Question

This problem has been solved!See the answerA 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, 10% were rated B, and 20% were rated C. Of the municipal bonds rated A, 50% were issued by cities, 20% by suburbs, and 30% by rural areas. Of the municipal bonds issued B, 40% were issued by cities, 30% by suburbs, and 30% by rural areas. Of the municipal bonds rated C, 60% were issued by cities, 15% by suburbs, and 25% 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. $\frac{35}{51}$

b. $\frac{51}{100}$

c. $\frac{1}{5}$

Step-by-step explanation:

Suppose cities represented by C', suburbs represented by S and rural represented by R,

Let x be the total number of bonds issued throughout the US,

According to the question,

n(A) = 70% of x = 0.7x,

n(B) = 10% of x = 0.1x,

n(C) = 20% of x = 0.2x,

n(A∩C') = 50% of n(A) = 0.5 × 0.7x = 0.35x,

n(A∩S) = 20% of n(A) = 0.2 × 0.7x = 0.14x,

n(A∩R) = 30% of n(A) = 0.3 × 0.7x = 0.21x,

n(B∩C') = 40% of n(B) = 0.4 × 0.1x = 0.04x,

n(B∩S) = 30% of n(B) = 0.3 × 0.1x = 0.03x,

n(B∩R) = 30% of n(B) = 0.3 × 0.1x = 0.03x,

n(C∩C') = 60% of n(C) = 0.6 × 0.2x = 0.12x,

n(C∩S) = 15% of n(C) = 0.15 × 0.2x = 0.03x,

n(C∩R) = 25% of n(C) = 0.25 × 0.2x = 0.05x,

n(C') = n(A∩C')  + n(B∩C')  + n(C∩C')  = 0.35x + 0.04x + 0.12x = 0.51x

n(S) = n(A∩S) + n(B∩S) + n(C∩S) = 0.14x + 0.03x + 0.03x = 0.20x

a. The probability that it will receive an A rating, if a new municipal bond is to be issued by a city,

$P(\frac{A}{C'})=\frac{P(A\cap C')}{P(C')}=\frac{0.35x/x}{0.51x/x}=\frac{0.35}{0.51}=\frac{35}{51}$

b. The proportion of municipal bonds are issued by cities = $\frac{n(C')}{x}$

$=\frac{0.51x}{x}$

$=\frac{51}{100}$

c. The proportion of municipal bonds are issued by suburbs = $\frac{n(S)}{x}$

$=\frac{0.20x}{x}$

$=\frac{20}{100}$

$=\frac{1}{5}$