Bayes With Frequencies: The 1,000-Person Grid

Suppose you take a screening test for a disease and it comes back positive. The doctor says the test catches 90% of people who have the disease. So what’s the chance you actually have it? Most people’s first instinct is “about 90%.” The real answer might be closer to one in ten. Where does the gap come from?

Turn percentages into people

Percentages are abstract; head counts are easy to picture. Imagine 1,000 people in your situation taking the test, and you know that:

• Base rate: 1% of people have the disease;
• Sensitivity: 90% of people who have it test positive;
• False-positive rate: 9% of healthy people also test positive.

Now split them step by step:

• Of 1,000 people, 1000 × 1% = 10 have the disease and 990 are healthy.
• Of the 10 who are sick, 10 × 90% = 9 test positive.
• Of the 990 who are healthy, 990 × 9% ≈ 89 test positive anyway.

So about 9 + 89 = 98 people test positive, and only 9 of them are actually sick: 9 ÷ 98 ≈ 9.2%.

The key is that there are so many healthy people. Even a 9% false-positive rate, applied to 990 of them, produces far more false alarms than there are real cases.

Four terms

• Prior: the probability before you see the evidence — here, the 1% base rate.
• Sensitivity: how likely the evidence is when the thing is true, P(positive | sick).
• False-positive rate: how likely the evidence is when the thing is false, P(positive | healthy).
• Posterior: the updated probability after seeing the evidence, P(sick | positive).

Notice that “catches 90%” is P(positive | sick), while what you want is P(sick | positive). That’s exactly the lesson from 1.1: the two directions aren’t interchangeable.

The dot grid below shows those 1,000 people. Drag the base rate, sensitivity, and false-positive sliders and watch how the share of positives who are truly sick changes.

Not just medicine

Is this message a scam? Is this odd number in the report a real problem? Same structure: get the base rate straight first, then ask how common the evidence is when the answer is “yes” versus “no.”