Bayes' Theorem

The question that breaks most people's intuition

A disease affects 1% of the population. A test for it correctly identifies 95% of people who have the disease (its sensitivity), and incorrectly flags 10% of people who don't (its false positive rate).

You test positive. What is the probability you actually have the disease?

Most people guess somewhere around 90%. The correct answer is about 8.8%.

Switch between scenarios above. The key insight is the same in every case: a positive result sounds alarming, but whether it actually means something depends on how rare the condition is to begin with.

Why the answer is so low

Imagine 1,000 people tested:

• 10 have the disease (1% prevalence)
• Of those 10: about 9.5 test positive (true positives)
• 990 don't have the disease
• Of those 990: about 99 also test positive (false positives — 10% of 990)

Total positive results: roughly 108. Of those, only 9.5 actually have the disease.

That's 9.5 out of 108 — about 8.8%.

The formula

Bayes' theorem gives us the exact calculation. We want P(disease | positive test) — the probability of having the disease given that you tested positive:

In our disease example:

• $P(A)$ = prior probability of disease = 0.01
• $P(B \mid A)$ = probability of positive test given disease = 0.95 (sensitivity)
• $P(B)$ = total probability of a positive test = $0.95 \times 0.01 + 0.10 \times 0.99 = 0.1085$

$P(\text{disease} \mid \text{positive}) = \frac{0.95 \times 0.01}{0.1085} \approx 0.088$

Prior, likelihood, and posterior

The three moving parts of Bayes' theorem:

• Prior — your belief before seeing any evidence. In this case, the 1% base rate.
• Likelihood — how probable the evidence is if the hypothesis is true. The 95% sensitivity.
• Posterior — your updated belief after the evidence. The ~8.8% we calculated.

Bayesian reasoning is a machine for updating beliefs. You start with a prior, observe evidence, and compute a posterior. That posterior becomes the prior for the next piece of evidence.

Real-world consequences

• Medical screening: population-wide screening for rare diseases produces many false positives. This is why mass screening for low-prevalence conditions is often a bad idea without confirmatory testing.
• Spam filters: spam is common (high prior), so even a modest classifier performs well. The math works in its favor.
• Fraud detection: fraud is rare (low prior), so flags trigger investigations rather than automatic blocks — because even a good detector will be wrong most of the time.
• Legal reasoning: "the DNA test is 99.9% accurate" is not the same as "there's a 99.9% chance the defendant is guilty." The prior matters.