Question

Dandelions are studied for their effects on crop production and lawn growth. In one region, the mean number of dandelions per square meter was found to be 2. We are interested in the number of dandelions in this region. (a) Find the probability of no dandelions in a randomly selected area of 1 square meter in this region. (Round your answer to four decimal places.) (b) Find the probability of at least one dandelion in a randomly selected area of 1 square meter in this region. (Round your answer to four decimal places.)

Answer 1

Answer:

(a) The probability that there are no dandelions in a randomly selected area of 1 square meter in this region is 0.1353.

(b) The probability that there are at least one dandelion in a randomly selected area of 1 square meter in this region is 0.8647.

Step-by-step explanation:

Let X = number of dandelions per square meter.

The average number of dandelions per square meter is, λ = 2.

The random variable X follows a Poisson distribution with parameter λ = 2.

The probability mass function of X is:

$P(X=x)=\frac{e^{-2}2^{x}}{x!};\ x=0,1,2,3...$

(a)

Compute the value of P (X = 0) as follows:

$P(X=0)=\frac{e^{-2}2^{0}}{0!}=\frac{0.13534\times 1}{1}=0.1353$

Thus, the probability that there are no dandelions in a randomly selected area of 1 square meter in this region is 0.1353.

(b)

Compute the value of P (X ≥ 1) as follows:

P (X ≥ 1) = 1 - P (X < 1)

             = 1 - P (X = 0)

             $=1-0.1353\\=0.8647$

Thus, the probability that there are at least one dandelion in a randomly selected area of 1 square meter in this region is 0.8647.