Conditional probability + Bayes’ rule

The single most important idea in modern AI is updating beliefs in light of evidence.

Conditional probability

P(A | B) reads “the probability of A, given that B has happened.”

Example: P(rain | dark clouds) is much higher than P(rain).

Conditioning on evidence is how all modern AI thinks.

Bayes’ rule

P(A | B) = P(B | A) · P(A) / P(B)

In plain English: your belief in A after seeing B equals your prior belief in A, scaled by how well A explains B.

Total probability

When several paths can produce the same evidence, account for all of them before updating.

Every learning system you will hear about is a Bayes-rule descendant.

Applied frame

Translate the claim into events, probabilities, distributions, and simulation checks before trusting a single answer.

Diagram cue

The diagram separates event space, probability mass, evidence, and update steps.