Consider this scenario:
You have almost won at a game show! Your host, Monty, gives you one final task. He lifts the curtains to reveal three closed doors, labeled A, B, and C. He informs you that behind the doors, in some order, there are two goats and a grand prize. The doors are identical and there is no way to tell what is behind them. He asks you to pick a door, which you do, picking one at random (let’s say A). Monty turns around and, in a fit of generosity, tells you that he will help you out by opening one of the unselected doors that contains a goat (let’s say B). Having taken one of the doors out of the running, he now gives you a choice. “Would you like to switch your selection to C or stay with A?”, he asks.
As you stand there, what should you do to have the best chance of winning the grand prize? Is there a systematic way to find out?
Consider this. You feel ill, are running a fever, and feel dizzy. You go to a nearby hospital and the attending physician suspects that you are suffering from a rare illness called Beetleguese fever. He orders a test to check his hypothesis which comes back positive for Beetleguese fever. Having seen this test result, you would expect the physician to start treating you immediately. Instead, he orders another test to confirm the diagnosis. Why? Isn’t that just a waste of time and money? Maybe the test was not accurate enough. You ask the physician how accurate the test is. He tells you that it is 99.999% accurate. “That’s great!”, you exclaim. The physician retorts, “It’s not good enough. We need to confirm it independently”. Why would he say that? How much more accurate can you get?
If Beetleguese fever is a very rare disease, then the physician is right. Let us assume that you live in a country with a population of 10 million people. At any given time, let us assume that 100 individuals in the population have Beetleguese fever (the physician mentioned that it was very rare). If a random person from this population was chosen and tested for Beetleguese fever using our 99.999% accurate test, and the test result came back as positive for the disease, the probability that the subject actually has Beetleguese fever is only around 50%! If you find this fact hard to believe, read on and I will show you exactly how we arrive at this number and why this is the case.