Answer to Question #183168 in Statistics and Probability for olga

Question

Question 2. A dairy farmer accidentally allowed some of his cows to graze in a pasture containing weeds that would contaminate the milk from this herd. The farmer estimates that there’s a 10% chance of a cow grazing on some of the flavorful weeds.

(a) Under these conditions, what is the probability that none of the 12 animals in this herd ate the tasty weeds?

(b) Does the Poisson model give a good estimate of the probability that no animal ate the weed

Answer 1

(a) p(none ate the tasty weed)

The probability that an animal ate is 10%

The probability that the 12 animals did not eat the weed is,

$P(X=x) ={n \choose x} p^x(1-p)^{n-x}$ $={12\choose 0}0.1^0*0.9^{12}$ = 0.2824

(b) using a poisson model

P(X = x) = $\frac{e^{-\lambda} \lambda^x} {x!}$

P(X = 0) = $\frac {e^{-0.1*12}{(0.1*12)}^0}{0!}$ = 0.3012

The values are nearly the same hence Poisson gives a good estimate