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# Thread: Need help with Probability and Distribution Problem

1. ## Need help with Probability and Distribution Problem

A municipal bond rating service has three rating categories (A,B,C). Suppose that in the past year, of the municipal bonds issued throughout the country, 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 rated 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, 5% by rural areas.

If a new municipal bond is to be issued by a city, what is the probability that it will receive an A rating?
What proportion of the municipal bonds are issued by cities?
What proportion of the municipal bonds are issued by suburbs?

2. Originally Posted by findsujit
A municipal bond rating service has three rating categories (A,B,C). Suppose that in the past year, of the municipal bonds issued throughout the country, 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 rated 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, 5% by rural areas.

If a new municipal bond is to be issued by a city, what is the probability that it will receive an A rating?
What proportion of the municipal bonds are issued by cities?
What proportion of the municipal bonds are issued by suburbs?
There are a couple of ways (as always) to go about this, but the easiest IMO is to break it down iwith Bayes' theorem:

$P(A | city) = P(city | A)\frac{P(A)}{P(city)}$

$P(A | city) = 0.5\frac{0.7}{0.56}$

$P(A | city) = 0.625$

Percentage of bonds issued to cities is P(city) in the previous calculation...

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