Bayes' theorem and the base-rate trap

Here's a puzzle that fools doctors. A disease affects 1 in 100 people. A test for it is 99% accurate. You test positive. How worried should you be? Most people say "99%." The real answer is closer to 17%. Understanding why is understanding Bayes' theorem — the rule for updating what you believe when new evidence arrives, and the engine behind spam filters and medical screening. Let's derive it and then compute the puzzle.

Updating a belief

Bayes' theorem connects two "backwards" questions. You often know P(evidence | cause) — e.g., "if you have the disease, the test is positive 99% of the time." But you want P(cause | evidence) — "given a positive test, do I have the disease?" Bayes flips one into the other:

$P(A \mid B) = \frac{P(B \mid A)\,P(A)}{P(B)}$

The pieces have names you'll meet everywhere in AI:

• $P(A)$ — the prior: your belief before the evidence (here, 1% — the disease is rare).
• $P(B \mid A)$ — the likelihood: how well the cause explains the evidence (a sick person testing positive).
• $P(A \mid B)$ — the posterior: your updated belief after the evidence.
• $P(B)$ — the evidence: how common the evidence is overall.

That bottom term is found by adding up every way the evidence can happen:

$P(B) = P(B\mid A)\,P(A) + P(B\mid \lnot A)\,P(\lnot A)$

(Here $\lnot A$ means "not A" — being healthy.) In one line: posterior ∝ likelihood × prior.

Solving the puzzle

Let's plug in the numbers. Disease is rare, test is accurate:

• Prior: $P(\text{disease}) = 0.01$ (1%)
• Sensitivity: $P(+ \mid \text{disease}) = 0.99$ (a sick person tests positive 99% of the time)
• False-positive rate: $P(+ \mid \text{healthy}) = 0.05$ (5% of healthy people wrongly test positive)

Step 1 — the top (likelihood × prior):

P(+ | disease) × P(disease) = 0.99 × 0.01 = 0.0099

Step 2 — the bottom (all the ways to test positive):

sick & positive    = 0.99 × 0.01 = 0.0099
healthy & positive = 0.05 × 0.99 = 0.0495
P(+) total         = 0.0099 + 0.0495 = 0.0594

Step 3 — divide:

P(disease | +) = 0.0099 / 0.0594 ≈ 0.167   →  about 17%

Even after a positive result on a 99%-accurate test, you're only ~17% likely to actually have the disease. Surprising? Here's the picture, per 10,000 people:

100 people have it       → 99 test positive   (true positives)
9,900 are healthy        → 495 test positive  (false positives, 5% of 9,900)

positives in total = 99 + 495 = 594
truly sick among them = 99 / 594 ≈ 17%

The disease is so rare that the flood of false positives from the huge healthy majority outnumbers the true positives. This is the base-rate trap: the prior matters as much as the test's accuracy. Ignoring it is one of the most common reasoning mistakes there is.

Why this matters for AI

• Naive Bayes classifiers pick the category with the highest posterior — spam vs. not-spam is literally P(spam | these words) computed with Bayes.
• Any rare-event model (fraud, disease, defects) lives and dies by the base rate — which is why "99% accuracy" can be a terrible model when the thing you're detecting is rare.
• It's the mathematical heart of updating beliefs with data, which is what learning is.

Now feel it. Drag the three sliders below — prevalence, sensitivity, false-positive rate — and watch the posterior swing. Push the disease rarer and see the "99% accurate" test's answer collapse, exactly as we computed.

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This is one idea from a curriculum that builds probability from the ground up. Try a free lesson and compute a posterior yourself.