Conditional Probability

Checkpoints

• Bayes' Formula Essential Bayes' formula inverts a conditional probability: given P(B | A_i) for each cell A_i of a partition and the prior probabilities P(A_i), the posterior P(A_i | B) is P(B | A_i) P(A_i) / Σ_j P(B | A_j) P(A_j). This checkpoint derives the formula from the multiplication rule and the law of total probability, frames it as the foundation of Bayesian inference (prior × likelihood ∝ posterior), and works through the canonical diagnostic-test example to show how base rates dominate intuitively surprising answers.
• Conditional Expectation Essential The conditional expectation E[X | B] is the expectation of X under the conditional probability P(· | B); more generally, E[X | Y] is the random variable that takes value E[X | Y = y] when Y = y. This checkpoint defines conditional expectation in the discrete and absolutely continuous cases, presents it as a function of the conditioning variable, and develops the linearity, monotonicity, and 'taking out what is known' properties that make it the natural projection onto the σ-algebra generated by Y.
• Conditional Probability Essential The conditional probability of A given B is P(A | B) = P(A ∩ B) / P(B), defined whenever P(B) > 0. This checkpoint motivates the definition as the renormalised probability on the restricted sample space B, derives the multiplication rule P(A ∩ B) = P(A | B) P(B), and proves the law of total probability via a partition of Ω.
• Independence of Events Essential Two events A and B are independent when P(A ∩ B) = P(A) P(B) — equivalently, when P(A | B) = P(A) for events with P(B) > 0. This checkpoint defines independence of two events, generalises it to pairwise and mutual independence of arbitrary collections, illustrates with examples why pairwise independence is strictly weaker than mutual, and contrasts independence with the (similar-sounding but distinct) notion of being mutually exclusive.
• Law of Total Expectation Essential The law of total expectation states E[X] = E[E[X | Y]] — averaging the conditional expectation over the conditioning variable recovers the unconditional expectation. This checkpoint proves the identity in the discrete and absolutely continuous cases, presents the analogous law of total variance Var(X) = E[Var(X | Y)] + Var(E[X | Y]), and applies the technique to compute expectations by conditioning on an auxiliary variable that simplifies the problem.