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, 30% were issued by​ cities, 30% by​ suburbs, and 40% 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, 60% were issued by​ cities, 35% 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? ​(Round to three decimal places as​ needed.)

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

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

Answer 1

We are given here that:

P(A) = 0.6, P(B) = 0.3 and P(C) = 0.1

Also, we are given here that:
P( cities | A) = 0.3, P(suburbs | A) = 0.3 and P(rural | A) = 0.4

P(cities | B) = 0.4, P(suburbs | B) = 0.5 and P(rural | B) = 0.1

P(cities | C) = 0.6, P(suburb | C) = 0.35 and P(rural | C) = 0.05

a) Using law of total probability, we have here:
P( cities ) = P( cities | A)P(A) +  P( cities | B)P(B) +  P( cities | C)P(C)

P(cities) = 0.3*0.6 + 0.4*0.3 + 0.6*0.1 = 0.18 0.12 + 0.06 = 0.36

Given that the bond is issued by a city, probability that it will receive an A rating is computed using Bayes theorem here as:

P( A | cities ) = P( cities | A)P(A) / P( cities ) = 0.18 / 0.36 = 0.5

Therefore 0.5 is the required probability here.

b) Using law of total probability, we have here:
P( cities ) = P( cities | A)P(A) +  P( cities | B)P(B) +  P( cities | C)P(C)

P(cities) = 0.3*0.6 + 0.4*0.3 + 0.6*0.1 = 0.18 0.12 + 0.06 = 0.36

Therefore 0.36 is the required probability here.

c) Using law of total probability, we have here:
P( suburb ) = P( suburb | A)P(A) +  P( suburb | B)P(B) +  P( suburb | C)P(C)

P(suburb) = 0.3*0.6 + 0.5*0.3 + 0.35*0.1 = 0.18 + 0.15 + 0.035 = 0.365

Therefore 0.365 is the required probability here.