Question

A dairy farmer accidentally allowed some of his cows to graze in a pasture containing weeds that would contaminate the flavor of the milk from this herd. The farmer estimates that there's a 9% chance of a cow grazing on some of the flavorful weeds (a) Under these conditions, what is the probability that none of the 14 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? (a) The probability that none of the 14 animals in this herd ate the tasty weeds is 0.2670 (Round to three decimal places as needed.) (b) Does the Poisson model give a good estimate of the probability that no animal ate the weed? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. (Round to three decimal places as needed.) A. No, because the probability found using the Poisson model, is not close to the probability found using the binomial model B. Yes, because the probability found using the Poisson model, is close to the probability found using the binomial model.

Answer 1

Answer b)

For approximating binomial distribution to Poisson, we can use

λ = n*p

Here, n = 14 and p = 0.09

λ = 14*0.09 = 1.26

For finding probability of no animal, we need to calculate P(X = 0)

Pr(X=k) = (e^−λλ^k​)/k!

Pr(X=0) = (e^−1.26*1.26⁰​)/0!

Pr(X=0) = 0.284 (Using Binomial Poisson model)

P(X = 0) = 0.267 (Using Binomial model obtained from From Part a)

Correct Answer:

A. No, because the probability using the Poisson model, 0.284, is not close to probability found using binomial model.