A fraud detection system at a payment processor has a 90% true positive rate (detected transaction is actually fraud) and a 1% false positive rate (detected transaction is not fraud). Over the past year, a review shows that 1460 of 1,000,000 transactions were actually fraudulent. 

(a)  What is the probability that the fraud detection system signals fraud for a randomly selected transaction?

(b)  What is the probability a transaction is fraudulent if the detection signals fraud? 

(c) Using the past year's data, what is the average fraud transaction rate per day?

(d) Assuming fraudulent transactions are independent of each other with the constant average rate from (c), what is the probability of observing 0 fraud transactions in a day?