Question

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.
Required:
a. What is the probability of more than 1 death in a corps in a year?
b. What is the probability of no deaths in a corps over five years?

Answer 1

Answer: a. 0.1252

b. 0.04736

Step-by-step explanation:

Let X denotes a  the number of soldiers killed by horse kicks in a year such that it has a Poisson distribution with mean $\lambda=0.61$.

Poisson distribution formula:

$P(x; \lambda) = \dfrac{(e^{-\lambda}) (\lambda^ x)}{ x!}$

a.    $P(X>1)=1-P(X\leq1)$

$=1- (P(X=0)+P(X=1))\\\\=1-( \dfrac{(e^{-0.61}) ((0.61)^0)}{ 0!}+\dfrac{(e^{-0.61}) ((0.61)^1)}{ 1!})\\\\= 1-( \dfrac{(0.54335) }{ 1}+\dfrac{(0.54335) (0.61)}{ 1})\\\\\approx0.1252$

Hence, the probability of more than 1 death in a corps in a year= 0.1252

b.  For five years, Mean = $\mu =\lambda \times 5= 0.61\times5 =3.05$

$P(X=0)=e^{-3.05}\dfrac{3.05^0}{0!}$

$\\\\=(0.04736)\dfrac{1}{1}\approx0.04736$

Hence,  the probability of no deaths in a corps over five years = 0.04736