Question

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) p(exactly five typographical errors are found on a page) =
(round to four decimal places as needed.)

Explanation:

Step1: Identify Distribution Type

The problem involves the number of typographical errors per page, which follows a Poisson distribution. The Poisson probability formula is \( P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \), where \( \lambda = 8 \) (mean errors per page) and \( k = 5 \) (exact errors we want).

Step2: Substitute Values into Formula

Substitute \( \lambda = 8 \), \( k = 5 \), \( e \approx 2.71828 \) into the formula:

$$ P(X = 5) = \frac{8^5 \times e^{-8}}{5!} $$

Calculate \( 8^5 = 32768 \), \( 5! = 120 \), \( e^{-8} \approx 0.00033546 \).

Step3: Compute the Probability

$$ P(X = 5) = \frac{32768 \times 0.00033546}{120} $$

First, multiply \( 32768 \times 0.00033546 \approx 10.995 \). Then divide by \( 120 \): \( \frac{10.995}{120} \approx 0.0916 \).

Answer:

\( 0.0916 \) (rounded to four decimal places)