Question

A municipal bond service has three rating categories​ (A, B, and​ C). Suppose that in the past​ year, of the municipal bonds issued thoughout a​ country, 60 % were rated​ A, 30 % were rated​ B, and 10 % were rated C. Of the municipal bonds rated​ A, 60 % were issued by​ cities, 10 % by​ suburbs, and 30 % by rural areas. Of the municipal bonds rated​ B, 40 % were issued by​ cities, 50 % by​ suburbs, and 10 % by rural areas. Of the municipal bonds rated​ C, 70 % were issued by​ cities, 25 % by​ suburbs, and 5 % by rural areas.

Complete​ (a) through​ (c) below.

a. If a new municipal bond is to be issued by a​ city, what is the probability that it will receive an A​ rating?

b. What proportion of municipal bonds are issued by cities?

c. What proportion of municipal bonds are issued by suburbs?

Answer 1

a)

P(   A   ) =    0.6
P(   B   ) =    0.3
P(    C   ) =    0.1
P(   cities   | A)=   0.6
P(   cities   | B)=   0.4
P(   cities   | C)=   0.7

P(cities) = P(A) * P(cities| A) + P(B) *P(cities| B) + P( C)*P(cities| C) =                                    0.6*0.6+0.3*0.4+0.1*0.7=               0.55

P(A| cities) = P(A)*P(cities| A)/P(cities)=               0.6*0.6/0.55=       0.6545

b)

P(cities) = P(A) * P(cities| A) + P(B) *P(cities| B) + P( C)*P(cities| C) = 0.6*0.6+0.3*0.4+0.1*0.7= 0.55

c)

P(suburbs) = P(A) * P(suburbs| A) + P(B) *P(suburbs| B) + P( C)*P(suburbs| C) =                                    0.6*0.1+0.3*0.5+0.1*0.25=               0.235