You've taken your prized sports car into a repair shop for a routine maintenance check. As part of a standard set of diagnostics, the car's onboard computer is checked for logged problems. The mechanic tells you that they have good and bad news. The bad news is that the car's computer has reported that it has detected a failure in the electrical system that will definitely cause an electrical fire in the near future. The probability of the computer reporting this warning given that there is actually a failure in the electrical system is 0.95, as is the probability that the computer will correctly not report anything if the failure isn't present. The good news is that this type of failure and the consequent electric fire will only occur in one out of every ten thousand cars of this model on the road.
(a) What are the chances that an electrical fire will occur in your car if don't opt to repair it? (Show your calculations as well as giving the final result.)
(b) Repeat the calculation of part (a), but for the case where the failure occurs in one out of every hundred cars of this model.
(c) Given the logged report of a problem, why is it good news that the issue is so rare in part (a)?