P(H|E) · Bayes' Theorem
How far new evidence should move your belief — Bayes' theorem.
A 99%-accurate test comes back positive. What's the probability you actually have the disease?
The probability trap that even doctors fall into.
Open the immersive version →Why does this question matter?
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.
Pause and try it yourself
Make a prediction first. Change one input at a time and compare what happens with what you expected.
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 the idea from another angle
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.
Now read the definitions with their conditions
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.
Make one more distinction
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.
Where else can this idea help?
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.
Explain it in your own words
Try another question
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.
What question comes next?
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 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 →