Question

A forest has three species of trees: 30% of the trees are maple trees, 50% of the trees are oak trees, and 20% of the trees are birch trees. Following a powerful storm, there is a disease outbreak; 10% of the maple trees have the disease, as do 4% of the oak trees, and 25% of the birch trees.

If you know that a given tree in the forest has a disease, what is the probability it is a maple tree? What is the probability it is an oak tree? What is the probability it is a birch tree?

Answer 1

P(maple tree) = 0.3

P(oak tree) = 0.5

P(birch tree) = 0.2

P(disease|maple tree) = 0.1

P(disease|oak tree} = 0.04

P(disease|birch tree} = 0.25

P(disease) = P(disease|maple tree)*P(maple tree) + P(disease|oak tree)*P(oak tree) + P(disease|birch tree)*P(birch tree)

P(disease) = 0.3* 0.1 + 0.5*0.04 + 0.2*0.25

P(disease) = 0.03 + 0.02 + 0.05

P(disease) = 0.1

P(maple tree|disease) = P(maple tree and disease)/P(disease)

P(maple tree|disease) = 0.3*0.1/0.1

P(maple tree|disease) = 0.3

P(oak tree|disease) = P(oak tree and disease)/P(disease)

P(oak tree|disease) = 0.5*0.04/0.1

P(oak tree|disease) = 0.2
P(birch tree|disease) = P(birch tree and disease)/P(disease)

P(birch tree|disease) = 0.2*0.25/0.1

P(birch tree|disease) = 0.5