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For 200 days with a 1.02% chance of low seismic activity per day in Alaska, the binomial model gives P(x=10) ≈ 0.000039 for ten such days, while the Poisson model with μ = np = 2.04 gives P(x=10) ≈ 0.000045. Why should the approximation be good here?
ABecause n here is small and p is close to one
BBecause μ exactly equals the sample size n
CBecause the binomial model is undefined for rare events
DBecause n here is large and p is small
Answer & Solution
Correct answer: D. Because n here is large and p is small
1. Here n = 200 days, which is much larger than the rule-of-thumb threshold of 20.
2. p = 0.0102 per day, well under the 0.05 threshold for a small probability.
3. Both conditions of the Poisson-approximates-binomial rule are satisfied, so the two computed probabilities land close together.
4. Option A states the opposite of the actual n and p here, and option D is simply false, since the binomial model is used first in this very example.
_Source: OpenStax Introductory Statistics (CC BY 4.0), Ch 4 "Discrete Random Variables", section 4.6 Poisson Distribution_
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