Probability on finite sample spaces

Mathematical statement

Let S be a nonempty finite sample space and F=𝒫(S) its event family, where 𝒫 denotes the full powerset and P denotes the probability law. Assign rational weights w_s≥0 to all s∈S with Σ_(s∈S)w_s=1. For E⊆S, P(E)=Σ_(s∈E)w_s. The formula P(E)=|E|/|S| applies only when all outcomes are explicitly equiprobable.

Representative formula (exact TeX)

P(E)

Assumptions

• S is finite and nonempty, its elementary outcomes are distinct, and the event family is the full powerset.
• Weights are exact nonnegative rational numbers summing exactly to one, and E must be a subset of S.

Symbol scope

• P — finite probability law
• E — event subset of the sample space

Intuition, definition, and misconceptions

First validate every weight and their total, then sum the weights of the event outcomes. Event cardinality is not a probability in general; elementary outcomes may have different weights.

P(∅)=0 and P(S)=1. The complement is taken relative to the same S, with P(S∖E)=1−P(E). A nonempty event can have probability zero when all its outcomes have zero weight.

The exercises use explicit integer outcome labels and rational weights with a common denominator for exact calculation and model-validity checks. They do not infer real-world experimental probabilities, establish independence or conditional probability, or extend to a general measure-theoretic definition.

Common errors include dividing favorable counts by total counts without equiprobability, or inferring that E is empty from P(E)=0.

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

• Event — Fix the sample space and event subsets, keeping outcomes distinct from events.
• Ratio and proportion — Equiprobable counts and rational-weight arithmetic require valid ratios with nonzero denominators.

Verification boundary

Explicit finite rational weights can be recomputed completely; formal Evidence for general probability measures and countable additivity has not been established.

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 PMFs, rational-weight normalization, event sums, and relative complements; use a separate semantic bridge for the measure layer.

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.