Conditional probability in finite models

Mathematical statement

In a nonempty finite sample space S with event family F=𝒫(S) and normalized nonnegative rational weights, let A,B⊆S with P(B)>0. Define C(A,B)=P(A|B)=P(A∩B)/P(B). If P(B)=0, this entry leaves that conditional probability undefined, even when B is nonempty.

Representative formula (exact TeX)

C(A,B)

Assumptions

• The finite weighted model is valid, and A and B are subsets of S.
• The denominator P(B) must be strictly positive. An equiprobable model requires a separate explicit declaration.

Symbol scope

• C — ordered conditional probability
• A — event being measured
• B — conditioning event with positive probability

Intuition, definition, and misconceptions

Compute the probability mass of B and then that of A∩B before taking their ratio. The common positive denominator cancels, after the model and event eligibility have been checked.

Nonempty B does not imply P(B)>0: if all its outcomes have zero weight, it still cannot be a conditioning event here. Only an equiprobable finite model gives the count ratio |A∩B|/|B|, with B nonempty.

P(A|B) and P(B|A) generally differ, and each direction needs its own positive denominator. Their numerical equality or inequality alone does not establish independence. Conditional probability is not computed by multiplying P(A) and P(B).

Common errors include treating nonemptiness as positive probability, assigning zero to an undefined zero-probability condition, or confusing the two directions.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Probability on finite sample spaces — First specify normalized weights and event probabilities.
• Set operations — State the set semantics of A∩B and event restrictions.

Verification boundary

Finite rational models can be recomputed exactly. General regular conditional probability, zero-probability conditioning, Bayes theorems, and independence theorems are outside this pulse.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Bind finite event intersections, conditioning with a positive denominator, and rational mass ratios; zero-probability conditioning has separate semantic obligations.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.