Probability and Statistics/Operations Research. Please help!!!

Question

You are trying to choose a reliable insurance company for fire insurance for your factory. There are only two fire insurance companies in your area. 60% of businesses have coverage from company A. 40% of businesses have coverage from company B. 1% of the businesses have coverage from company A and have fire claims that are settled without fuss. 0.2% of the businesses have coverage from company A and have fire claims that are not settled without fuss. 0.8% of businesses have coverage from company B and have fire claims that are settled without fuss. 0.1% of businesses have coverage from company B and have fire claims that are not settled without fuss. Use Bayes's formula to compute the probability of having a fire claim that is not settled without fuss given that your are covered by company A, and to compute the probability of having a fire claim that is not settled without fuss given that you are covered by company B.

Answer 1

Based on the wording in the question, I'm not sure that Baye's formula needs to be used.  Typically, if we have P(A|B) and need to get P(B|A), we will use Bayes.  But in the problem above, none of the statements seem to have that flavor.  Based on the current wording I would solve it like this:

S = Settled without Fuss
S- = Not Settled without Fuss
A = Insured by company A
B = Insured by company B

P(A) = .4
P(B) = .6
P(A and S) = .01
P(A and S-) = .002
P(B and S) = .008
P(B and S-) = .001

We want P(S-|A) = P(S- and A)/P(A) = .002/.4 = .005
Next, P(S-|B) = P(S- and B)/P(B) = .001/.6 = .0025

 

IF you changed the wording so that it said "Of the fire claims settled without fuss, 1% are from company A," it would change the problem for sure.