Instrument 1 · Bayes' theorem

The test that is 99% right

A test detects a condition 99% of the time and falsely flags 1% of healthy people. You test positive. Intuition says 99%. The answer is 50%. The gap between those two numbers is the bars below: how many people already had it before you were tested.

The population

What a positive result is worth

has it, tested positive has it, tested negative doesn't have it, tested positive doesn't have it, tested negative

The same statement, with the numbers in it

P (H ∣ E) = \frac{P (E ∣ H) \cdot P (H)}{P (E)}

0.50 = \frac{0.99 \cdot 0.01}{0.0198}

The denominator is the part everyone drops: P(E) = 0.99·0.01 + 0.01·0.99 = 0.0198 — the chance of a positive result from any cause, not just the condition.

Fine tune

How common is it? 1 in 100

Sensitivity — detects it when present 99%

Specificity — clears a healthy person 99%

People in the picture 10,000

Specificity 99% means 1% of healthy people are flagged anyway.