Bayes' Theorem, Explained Without the Fear
Updated 2026-07-12
Bayes' theorem updates a probability when you get new evidence. It relates the probability of a cause given the evidence to the probability of the evidence given the cause. In words: posterior equals prior times likelihood, divided by the total probability of the evidence.
Bayes' theorem feels abstract until you see it as one idea: how should a belief change after new evidence? The formula is just bookkeeping for that update, and the classic medical-test example makes it concrete.
The formula in words
The probability of A given B equals the probability of B given A, times the probability of A, divided by the probability of B. The starting belief is the prior, the update factor is the likelihood, and the result is the posterior.
Why the base rate matters
The most counterintuitive part is that a very accurate test can still give a mostly wrong positive, if the condition is rare. The rarity, the base rate, dominates. Bayes' theorem is what forces you to include it instead of ignoring it.
Worked example
A disease affects 1 percent of people. A test is 99 percent accurate. You test positive. What is the chance you have the disease?
- Out of 10,000 people, 100 have the disease and 9,900 do not.
- Of the 100 sick, 99 test positive. Of the 9,900 healthy, 1 percent, which is 99, also test positive.
- So 99 true positives and 99 false positives, 198 positives in total.
Answer: The chance is 99 out of 198, which is 50 percent, not 99 percent.
The common mistake
Wrong: Assuming a 99 percent accurate test means a positive is 99 percent likely to be real.
Fix: You must weigh in the base rate. When the condition is rare, false positives from the large healthy group can equal the true positives, so the real probability is far lower.
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Frequently asked questions
What is Bayes' theorem in simple terms?
It is a rule for updating a probability when you get new evidence. You start with a prior belief, multiply by how likely the evidence is under that belief, and divide by the overall probability of the evidence.
Why can an accurate test still be wrong?
Because of the base rate. If a condition is rare, the small false-positive rate applied to the large healthy population can produce as many false positives as true positives, so a positive result is far less certain than the accuracy number suggests.