# Probability Rules, Complements, Unions, and Intersections

`D00-F06-A02` · Financial Mathematics, Statistics, and Data Foundations → Probability and Random Variables · archetype `record-transform` · difficulty 1/5 · verification **verified**

Full page: https://docs.thefintechbuilder.com/foundations/probability-and-random-variables/probability-rules-complements-unions-and-intersections/
Agent skill: `npx skills add IslamBaraka90/Fintech-Algorithms-Library` — https://docs.thefintechbuilder.com/guides/agent-skill/

## Install and import

```bash
npm install fintech-algorithms
```

```ts
import { probabilityRulesComplementsUnionsAndIntersections } from "fintech-algorithms/foundations/probability-and-random-variables/probability-rules-complements-unions-and-intersections";
```

## Signature

```ts
probabilityRulesComplementsUnionsAndIntersections(input)
```

Applies the complement and addition rules to a pair of events, deriving the probability that A does not happen and the probability that at least one of A or B happens.

## Parameters

| Name | Type | Required | Notes |
| --- | --- | --- | --- |
| `input` | `{ pA: number; pB: number; pAB: number }` | yes | `pA` and `pB` are the marginal probabilities of the two events and `pAB` the probability of both together. The joint probability is taken as given, not inferred, so it must be consistent with the two marginals. · pA: 0 <= pA <= 1, pB: 0 <= pB <= 1, pAB: 0 <= pAB <= min(pA, pB) |

## Returns

`{ complementA: number; union: number; intersection: number }`

`complementA` is `1 - pA`, `union` is `pA + pB - pAB`, and `intersection` echoes the supplied `pAB` so all three rule terms sit side by side.

## Errors

- When `input` is null, an array, or not an object — throws TypeError
- When `pA`, `pB`, or `pAB` falls outside [0, 1], or `pAB` exceeds `min(pA, pB)` — throws RangeError

## Complexity

Time `O(1)`, space `O(1)`.

## Worked example

This is the worked example published in the article, replayed by the test suite on every run. The output cannot drift.

### Input

`input`:

```json
{
  "pA": 0.6,
  "pB": 0.5,
  "pAB": 0.3,
  "outcomes": ["up", "flat", "down"],
  "event": ["up", "flat"],
  "prior": 0.01,
  "sensitivity": 0.9,
  "falsePositiveRate": 0.05,
  "randomValues": [0, 1, 2],
  "probabilities": [0.2, 0.5, 0.3],
  "randomVariableKind": "discrete",
  "joint": [
    {
      "x": 0,
      "y": 0,
      "p": 0.3
    },
    {
      "x": 0,
      "y": 1,
      "p": 0.2
    },
    {
      "x": 1,
      "y": 0,
      "p": 0.1
    }
  ],
  "conditionY": 1
}
```

### Call

```ts
probabilityRulesComplementsUnionsAndIntersections(input)
```

### Returns

object with 2 fields: complementA, union

```json
{
  "complementA": 0.4,
  "union": 0.8
}
```

## Verification and provenance

Tier: **verified** (via input-expected).

The worked example below is the figure published in this algorithm's article, replayed and asserted by the test suite on every build. The arithmetic cannot drift without the build failing.

Both tiers guarantee the signature. Full explanation: https://docs.thefintechbuilder.com/guides/verification/

Generated from the docs.json payload shipped inside fintech-algorithms@0.13.0.
The signature and parameter list are checked against the compiled implementation at build time,
so a description that contradicts the code fails the build rather than reaching this file.

## Links

- Article (how it works, step by step): https://thefintechbuilder.com/foundations/probability-and-random-variables/probability-rules-complements-unions-and-intersections/
- Implementation source: https://github.com/IslamBaraka90/Fintech-Algorithms-Library/blob/main/src/foundations/probability-and-random-variables/probability-rules-complements-unions-and-intersections/impl.ts
- Package on npm: https://www.npmjs.com/package/fintech-algorithms
- Domain index for agents: https://docs.thefintechbuilder.com/foundations/llms.txt