''Suppose the police pull someone over at random at a drunk-driving checkpoint and administer a Breathalyzer test that indicates they are drunk. Further, suppose the test is wrong on average 5 percent of the time, saying that a sober person is drunk. What is the probability that this person is wrongly accused of drunk driving?
Your first inclination might be to say 5 percent. However, you have been given the probability that the test says someone is drunk given they are sober, or P(Test=drunk | Person=sober) = 5 percent. But what you have been asked for is the probability that the person is sober given that the test says they are drunk, or P(Person=sober | Test=drunk). These are not the same probabilities!
What you haven’t considered is how the results depend on the base rate of the percentage of drunk drivers. Consider the scenario where everyone makes the right decision, and no one ever drives drunk. In this case the probability that a person is sober is 100 percent, regardless of what the Breathalyzer test results say. When a probability calculation fails to account for the base rate (like the base rate of drunk drivers), the mistake that is made is called the base rate fallacy.''
-Gabriel Weinberg and Lauren McCann, Super Thinking













