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 0.61.
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 five years?

The answer is:

a) 0.12521 or 12.521%

b) 0.04736 or 4.736%

Explanation:

a) Because 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 0.61.

The probability of one death in a corps in a year:

P(k = 1 in a year) = (0.61^1 x e^(-0.61))/ 1! = 0.33144

The probability of no death in a corps in a year:

P(k = 0 in a year) = (0.61^0 x e^(-0.61))/ 0! = 0.54335

The probability of more than one death in a corps in a year:

P(k > 1 in a year) = 1 - P(k = 1 in a year) - P(k = 0 in a year)

= 1 - 0.33144 - 0.54335 = 0.12521 or 12.521%

b) The probability of no deaths in a corps over five years:

P(k=0 in 5 years) = e^(-λ) with λ as the average number of soldiers killed by horse kicks in a 5 year interval.

λ = 0.61 x 5 = 3.05 => P(k=0 in 5 years) = e^(-3.05) = 0.04736 or 4.736%

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