Bayes Theorem Formula: What It Is and How It Works

Conditional probability answers a precise question: how likely does an event become once we know that another one has already happened? It is written P(A|B), where the vertical bar reads “given that”, and it is found by dividing the probability that A and B both occur by the probability of B taken on its own. The idea comes up as soon as you learn how to calculate probability, and it often leans on counting rules from combinatorics, because counting the favorable and the possible outcomes is the first step toward any conditional probability.

Bayes theorem exists to flip that conditional probability around. According to the Stanford Encyclopedia of Philosophy, the theorem starts from how likely a given piece of evidence is if a hypothesis were true — P(evidence|hypothesis) — and arrives at the opposite probability: how likely the hypothesis is, given that evidence — P(hypothesis|evidence). These are two different conditional probabilities: knowing how likely a symptom is if someone is sick does not automatically tell you how likely someone is to be sick if they have that symptom.

The formula, in the notation used by Wikipedia, is:

P(A|B) = P(B|A) · P(A) / P(B)

Here A is the hypothesis being tested and B is the evidence observed. Using it takes three ingredients: the known conditional probability P(B|A), the “starting” probability of the hypothesis P(A), and the overall probability of the observed evidence P(B).

Conditional probability in a diagnostic test

The most common way to grasp Bayes theorem is through a medical test. A real application, cited by the Stanford Encyclopedia of Philosophy, is screening: a test’s sensitivity and specificity — its rates of true positives and true negatives — combined with how common the disease is in the tested population, let you work out how likely someone really is to be sick when the test comes back positive. The same logic runs through plenty of problems in probability and statistics where an observed result has to be traced back to a cause.

Worked example: picture, as in Lumen Learning / OpenStax’s example, a sample of 1,000 people with a disease incidence of 0.1%. That means expecting only 1 person in the sample to be truly sick, with the other 999 healthy. Say the test has a 5% false-positive rate: applied to the 999 healthy people, it wrongly flags about 50 of them as positive. The test ends up producing roughly 51 positive results in total — the 50 false alarms plus, in most cases, the one person who is truly sick. Out of those 51 positives, only 1 is truly sick: the probability of actually being sick given a positive result works out to about 1/51, around 2%.

A positive result instinctively sounds like near-certain illness. The test is run on a huge number of healthy people — 999 out of 1,000 — so even a small error margin, applied to such a large group, ends up producing more wrong flags than real cases. It is what happens when a disease is rare in the population being tested, and working that out with real numbers rather than gut feeling is exactly what Bayes theorem is for.

A bit of history, with one uncertain date

According to the Stanford Encyclopedia of Philosophy, the English minister Thomas Bayes was the first to grasp the theorem’s importance, in an essay published after his death titled “An Essay Toward Solving a Problem in the Doctrine of Chances,” dated 1764. Wikipedia instead gives 1763 as the publication year, with the title spelled “Towards” instead of “Toward”: the two sources disagree on the publication year.

Bayes never published the work during his life. After his death, his family passed the papers to his friend Richard Price, a minister and philosopher with a solid grounding in mathematics. Price spent two years editing the unpublished manuscript before it was read to the Royal Society on 23 December 1763. A few years later, in 1774, the French mathematician Pierre-Simon Laplace arrived at the same result on his own, unaware of Bayes’s earlier essay.

Readers who want to review the basics of number sets that probability and statistics build on will find a good complementary starting point there.