Let's solve each part step by step. ## **Step 1: Find the Poisson Rate (λ)** A 500-page manuscript has 30 typos. \[ \text{Average typos per page} = \frac{30}{500} = .06 \] For \( n \) pages, the Poisson mean (λ) is: \[ \lambda = .06 \times n \] --- ## **Step 2: Poisson Probability Formula** The Poisson probability of \( k \) events (typos) with mean \( \lambda \): \[ P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \] --- ### **a. Probability of exactly 1 typo in the first 20 pages** - \( n = 20 \) - \( \lambda = .06 \times 20 = 1.2 \) - \( k = 1 \) \[ P(X = 1) = \frac{e^{-1.2} \cdot 1.2^1}{1!} \] \[ = e^{-1.2} \cdot 1.2 \] \[ = .3012 \cdot 1.2 = .3614 \] **Final Answer (rounded to four decimal places):** \[ \boxed{.3614} \] --- ### **b. Probability of exactly 1 typo in the first 40 pages** - \( n = 40 \) - \( \lambda = .06 \times 40 = 2.4 \) - \( k = 1 \) \[ P(X = 1) = \frac{e^{-2.4} \cdot 2.4^1}{1!} \] \[ = e^{-2.4} \cdot 2.4 \] \[ = .0907 \cdot 2.4 = .2177 \] **Final Answer (rounded to four decimal places):** \[ \boxed{.2177} \] --- ### **c. Probability of exactly 6 typos in the first 80 pages** - \( n = 80 \) - \( \lambda = .06 \times 80 = 4.8 \) - \( k = 6 \) \[ P(X = 6) = \frac{e^{-4.8} \cdot 4.8^6}{6!} \] \[ = .0082 \cdot 12288.037 \div 720 \] \[ = .0082 \cdot 17.067 = .140 \] **Final Answer (rounded to four decimal places):** \[ \boxed{.140} \] --- ### **d. Probability of exactly 12 typos in the first 140 pages** - \( n = 140 \) - \( \lambda = .06 \times 140 = 8.4 \) - \( k = 12 \) \[ P(X = 12) = \frac{e^{-8.4} \cdot 8.4^{12}}{12!} \] Calculate step by step: - \( e^{-8.4} \approx .00022487 \) - \( 8.4^{12} \approx 2,214,700,000 \) - \( 12! = 479001600 \) \[ P(X = 12) = \frac{.00022487 \times 2,214,700,000}{479001600} \] \[ = \frac{498,065.989}{479001600} \approx .001039 \] **Final Answer (rounded to four decimal places):** \[ \boxed{.001} \] --- ## **Summary Table** | Part | Pages | \( k \) | \( \lambda \) | Probability | |------|-------|---------|---------------|--------------| | a | 20 | 1 | 1.2 | .3614 | | b | 40 | 1 | 2.4 | .2177 | | c | 80 | 6 | 4.8 | .140 | | d | 140 | 12 | 8.4 | .001 | --- **If you need explanations for the calculations or further breakdown, let me know!**