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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 good when your signal tell you that he is good?
A)0.36
B)0.55
C)0.64
D)0.70
E)Insufficient information

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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 good when your signal tell you that he is good?
A)0.36
B)0.55
C)0.64
D)0.70
E)Insufficient information

Quizplus · Accepted Answer

To calculate the posterior probability that an applicant is good given that your signal tells you the applicant is good, you can use Bayes' theorem. The formula for Bayes' theorem in this context is:$$ P(\text{Good}|\text{Signal Good}) = \frac{P(\text{Signal Good}|\text{Good}) \times P(\text{Good})}{P(\text{Signal Good})} $$Given:- $P(\text{Signal Good}|\text{Good}) = 0.7$- $P(\text{Signal Good}|\text{Bad}) = 0.4$- $P(\text{Good}) = P(\text{Bad}) = 0.5$ (since it's stated the belief is equal)First, calculate $P(\text{Signal Good})$, which is the total probability of receiving a good signal:$$ P(\text{Signal Good}) = P(\text{Signal Good}|\text{Good}) \times P(\text{Good}) + P(\text{Signal Good}|\text{Bad}) \times P(\text{Bad}) $$$$ P(\text{Signal Good}) = 0.7 \times 0.5 + 0.4 \times 0.5 = 0.35 + 0.2 = 0.55 $$Then, apply Bayes' theorem:$$ P(\text{Good}|\text{Signal Good}) = \frac{0.7 \times 0.5}{0.55} = \frac{0.35}{0.55} = \frac{7}{11} \approx 0.636 $$Rounded to two decimal places, the posterior probability is approximately 0.64, making option C the correct answer.

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