To reduce fraud, businesses and credit card vendors have devised systems that recognize characteristics of fraudulent transactions. These systems are not perfect. They sometimes flag as Bad an honest transaction and sometimes miss flagging a fraudulent transaction. A business has been offered a fraud detection system to protect its online retail site. The system promises high accuracy. It claims it catches 99% of fraudulent transaction; on 99% of transactions that are fraudulent, the system correctly flags them Bad. The system incorrectly flags only 2% of the honest transactions; on 2% of the honest transaction, the system incorrectly flags them as Bad. Suppose that the prevalence of fraud among transactions at the retail site is 1%. Please show all work to answer the following questions. For each question, include in your answer the notation representing your result, such as P(A) or P(A|B). And, please use notation: fraud and honest to indicate fraudulent and honest transactions, and Bad and Good, to indicate characterization of transactions by the fraud detection system.
a. What is the probability that a transaction is flagged as Bad?
b. If a transaction if flagged as Bad, what is the probability the transaction is fraudulent?
c. If a transaction if not flagged as Bad (hence it was flagged as Good), what is the probability the transaction is honest?