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, 50 % were rated​ A, 20% were rated​ B, and 30% 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, 50% were issued by​ cities, 10% by​ suburbs, and 40% by rural areas. Of the municipal bonds rated​ C, 80% were issued by​ cities,15 % 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

P(A) = 0.50, P(B) = 0.20, P(C) = 0.30

P(cities|A) = 0.30, P(suburbs|A) = 0.30, P(rural|A) = 0.40

P(cities|B) = 0.50, P(suburbs|B) = 0.10, P(rural|B) = 0.40

P(cities|C) = 0.80, P(suburbs|C) = 0.15, P(rural|C) = 0.05

P(Cities) = P(A)*P(cities|A) +P(B)*P(cities|B) +P(C)*P(cities|C) = 0.50*0.30 + 0.20*0.50 +0.30*0.80 = 0.49

a) probability that a new municipal bond will receive an A rating given that it is from cities, P(A|cities) =

b) proportion of municipal bonds are issued by cities, P(Cities) =

=P(A)*P(cities|A) +P(B)*P(cities|B) +P(C)*P(cities|C)

= 0.50*0.30 + 0.20*0.50 +0.30*0.80 = 0.49

c) proportion of municipal bonds are issued by suburbs , P(Suburbs) =

=P(A)*P(suburbs|A) +P(B)*P(suburbs|B) +P(C)*P(suburbs|C)

= 0.50*0.30 + 0.20*0.10 +0.30*0.15 = 0.215