Question

A municipal bond service has three rating categories (A, B, and C). Suppose that in the past year, of the municipal bonds issued throughout the United States, 70% were rated A, 20% were rated B, and 10% were rated C. Of the municipal bonds rated A, 50% were issued by cities, 40% by suburbs, and 10% by rural areas. Of the municipal bonds issued B, 60% were issued by cities, 20% by suburbs, and 20% by rural areas. Of the municipal bonds rated C, 90% were issued by cities, 5% by suburbs, and 5% by rural areas. 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

Answer:

(a) The probability that the new municipal bond issued will receive an A rating given that it was issued by a city is 0.196.

(b) Thus, the probability of municipal bonds issued by suburbs is 0.325.

Step-by-step explanation:

The data provided is as follows:

P (A) = 0.70

P (B) = 0.20

P (C) = 0.10

P (c | A) = 0.50

P (s | A) = 0.40

P (r | A) = 0.10

P (c | B) = 0.60

P (s | B) = 0.20

P (r | B) = 0.20

P (c | C) = 0.90

P (s | C) = 0.05

P (r | C) = 0.05

(a)

The Bayes' theorem states that the conditional probability of an event Y$_{i}$ given that another event X has already occurred is:

$P(Y|X)=\frac{P(X|Y)P(Y)}{\sum\limits_{i} {P(X|Y_{i})P(Y_{i})}}$

Compute the probability that the new municipal bond issued will receive an A rating given that it was issued by a city as follows:

$P(A|c)=\frac{P(c|A)P(A)}{P(c|A)P(A)+P(c|B)P(B)+P(c|C)P(C)}$

           $=\frac{(0.50\times 0.70)}{(0.50\times 0.70)+(0.60\times 0.20)+(0.10\times 0.90)}\\\\=\frac{0.35}{0.56}\\\\=0.196$

Thus, the probability that the new municipal bond issued will receive an A rating given that it was issued by a city is 0.196.

(b)

The law of total probability states that:

$P(X)=\sum\limits_{i} {P(X|Y_{i})}$

Compute the probability of municipal bonds issued by suburbs as follows:

$P(s)=P(s|A)P(A)+P(s|B)P(B)+P(s|C)P(C)$

        $=(0.40\times 0.70)+(0.20\times 0.20)+(0.10\times 0.05)\\\\=0.28+0.04+0.005\\\\=0.325$

Thus, the probability of municipal bonds issued by suburbs is 0.325.