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) $P$ (at most five typographical errors are found on a page) $=$ (Round to four decimal places as needed.)

Step 1 ：The problem involves a Poisson distribution, as it deals with the number of events (typographical errors) happening in a fixed interval (a page). The mean number of typographical errors per page is given as 8.

Step 2 ：For part (b), we need to find the probability of at most five typographical errors. This means we need to find the sum of the probabilities of getting 0, 1, 2, 3, 4, and 5 typographical errors.

Step 3 ：We can use the formula for the Poisson distribution to calculate these probabilities: \(P(X=k) = (λ^k * e^-λ) / k!\) where λ is the mean number of events (8 in this case), k is the number of events we are interested in, and e is the base of the natural logarithm (approximately 2.71828).

Step 4 ：After running this code, we can observe the result and round it to four decimal places as needed. The probability is approximately 0.19123606207962526.

Step 5 ：Final Answer: The probability that at most five typographical errors are found on a page is approximately \(\boxed{0.1912}\).