors are found on a page.
(a) $\mathrm{P}$ (exactly five typographical errors are found on a page $)=0.0916$
(Round to four decimal places as needed.)
(b) $\mathrm{P}$ (at most five typographical errors are found on a page $)=0.1912$
(Round to four decimal places as needed.)
(c) $\mathrm{P}$ (more than five typographical errors are found on a page) $=$
(Round to four decimal places as needed.)

Step 1 ：The problem involves finding probabilities related to the number of typographical errors on a page. The mean number of errors is given as 8. This is a Poisson distribution problem because we are dealing with the number of events (errors) that occur in a fixed interval (a page).

Step 2 ：The Poisson probability formula is given by: \(P(x; μ) = (e^-μ) * (μ^x) / x!\) where: \(P(x; μ)\) is the Poisson probability, x is the actual number of successes, e is approximately equal to 2.71828, μ is the mean number of successes, and x! is the factorial of x.

Step 3 ：For part (a), we need to find the probability of exactly 5 errors. We can plug x=5 and μ=8 into the Poisson formula. The calculated probability is approximately 0.0916.

Step 4 ：For part (b), we need to find the probability of at most 5 errors. This is the sum of the probabilities of 0, 1, 2, 3, 4, and 5 errors. The calculated probability is approximately 0.1912.

Step 5 ：For part (c), we need to find the probability of more than 5 errors. This is 1 minus the probability of at most 5 errors. The calculated probability is approximately 0.8088.

Step 6 ：Final Answer: \(\boxed{0.8088}\) (Round to four decimal places as needed.)