There are many websites which allow patients to rate their experience with doctors. In this problem, we're going to try to understand what a rating might tell us about a doctor.
Suppose doctors are either good or bad. If the doctor is actually bad, there is a 95% chance that a patient will have a negative experience and rate the doctor as bad. However, even when the doctor is good, there is a 1% chance that the patient will have a negative experience and rate the doctor as bad.
(a) Suppose the doctor on a website are known to be mostly good- only 0.5% of all registered practitioners on the website are bad. If a patients rates a doctor as bad, what is the probability the doctor is actually bad? Now suppose a doctor gets 2 bad ratings instead of 1- what is the probability the doctor is actually bad?
(b) Suppose the website is instead full of second-rate doctors, and 70% of all registered practitioners on the website are bad. Now what is the probability the doctor is actually bad if a single patient rates a doctor as bad?