Enterprise risk management software identifies patterns in credit card fraud by analyzing user transaction history and inferring probabilities through algorithms that utilize Bayesian neural networks. Unfortunately, credit card fraud detection algorithms may also result in false positives due to incomplete information.

Define events
F : Transaction is fraudulent,
N : Transaction is normal,
where F is the complement of N with P(F) = 1 – P(N).

The probability of a credit card transaction is fraudulent is P(F)=0.01.

Given baseline credit card transaction characteristics, the bank's fraud detection system will recognise a transaction as either being fraudulent, or not fraudulent. If the transaction is identified as fraudulent the transaction is cancelled and the customer notified. It is worth noting that the system is imperfect and that such recognition may be incorrect.

The probability of correct identification of fraudulent and non-fraudulent transactions are:
P(S | F) = 0.99 (true-positive) and P(C | N) = 0.99 (true-negative) respectively,
where the above events S and C are defined as;
S: the transaction is identified by the detection system as fraudulent and a 'stop' is put on the transaction,
C: the transaction is identified by the detection system as non-fraudulent, and the transaction continues as normal.

(a) If the fraud detection system claims that a transaction is fraudulent, what is the probability of it actually being fraudulent? Present all relevant probabilities and each step of your calculations. In doing your calculations, consider building a model in Excel as this will come handy in part (b). You do not need to present you Excel model in your solutions. You do not need to present you Excel model in your solutions and Excel is not necessary to answer either question.

(b) If the bank desires to improve/increase the probability calculated in (a), should it prioritize tweaking the algorithm to improve the true-positive rate or the true-negative rate? Hint: You can numerically resolve this by varying the true-positive rate and the true-negative rate by the same amount (e.g. by 0.005) and observing the impact on the probability of interest. Present your results.