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) = 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.)

Explanation:

Step1: Recall Poisson Probability Formula

The Poisson probability formula is \( P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \), and for \( P(X \leq k) \), we sum \( P(X = i) \) for \( i = 0 \) to \( k \), where \( \lambda = 8 \) (mean number of errors per page).

Step2: Calculate \( P(X \leq 5) \)

We need to find \( P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) \).

• \( P(X = 0) = \frac{8^0 e^{-8}}{0!} = e^{-8} \approx 0.000335 \)
• \( P(X = 1) = \frac{8^1 e^{-8}}{1!} = 8e^{-8} \approx 0.002684 \)
• \( P(X = 2) = \frac{8^2 e^{-8}}{2!} = \frac{64e^{-8}}{2} \approx 0.010737 \)
• \( P(X = 3) = \frac{8^3 e^{-8}}{3!} = \frac{512e^{-8}}{6} \approx 0.028635 \)
• \( P(X = 4) = \frac{8^4 e^{-8}}{4!} = \frac{4096e^{-8}}{24} \approx 0.057270 \)
• \( P(X = 5) = \frac{8^5 e^{-8}}{5!} = \frac{32768e^{-8}}{120} \approx 0.091632 \) (Wait, but part (a) is 0.0916, maybe using calculator for precision)

Now sum these up:
\( 0.000335 + 0.002684 + 0.010737 + 0.028635 + 0.057270 + 0.091632 \approx 0.191293 \). Wait, maybe using a Poisson table or calculator for more accuracy. Alternatively, using a calculator, the cumulative probability \( P(X \leq 5) \) for \( \lambda = 8 \) is approximately 0.1912 (rounded to four decimals). Wait, maybe my manual calculation has errors. Let's use a better approach: using the Poisson cumulative distribution function.

Using a calculator or software, the cumulative probability \( P(X \leq 5) \) when \( \lambda = 8 \) is calculated as follows:

\( P(X \leq 5) = \sum_{i=0}^{5} \frac{8^i e^{-8}}{i!} \)

Using a calculator (like TI-84 or online Poisson calculator), we find that \( P(X \leq 5) \approx 0.1912 \) (rounded to four decimal places).

Answer:

0.1912