In 1898 L. J. Bortkiewicz published a book entitled The Law of Small Numbers. He used data collected over 20 years to show that

Question

7 days ago

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In 1898 L. J. Bortkiewicz published a book entitled The Law of Small Numbers. He used data collected over 20 years to show that

the number of soldiers killed by horse kicks each year in each corps in the Prussian cavalry followed a Poisson distribution with a mean of 0.62.
(a) What is the probability of more than one death in a corps in a year?

(b) What is the probability of no deaths in a corps over 7 years?

Round your answers to four decimal places (e.g. 98.7654).

Mathematics

1 answer:

Svet_ta [4.3K] · Answer 1 · 2026-02-26T06:33:10+03:00

(a) The likelihood of having more than one death in a corps within a single year is 0.1252. (b) The likelihood of experiencing no deaths in a corps throughout 7 years is 0.0130. Detailed Explanation: Define X as the number of soldiers fatally injured by horse kicks in one year. The random variable $X\sim Poisson(\lambda = 0.62)$ . The mathematical function following a Poisson distribution is: $P(X=x)=\frac{e^{-\lambda}\lambda^{x}}{x!};\ x=0,1,2,...$ (a) To calculate the probability of exceeding one death in a corps during a year: P (X > 1) = 1 - P (X ≤ 1) = 1 - P (X = 0) - P (X = 1) $=1-\frac{e^{-0.62}(0.62)^{0}}{0!}-\frac{e^{-0.62}(0.62)^{1}}{1!}\\=1-0.54335-0.33144\\=0.12521\\\approx0.1252$ Thus, the probability of witnessing more than one death within a corps yearly is 0.1252. (b) The average number of deaths in a span of 7 years is: $\lambda=7\times0.62=4.34$ To assess the probability of zero deaths in a corps over 7 years: $P(X=0)=\frac{e^{-4.34}(4.34)^{0}}{0!}=0.01304\approx0.0130$ Therefore, the chance of encountering no fatalities within a corps over 7 years is 0.0130.