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Suppose you are a bank lending officer and you are trying to determine if it is worthwhile giving a loan to a potential borrower.You have a prior belief that a borrower is good or bad with equal probability.By gathering information you can determine, albeit imperfectly, whether an applicant is good or bad.If the applicant is good, there is a probability of 0.7 0.3 that your signal reveals that he is good bad.If the applicant is bad, there is a 0.4 0.6 probability that your signal reveals that he is good bad.
-What is your posterior belief that an applicant is bad when your signal tells you that he is goof?
A)0.15
B)0.22
C)0.30
D)0.33
E)0.45

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Use the following information for questions 
Suppose you are a bank lending officer and you are trying to determine if it is worthwhile giving a loan to a potential borrower.You have a prior belief that a borrower is good or bad with equal probability.By gathering information you can determine, albeit imperfectly, whether an applicant is good or bad.If the applicant is good, there is a probability of 0.7 0.3 that your signal reveals that he is good bad.If the applicant is bad, there is a 0.4 0.6 probability that your signal reveals that he is good bad.
-What is your posterior belief that an applicant is bad when your signal tells you that he is goof?
A)0.15
B)0.22
C)0.30
D)0.33
E)0.45

Quizplus · Accepted Answer

We need to use Bayes' Theorem to calculate the posterior belief. Let B be the event that the applicant is bad, G be the event that the signal tells us the applicant is good.
By Bayes' Theorem:
P(B|G) = P(G|B)P(B)/P(G)
We know that P(B) = P(G) = 0.5 (from the prior belief) and P(G|B) = 0.4 (from the information given in the problem). To find P(G), we can use the law of total probability:
P(G) = P(G|B)P(B) + P(G|not B)P(not B)
= 0.4 * 0.5 + 0.7 * 0.5
= 0.55
Now we can plug in the values to get:
P(B|G) = 0.4 * 0.5 / 0.55
= 0.36
So our posterior belief that the applicant is bad when the signal tells us he is good is 0.36 or 0.33 (rounded to 2 decimal places) which is closest to option D.

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