Five Cs Bank issues credit cards to its customers. In the past 3.4% of the out-of-state nonfraudulent credit card transactions were made at gasoline stations and 9.2% of the out-of-state fraudulent credit card transactions were made at gasoline stations. Based on historical transactions 97.7% of out-of-state credit card transactions are legitimate while the rest of its out-of-state credit card transactions are fraudulent. An out-of-state credit card transaction took place at a gasoline station, what is the probability the transaction is fraudulent?

0.0838 (8.62%)

Step-by-step explanation:

defining the event G = an out-of-state transaction took place in a gasoline station, then the probability is

P (G) = probability that the transaction is fraudulent * probability that took place in a gasoline station given that is fraudulent + probability that the transaction is not fraudulent * probability that took place in a gasoline station given that is not fraudulent = 0.033 * 0.092 + 0.977 * 0.034 = 0.0362

then we use the theorem of Bayes for conditional probability. Defining also the event F = the transaction is fraudulent, then

P (F/G) = P (F∩G) / P (G) = 0.033 * 0.092 / 0.0362 = 0.0838 (8.62%)

where

P (F∩G) = probability that the transaction is fraudulent and took place in a gasoline station

P (F/G) = probability that the transaction is fraudulent given that it took place in a gasoline station

Step-by-step explanation:

let F = fraud credit card transaction and G = legit credit card transaction C = credit card transaction at gasoline station P (G) = 0.977 P (F) = 1 - P (G) = 0.023 P (C|G) = 0.034 P (C|F) = 0.092

To find what is the probability the transaction is fraudulent = P (F|C)

From application of baye's rule;

P (F|C) = P (F) P (C|F) / P (F) P (C|F) + P (G) P (C|G)

= 0.092 X 0.023 / 0.023 X 0.092 + 0.097 X 0.034

= 0.0599 is the probability the transaction is fraudulent