A tree in a forest is not growing properly. A botanist determines that there is

Question

A tree in a forest is not growing properly. A botanist determines that there is an 85% probability the tree has a disease or is being damaged by insects, a 45% probability it has a disease, and a 50% probability it is being damaged by insects.

a)What is the probability that the tree both has a disease and is being damaged by insects?

b)What is the probability that the tree damaged by insects but doesn't have any disease?

c)What is probability that the tree is damaged by insects given that it has a disease?

d)What is probability that the tree is damaged by insects given that it doesn't have a disease?

e)Find P(a tree has disease) X P( damaged by insects )

d)Compare the answer obtained in Step 5 to that of Step 1. Are two events "disease" and "damaged by insect" independent?

Explanation / Answer

P( D) =0.45

P(I)= 0.5

P( I or D) = 0.85

a) P( I and D)= P(I) +P(D) -P(A or D)= 0.45+ 0.5 -0.85 = 0.1

b) P(Only damaged by Insects)= P(I)- P( I and D)= 0.5- 0.1= 0.4

c) P( I | D)= P( I & D) / P(D) = 0.1 / 0.45= 0.2222

d) P( I | D')

P( I & D')= P(I) - P( I & D) = 0.5 - 0.1 =0.4

P( D' )= 1- P(D) =0.65

P( I | D') = P( I & D') / P(D') = 0.4 / 0.65 = 0.615385

e) P( D) * P( I)= 0.45*0.5= 0.225

f) if P( I & D) = P(I) * P(D) for independent but both are not equal. So both are dependent.

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