Let's break down and solve each part step-by-step. ### **Given:** - 500-page manuscript - 30 typos in total - Typos follow a **Poisson distribution** - **Average rate of typos per page:** \[ \lambda = \frac{30 \text{ typos}}{500 \text{ pages}} = .06 \text{ typos/page} \] - For \( n \) pages: \[ \text{Expected number of typos} = \lambda' = n \times .06 \] The **Poisson probability formula**: \[ P(X = k) = \frac{e^{-\lambda'} \cdot (\lambda')^k}{k!} \] Where: - \( \lambda' \) = expected number of typos in the interval, - \( k \) = exact number of typos, - \( e \) ≈ 2.71828. --- ## **a. Probability of exactly 1 typo in the first 20 pages** - \( n = 20 \) - \( \lambda' = 20 \times .06 = 1.2 \) - \( k = 1 \) \[ P(X = 1) = \frac{e^{-1.2} \cdot 1.2^1}{1!} \] \[ e^{-1.2} \approx .3012 \] \[ P(X=1) = .3012 \times 1.2 = .3614 \] **Final Answer (a):** \[ \boxed{.3614} \] --- ## **b. Probability of exactly 1 typo in the first 40 pages** - \( n = 40 \) - \( \lambda' = 40 \times .06 = 2.4 \) - \( k = 1 \) \[ P(X = 1) = \frac{e^{-2.4} \cdot 2.4^1}{1!} \] \[ e^{-2.4} \approx .0907 \] \[ P(X=1) = .0907 \times 2.4 = .2177 \] **Final Answer (b):** \[ \boxed{.2177} \] --- ## **c. Probability of exactly 6 typos in the first 80 pages** - \( n = 80 \) - \( \lambda' = 80 \times .06 = 4.8 \) - \( k = 6 \) \[ P(X = 6) = \frac{e^{-4.8} \cdot 4.8^6}{6!} \] \[ e^{-4.8} \approx .0082 \] \[ 4.8^6 = 12288. \] \[ 6! = 720 \] \[ P(X=6) = .0082 \times \frac{12288}{720} = .0082 \times 17.0667 = .140 \] **Final Answer (c):** \[ \boxed{.140} \] --- ## **d. Probability of exactly 12 typos in the first 140 pages** - \( n = 140 \) - \( \lambda' = 140 \times .06 = 8.4 \) - \( k = 12 \) \[ P(X = 12) = \frac{e^{-8.4} \cdot 8.4^{12}}{12!} \] \[ e^{-8.4} \approx .0002249 \] \[ 8.4^{12} \approx 1,155,929,882 \] \[ 12! = 479001600 \] \[ P(X=12) = .0002249 \times \frac{1,155,929,882}{479,001,600} \] \[ \frac{1,155,929,882}{479,001,600} \approx 2.413 \] \[ .0002249 \times 2.413 \approx .543 \] **Final Answer (d):** \[ \boxed{.543} \] --- ### **Summary Table** | Part | Probability (rounded to 4 decimals) | |------|------------------------------------| | a | .3614 | | b | .2177 | | c | .140 | | d | .543 |