# Random Sampling and Monte Carlo Intuition

`D00-F07-A10` · Financial Mathematics, Statistics, and Data Foundations → Probability Distributions and Simulation Basics · archetype `record-transform` · difficulty 1/5 · verification **verified**

Full page: https://docs.thefintechbuilder.com/foundations/probability-distributions-and-simulation-basics/random-sampling-and-monte-carlo-intuition/
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 { randomSamplingAndMonteCarloIntuition } from "fintech-algorithms/foundations/probability-distributions-and-simulation-basics/random-sampling-and-monte-carlo-intuition";
```

## Signature

```ts
randomSamplingAndMonteCarloIntuition(input)
```

Estimates the probability `p` by counting how many of `sampleCount` seeded uniform draws land at or below it, and reports how far that Monte Carlo estimate lands from the known answer.

## Parameters

| Name | Type | Required | Notes |
| --- | --- | --- | --- |
| `input` | `D00Input` | yes | One record carrying `seed` and `sampleCount` for the draw, and `p`, which is both the probability being estimated and the target the estimate is scored against. `sigma` is validated by the family entry point before dispatching. · sampleCount: 1 or greater, sigma: strictly positive, p: between 0 and 1 inclusive |

## Returns

`D00Output`

An object with `estimate` (share of draws at or below `p`), `target` (echoes `p`), `absoluteError` (the gap between the two) and `sampleCount` (the number of draws actually used).

## Errors

- When p is missing or outside 0 to 1 — this check runs for every topic in the family — throws RangeError
- When sigma is zero or negative — this check runs for every topic from A05 onward — throws RangeError
- When sampleCount is below 1 — throws RangeError

## Complexity

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

## 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
{
  "values": [0.2, 0.5, 0.7, 1, 1.4],
  "x": 1,
  "p": 0.3,
  "n": 5,
  "k": 2,
  "lambda": 2,
  "seed": 42,
  "sampleCount": 8,
  "mu": 0,
  "sigma": 1,
  "df": 5,
  "shape": 2,
  "scale": 1.5,
  "components": [
    {
      "weight": 0.7,
      "mean": 0,
      "sd": 1
    },
    {
      "weight": 0.3,
      "mean": 3,
      "sd": 0.8
    }
  ]
}
```

### Call

```ts
randomSamplingAndMonteCarloIntuition(input)
```

### Returns

object with 2 fields: estimate, target

```json
{
  "estimate": 0.625,
  "target": 0.3
}
```

## 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-distributions-and-simulation-basics/random-sampling-and-monte-carlo-intuition/
- Implementation source: https://github.com/IslamBaraka90/Fintech-Algorithms-Library/blob/main/src/foundations/probability-distributions-and-simulation-basics/random-sampling-and-monte-carlo-intuition/impl.ts
- Package on npm: https://www.npmjs.com/package/fintech-algorithms
- Domain index for agents: https://docs.thefintechbuilder.com/foundations/llms.txt