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Bayes' theorem in plain words: how evidence should change your mind

Last updated 10 November 2026

By Plax Labs · 4 min read · Published 10 November 2026

A test for a rare disease is 90 per cent accurate, and your result is positive. How likely is it that you have the disease? Most people say about 90 per cent. The correct answer can be less than 10 per cent, and understanding why is the heart of an idea named after the eighteenth-century English minister Thomas Bayes.

Bayes' theorem is a rule for how to change your mind in the light of evidence. It says that the answer depends not only on how good the evidence is, but also on how likely the thing was to be true before you saw it.

The rule

In symbols, the probability of A given that B has been observed is the probability of B given A, times the prior probability of A, divided by the overall probability of B.

P(A | B) = P(B | A) x P(A) / P(B)

A = the claim (for example, you have the disease)
B = the evidence (for example, a positive test)

A worked example

Suppose a disease affects 1 in 100 people. A test correctly detects it in 90 per cent of people who have it, and wrongly gives a positive result to 10 per cent of people who do not. Take 1,000 people. About 10 have the disease and 9 of them test positive. Of the 990 who do not, about 99 test positive by mistake. In all, 108 people test positive, and only 9 of them are really ill. A positive result therefore means about a 1 in 12 chance of having the disease, roughly 8 per cent.

The test is good, and the result is still more likely to be a false alarm than a true positive, because the disease is rare and the healthy are many. If the same test were used on people who already had symptoms, so that the disease was far more common among them, the same positive result would be much more convincing.

1,000 people tested
  10 have the disease    -> 9 test positive
 990 do not             -> 99 test positive (false alarms)
All positives: 9 + 99 = 108
Chance a positive is real: 9 / 108 = about 8%

Why it matters outside the clinic

The same reasoning applies whenever something rare is screened for. A system that flags suspicious transactions, faces in a crowd or messages that look like spam will produce many false alarms if the thing it is looking for is uncommon, even when the system is very accurate. Spam filters use Bayes' theorem directly: the presence of a word such as lottery raises the probability that a message is spam, because it is far more common in spam than in genuine mail.

In news stories, base rates are often missing. A report that a test, a camera or an algorithm is 99 per cent accurate says little until you know how common the thing being detected is.

Who was Bayes?

Thomas Bayes was a Presbyterian minister and mathematician who died in 1761. His essay on the problem was found among his papers and published in 1763 by his friend Richard Price. The French mathematician Pierre-Simon Laplace later developed the idea independently and made it a general tool. For much of the twentieth century it was overshadowed by other methods of statistics, and it is now widely used in medicine, machine learning and many other fields.

Common mistakes, and a habit

The classic error is to confuse the probability of the evidence given the claim with the probability of the claim given the evidence. A court that hears that the chance of a match by coincidence is one in a million may conclude that the chance of innocence is also one in a million, which does not follow; it is known as the prosecutor's fallacy. Overconfidence in a single piece of evidence is another.

A useful habit is to ask two questions whenever new evidence arrives. How common was this to begin with? And how often would I see this evidence even if the claim were false?

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Written by Plax Labs with the help of AI writing tools, and updated 10 November 2026. Found a mistake or something out of date? Tell us, and read our editorial policy. Read Mathematics stories on Plax.