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Poisson: error pages in 400-page novel

Short answer

Poisson approximation with λ=np=2. P(exactly one page with errors)=0.2707; P(at most three pages)=0.8571.

If a publisher of nontechnical books takes great pains to ensure that its books are free of typographical errors, so that the probability of any given page containing at least one such error is 0.005 and errors are independent from page to page, what is the probability that one of its 400-page novels will contain a) Exactly one page with errors? b) At most three pages with errors?

Let X = number of pages with errors in a 400-page book. n = 400 pages, p = 0.005 per page, independent, so X is binomial. Because n is large and p is small, use the Poisson approximation:

λ = np = 400 × 0.005 = 2, P(X = x) = e⁻² 2ˣ / x!, e⁻² = 0.135335.

a) Exactly one page with errors

P(X = 1) = e⁻² × 2 = 0.2707

b) At most three pages with errors

P(0) = 0.1353
P(1) = 0.2707
P(2) = 0.2707
P(3) = e⁻² · 8/6 = 0.1804

P(X ≤ 3) = 0.1353 + 0.2707 + 0.2707 + 0.1804 = 0.8571

Answer: (a) 0.2707, (b) 0.8571

✓ Answer checked by Claude Sonnet 5.5

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