A tax software company is considering offering an optional "Audit Protection" service to its customers. The policy will offer legal and administrative support for a client in the event the client in the event the client is audited after filing their taxes using the software. The customers must sign up ahead of time (i.e before they are audited) and the policy will last for 10 years as long as the customer continues to use the software. If someone is audited more than once, they have to renegotiate their policy moving forward depending on whether or not fraud has occurred.

The company knows the probability of an audit is heavily influenced by reported business losses. If a customer does not claim any bussiness losses, then their probability of an audit in any particular year is 0.015 (1.5%), independently of whether they are audited any other year. If a customer does claim business losses, the probability of an audit in any particular year is 5% (0.05), independently of whether they are audited any other year.

As it turns out, half (50%) of the company's customers always report business losses every year. The other half never reports business losses in any year.

a) Consider one of the customers who never claims business losses in any year. What is the probability that he/she is not audited over an entire decade (10 years?)

b) Consider one of the customers who always claims business losses every year. What is probability that he/she is not audited over an entire decade?

c) If it turns out that customer has been audited within the past 10 years, what is the probability that he/she is one of the customers who always claim business losses

d) Based on the probabilities calculated above the firm has concluded that audit protection service would be mildly profitable, so they will go ahead and launch the service as an option to all the customers. Common the wisdom of their use of the probabilities as estimates regarding their customers who buy audit protections.