Answer:
a. ![\frac{35}{51}](https://tex.z-dn.net/?f=%5Cfrac%7B35%7D%7B51%7D)
b. ![\frac{51}{100}](https://tex.z-dn.net/?f=%5Cfrac%7B51%7D%7B100%7D)
c. ![\frac{1}{5}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B5%7D)
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}](https://tex.z-dn.net/?f=P%28%5Cfrac%7BA%7D%7BC%27%7D%29%3D%5Cfrac%7BP%28A%5Ccap%20C%27%29%7D%7BP%28C%27%29%7D%3D%5Cfrac%7B0.35x%2Fx%7D%7B0.51x%2Fx%7D%3D%5Cfrac%7B0.35%7D%7B0.51%7D%3D%5Cfrac%7B35%7D%7B51%7D)
b. The proportion of municipal bonds are issued by cities = ![\frac{n(C')}{x}](https://tex.z-dn.net/?f=%5Cfrac%7Bn%28C%27%29%7D%7Bx%7D)
![=\frac{0.51x}{x}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B0.51x%7D%7Bx%7D)
![=\frac{51}{100}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B51%7D%7B100%7D)
c. The proportion of municipal bonds are issued by suburbs = ![\frac{n(S)}{x}](https://tex.z-dn.net/?f=%5Cfrac%7Bn%28S%29%7D%7Bx%7D)
![=\frac{0.20x}{x}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B0.20x%7D%7Bx%7D)
![=\frac{20}{100}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B20%7D%7B100%7D)
![=\frac{1}{5}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B1%7D%7B5%7D)