Question

Question (5) [12 Marks] Note: Do not use R, do the calculations by hand.

A very large (essentially infinite) number of butterflies is released in a large field. Assume the butterflies are scattered randomly, individually, and independently at a constant rate with an average of 6 butterflies on a tree.

(a) [3 points] Find the probability a tree (X) has > 3 butterflies on it.  

(b) [3 points] When 10 trees are picked at random, what is the probability 8 of these trees have > 3 butterflies on them?

(c) [3 points] Find the probability a tree with > 3 butterflies on it has exactly 6.

(d) [3 points] On 2 trees there are a total of t butterflies. Find the probability that x of these butterflies are on the first tree. Note: Use the conditional probability to solve this part.

Answer 1

The number of butterflies on a tree can be modeled by Poisson distribution

The Poisson PMF is

Here
a) The probability a tree has > 3 butterflies on it is

b) Here we use Binomial distribution. The number of trees N out of 10 which has >3 butterflies in it has distribution

The required probability is

c) The required conditional probability,

d) The be the random variables representing the number of butterflies on 2 trees. are independent.

The required probability is