The probability trap that even doctors fall into.
A 99%-accurate test comes back positive. What's the probability you actually have the disease?
How far new evidence should move your belief — Bayes' theorem.
Experiment
Hands-on experiment
Predict first — a disease with 0.1% prevalence, a 99%-accurate test, and your result is positive. Probability you're actually sick?
Run the test on 100,000 people
Read more — why it exists · insights · common mistakes · formulasExpand ▾
Why
Why does this exist?
Test results, courtroom evidence, spam words — every day we receive 'evidence' and must update our judgments. And human intuition gets this update systematically wrong.
The classic failure is base-rate neglect. Eyes lock onto the test's accuracy (99%) and forget how rare the disease itself is (0.1%). The rarer the disease, the more the positives are false.
Bayes' theorem performs the update mechanically and correctly. Prior belief × strength of evidence = posterior belief — modern AI, spam filters, and medical diagnosis all run on it.
Insight
Insights from the video
“Flipping a probability has a price — and the price is the base rate.”
Turning P(positive|disease)=99% into P(disease|positive) brings in the prevalence as a multiplicative factor. The number can change dramatically the moment you flip direction.
“Bayes isn't an answer machine — it's an update machine.”
It grants no absolute truth; it moves your current belief exactly as far as the evidence warrants. Stack enough evidence and even different priors converge to the same place.
Misconception
Common misconceptions
Positive on a 99%-accurate test means 99% chance of disease.
At 0.1% prevalence, most positives are false positives, so the real chance is about 9%. 'The test's accuracy' and 'the chance of disease given a positive' point in different directions.
Just raise the test's accuracy and the problem disappears.
Even at 99.9% accuracy, 0.1% prevalence gives only a 50% chance. As long as the prior (prevalence) is tiny, the sea of false positives remains — which is why retests exist.
Formula
Writing it as math
Write what you counted in the 100,000-person experiment as a formula and you have Bayes.
Bayes' theorem
The probability of hypothesis H after seeing evidence E. The prior P(H) sits multiplied in the numerator — why the base rate can't be ignored.
Applied to the test
The denominator's second term (healthy people's false positives) is 10x the first — the formula shows directly why most positives are false.
Odds form — the language of updating
Multiply belief (odds) by the evidence's strength (LR) and you get the new belief. Multiple pieces of evidence? Keep multiplying — exactly what a spam filter does.
In Real Life
Where you meet it in real life
Screening policy
Why mass-screening for rare diseases is done cautiously — it manufactures a sea of false positives. Retesting (adding evidence) is how the probability gets pulled up.
Spam filters
Words like 'free' and 'winner' are each a likelihood ratio. A naive Bayes filter multiplies them to update the spam probability.
DNA evidence in court
'A one-in-a-million match' is NOT 'a one-in-a-million chance of innocence' — you must multiply by the prior over possible suspects.
AI inference
A self-driving car updating 'probability that's a pedestrian' through sensor noise is running a Bayes filter.
Try Yourself
Test yourself
Q1If the disease's prevalence were 10%, roughly what would a positive on the same 99% test mean?Show answer ▾
About 92%. Of 100,000 people: 10,000 sick → 9,900 true positives; 90,000 healthy → 900 false positives. 9,900/10,800 ≈ 92%. Higher prevalence makes the same test trustworthy — a doctor ordering the test at all is itself an act that raises the prior.
Q2After one positive, an independent retest also comes back positive. Now what?Show answer ▾
Take 9% as the new prior and update again — it jumps to about 91%. Stacking evidence multiplies the odds by the likelihood ratio repeatedly. The mathematical reason retests exist.
Q3The forecast said '70% chance of rain', you took an umbrella, and it didn't rain. Was the forecast wrong?Show answer ▾
No — 70% is a statement that includes a 30% 'no rain'. Forecast quality is judged not by one day but by calibration: of the days it said 70%, did it rain about 70% of the time? A core habit of Bayesian thinking.
Try answering yourself before revealing it — getting it wrong is where learning starts.
Connection
Concepts connect
Previous concept
Conditional Probability
You must be able to read P(A|B) before the formula that flips it can mean anything.
← Conditional Probability labLeads to next
Expected Value
Belief updated — now decide. The next tool multiplies each option's probability by its value and puts them on one scale: expected value.
Go to the Expected Value lab →Related
Labs worth exploring together
Related lab
Conditional Probability
Bayes' raw material — build the sense of conditions changing denominators first, and this formula feels natural.
Go to the Conditional Probability lab →