Bayes' theorem is the rule for updating a belief when new evidence arrives. It rewrites a conditional probability by swapping what is given for what is unknown:

Read it as a flow of belief about a hypothesis H after seeing data D:

• P(H) — the prior: what you believed before.
• P(D \mid H) — the likelihood: how well H predicts the data.
• P(H \mid D) — the posterior: your updated belief.
• P(D) — the evidence: a normalizing constant.

In words: posterior ∝ likelihood × prior. This one proportionality is the backbone of all the Bayesian reasoning ahead.

The base-rate surprise

Expanding the evidence with the law of total probability, for a yes/no hypothesis,

The famous lesson: a very accurate test for a very rare condition still gives mostly false alarms, because the tiny prior drags the posterior down. The prior is not optional — it is half the answer.

How the prior steers the posterior

The curve is the posterior P(H\mid D) as a function of the prior P(H), for a test with sensitivity P(D\mid H) and false-positive rate P(D\mid\lnot H). The faint diagonal is "no update". Set the test's accuracy and read off the surprise: when the prior is small (left edge), even a strong positive leaves the posterior low.