Question

A botanist is studying the distribution of daisies in a field. The field is divided into a number of equal sized squares. The mean number of daisies per square is assumed to be 3.

The daisies are distributed randomly throughout the field. Find the probability that, in a randomly chosen square there will be,

a. More than 2 daisies

b. Either 5 or 6 daisies.

Answer 1

a. The mean no. of diasies are 3. It follows a poisson distribution.

X ~ Poi(λ = 3)

$P(X=x) = \frac{e^λλ ^x}{x!} \\ = \frac{e^{-3}3^x}{x!} \\ P(X>2) = 1 -P(X≤2) \\ = 1 - [P(X=0) + P(X=1) + p(X=2)] \\ = 1 -( \frac{e^{-3}3^0}{0!} + \frac{e^{-3}3^1}{1!} + \frac{e^{-3}3^2}{2!}) \\ = 1 -(0.0498+0.1494+0.224) \\ P(X>2) = 0.5768$

b. Either 5 or 6 daisies

$P(either \; 5 \; or \; 6 \; daisies ) = P(X=5) + P(X=6) \\ = 0.1008 + 0.0504 \\ = 0.1512$