Probability · step 3 of 13

Bayes’ theorem

Bayes’ theorem reverses a conditional: P(A|B) = P(B|A)P(A)/P(B), with P(B) usually expanded by the law of total probability. Its famous consequence is that a highly accurate test for a rare condition still produces mostly false positives — the base rate P(A) dominates, and ignoring it is the single most common probabilistic error in medicine and law.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Name the events and list what you are givenTypically P(disease), P(positive | disease) and P(positive | no disease). Write them down before computing.
Compute the denominator by total probabilityP(B) = P(B|A)P(A) + P(B|Aᶜ)P(Aᶜ). This is where the base rate enters.
Assemble BayesP(A|B) = P(B|A)P(A)/P(B). The numerator is one of the terms in the denominator.
Sanity check against the base rateThe posterior should move in the direction the evidence points, but stay anchored by a small prior.

Worked example

A disease affects 1% of people. A test is 99% sensitive and 95% specific. Given a positive result, what is the probability of having the disease?

P(D) = 0.01, P(+|D) = 0.99, P(+|Dᶜ) = 1 − 0.95 = 0.05.
P(+) = 0.99(0.01) + 0.05(0.99) = 0.0099 + 0.0495 = 0.0594.
P(D|+) = 0.0099 / 0.0594.
= 1/6 ≈ 0.1667.

Answer. About 16.7% — despite a 99% sensitive test, most positives are false, because healthy people vastly outnumber sick ones.

Where marks get dropped

These are the specific errors that cost credit on bayes’ theorem questions — QED's rubric penalises each of them separately.

Reporting the sensitivity as the answer. P(+|D) = 0.99 is not P(D|+), and confusing them is the base rate fallacy.
Forgetting the false positives from the healthy majority. With a 1% prevalence they outnumber true positives five to one here.
Mixing up specificity with the false positive rate. Specificity is P(−|Dᶜ); the false positive rate is its complement.

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Bayes’ theorem — frequently asked questions

Why is the answer so low?

Because 99% of people are healthy, and 5% of them test positive — that is 4.95% of the population, versus 0.99% true positives. Sheer numbers dominate.

What are prior and posterior?

The prior P(A) is the probability before the evidence; the posterior P(A|B) is after. Bayes is the rule for updating one into the other.

How do I handle repeated tests?

Use the posterior from the first test as the prior for the second, assuming conditional independence given the true state. Two independent positives raise the probability sharply.

The rest of Probability

Events, conditional probability, random variables. Each subtopic below has its own method, worked example and mark-losing traps.

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