How to Apply Bayes' Theorem

Finds the probability of really having a condition after testing positive. From the prior P(A), the true positive rate P(B|A) and the false positive rate P(B|Ā), it calculates P(A|B).

Explanation

Bayes' theorem updates a probability in the light of new evidence. Its most famous use is working out how likely you are to be ill after testing positive.

P(A \mid B) = \dfrac{P(A)\,P(B \mid A)}{P(A)\,P(B \mid A) + P(\bar{A})\,P(B \mid \bar{A})}

$P(A)$ — the prior, the chance of being ill before testing (the prevalence)
$P(B \mid A)$ — the chance an ill person tests positive (the sensitivity)
$P(B \mid \bar{A})$ — the chance a healthy person tests positive (the false positive rate)

The denominator collects everyone who tests positive: the ill who are correctly flagged plus the healthy who are wrongly flagged.

Example

A disease affects 1% of people ( $P(A) = 0.01$ ). The test catches 99% of cases and wrongly flags 5% of healthy people. You test positive.

P(B) = 0.01 \times 0.99 + 0.99 \times 0.05 = 0.0099 + 0.0495 = 0.0594

P(A \mid B) = \dfrac{0.0099}{0.0594} = 0.1667

Despite the positive result, the chance of actually being ill is only 16.7%.

Why so low

Imagine 10,000 people. About 100 are ill, and 99 of them test positive. Of the 9,900 healthy people, 5% — that is 495 — also test positive.

That makes 594 positive results, of which only 99 are genuine: $99 \div 594 = 16.7\%$ .

Because the disease is rare, the false positives among the healthy majority simply outnumber the true positives. Even an accurate test produces mostly false alarms when screening for something uncommon. This is the base rate fallacy.