A newspaper finds that the mean number of typographical errors per page is fourfour. find the probability that​ (a) exactly fivefive typographical errors are found on a​ page, (b) at most fivefive typographical errors are found on a​ page, and​ (c) more than fivefive typographical errors are found on a page.

For this question, we are given a mean/average rate, that is 4 errors per page;
This suggests we need to use the Poisson distribution, so:
Let X be the random variable, the number of errors per page;
X~Po(4) ← X has a Poisson distribution with mean (λ) of 4
a)
What we want is the probability that X = 5, so we use the formula for Poisson:
[tex]P(X = 5) = \frac{e^{-(4)}. (4)^{5}}{5!}[/tex]
= 0.156 (to 3 s.f.)
b)
What we want to find is the probability that X ≤ 5, so we have to use the cumulative tables:
P(X ≤ 5) = 0.7851
c)
What we want to find is the probability that X > 5, so we have to do a bit of manipulation and then use the cumulative tables;
The tables give values for ≤, so we cannot directly look up a value for >, thus the manipulation:
P(X > 5) = 1 - P(X ≤ 5)
= 1 - (0.7581)
= 0.2419