Law of total probability

A probability identity expressing an event's probability as the sum or integral of its conditional probabilities over a mutually exclusive exhaustive partition.

Version

v2 · 2026-09-19 · History

Domain-specific #

5277

Domain group

Formal Sciences

Origin domain

Mathematics

Subdomains

Probability, Conditioning Identities → Mathematics

Core Idea

For a partition with positive-probability cells, P(A)=sum_i P(A|B_i)P(B_i), with conditional-expectation analogues in general settings. The partition decomposes A into disjoint intersections A∩B_i; countable additivity and the definition of conditional probability produce the weighted sum. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

The load-bearing residual is not the broad topic of probability. It is marginalization identity assembling a total probability from conditional cases. That residual remains recognizable when examples, notation, scale, or implementation change, but it disappears if the carrier is mistyped, the condition that partition events are disjoint and exhaustive and zero-probability conditioning is treated through an appropriate regular conditional framework fails, a neighboring object is substituted, or notation and topical resemblance replace the constitutive test.

How would you explain it like I'm…

Add Up Every Way

Say a toy is hidden in one of two boxes, and you know how likely each box is, and how likely you'd find the toy if you looked in each. To find the total chance, you figure out each box's share and add the shares together. The boxes can't overlap, and together they have to cover every possibility.

Chance by Cases

The law of total probability helps you find the chance of something by breaking it into cases. First, split all the possibilities into separate cases that don't overlap and that cover everything — like 'sunny', 'cloudy' or 'rainy' days. For each case, multiply the chance of that case by the chance your event happens in that case. Then add up all the answers. Because the cases don't overlap and cover everything, nothing is counted twice or left out.

Weighted Sum Over a Partition

The law of total probability says that if events B_1, B_2, ... form a partition — they don't overlap and together cover every outcome — and each has positive probability, then P(A) = sum of P(A | B_i)·P(B_i). It works because A splits into the disjoint pieces A∩B_i, and each piece has probability P(A | B_i)·P(B_i) by the definition of conditional probability. This is a weighted average of conditional probabilities, with weights P(B_i). It is often the denominator in Bayes' theorem. If the cases overlap or miss some outcomes, the formula gives the wrong answer.

 

The law of total probability is a marginalization identity. For a finite or countable partition {Bᵢ} of the sample space with P(Bᵢ) > 0, P(A) = Σᵢ P(A|Bᵢ)P(Bᵢ). The proof writes A as the disjoint union of A∩Bᵢ, applies countable additivity, and substitutes P(A∩Bᵢ) = P(A|Bᵢ)P(Bᵢ). The partition conditions are essential, since overlapping cells double-count and non-exhaustive cells miss mass. In general settings the analogue is P(A) = E[P(A|G)] for a σ-algebra G, or an integral over a continuous variable. Conditioning on zero-probability events requires a regular conditional probability framework rather than the elementary ratio definition. It is the mechanism behind marginalizing out latent variables and computing Bayes denominators.

Scope of Application

Law of total probability belongs to probability and is useful where the analyst can specify probability space, event A, finite, countable or continuous partition B_i, conditional probabilities P(A|B_i), weights P(B_i) and measurability conditions, then evaluate partition events are disjoint and exhaustive and zero-probability conditioning is treated through an appropriate regular conditional framework. The scope is broad within that domain but bounded by the need for partition events are disjoint and exhaustive and zero-probability conditioning is treated through an appropriate regular conditional framework. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity

The abstraction clarifies a crowded vocabulary by making partition events are disjoint and exhaustive and zero-probability conditioning is treated through an appropriate regular conditional framework the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Law of total probability can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Law of total probability. Law of total probability compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning

1. Identify the carrier. State what the elements, states, objects, or observations are: probability space, event A, finite, countable or continuous partition B_i, conditional probabilities P(A|B_i), weights P(B_i) and measurability conditions. Reject examples whose alleged carrier belongs to a different problem.
2. Lock the constitutive rule. Express partition events are disjoint and exhaustive and zero-probability conditioning is treated through an appropriate regular conditional framework independently of one notation or implementation.

Knowledge Transfer

Knowledge transfers strongly among subfields of probability because they reuse probability space, event A, finite, countable or continuous partition B_i, conditional probabilities P(A|B_i), weights P(B_i) and measurability conditions, The partition decomposes A into disjoint intersections A∩B_i; countable additivity and the definition of conditional probability produce the weighted sum., and type the carrier, state every parameter and convention in the definition, test that partition events are disjoint and exhaustive and zero-probability conditioning is treated through an appropriate regular conditional framework, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions

Current abstraction Law of total probability Domain-specific

Parents (1) — outgoing typed relations

• Law of total probability is a kind of Composition Prime

  The law composes disjoint conditional cases into a marginal probability.

Hierarchy path (1) — routes to 1 parentless root

• Law of total probability → is a kind of Composition → presupposes Gestalt Principles → is a decomposition of Holism

Neighborhood in Abstraction Space

Law of total probability sits in a crowded region of the domain-specific corpus (24^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Measure & Probability Foundations (42 abstractions)

Nearest neighbors

• Probability axioms — 0.90
• Unit measure — 0.90
• Complementary event — 0.90
• Discrete-time Markov chain — 0.89
• Continuous-time stochastic process — 0.89

Computed from structural-signature embeddings · 2026-10-25