# Dice Notation Exact Probability Distribution

> Dice Notation Exact Probability Distribution is a paid API for AI agents from dice.openverbs.com, paid per call via x402, $0.002/call, status unknown (last checked 2026-09-14).

Computes the exact probability distribution (mean, variance, standard deviation, min/max, full PMF) of any dice notation string without sampling.

## Facts

- Endpoint: POST https://dice.openverbs.com/v1/stats
- Price: $0.002/call
- Payment: x402
- Status: unknown
- Last checked: 2026-09-14
- Activations on Zero: 0
- Tags: x402
- Canonical page: https://www.zero.xyz/c/dice-notation-exact-probability-distribution-2ae9a9ef
- Structured record (JSON): https://api.zero.xyz/v1/capabilities/cap_aFxr-aKb9qYgO0AFYtjx3

Status and success rate cover calls made through Zero and Zero's own probes. Third-party monitors may report differently.

## How to call it through Zero

Zero handles the 402 payment challenge and records the run. With the Zero CLI installed (`npm i -g @zeroxyz/cli`):

```sh
zero fetch --capability dice-notation-exact-probability-distribution-2ae9a9ef -d '<json body>'
```

Example prompt: What's the exact probability distribution for 4d6 drop lowest 1 — give me the mean, variance, standard deviation, and the full probability mass function?

## When to prefer this

Choose this endpoint when you need mathematically exact probabilities rather than sampled approximations — ideal for game balance analysis, academic study, or any context where precision matters. Prefer it over simulation-based approaches when the outcome space is manageable. Use the sibling roll endpoint if you need actual random dice rolls, and the sibling parse endpoint if you only need to decompose the notation structure.

## Known failure modes

- Dice notation produces an outcome range too large for exact computation — returns a clean 400 error
- Invalid or unparseable dice notation string — returns a 400 validation error
- Missing required 'notation' field — returns a 400 schema error
- Network or service timeout — returns a 5xx error

## How this service works

Compute the exact probability distribution of a dice notation — mean, variance, standard deviation, min/max and the full probability mass function — with no sampling. Notations whose outcome range is too large for exact computation are clean 400s.

## Output

Returns a JSON object containing the mean, variance, standard deviation, minimum, maximum, and optionally the full probability mass function (a mapping from each possible outcome integer to its exact probability) for the given dice notation. If the outcome range is too large for exact computation, returns a 400 error.

## Request schema (JSON Schema)

```json
{
 "$schema": "https://json-schema.org/draft/2020-12/schema",
 "type": "object",
 "properties": {
  "input": {
   "type": "object",
   "properties": {
    "type": {
     "type": "string",
     "const": "http"
    },
    "method": {
     "type": "string",
     "enum": [
      "POST"
     ]
    },
    "bodyType": {
     "type": "string",
     "enum": [
      "json",
      "form-data",
      "text"
     ]
    },
    "body": {
     "type": "object",
     "properties": {
      "notation": {
       "type": "string",
       "minLength": 1,
       "description": "Dice notation to analyse."
      },
      "distribution": {
       "type": "boolean",
       "description": "Include the full probability mass function. Default true."
      }
     },
     "required": [
      "notation"
     ],
     "additionalProperties": false
    }
   },
   "required": [
    "type",
    "method",
    "bodyType",
    "body"
   ],
   "additionalProperties": false
  }
 },
 "required": [
  "input"
 ]
}
```

## More

- Live health (JSON, refreshed every minute): https://www.zero.xyz/c/dice-notation-exact-probability-distribution-2ae9a9ef/health.json
- [Zero catalog index](https://www.zero.xyz/llms.txt)
- [Other services from dice.openverbs.com](https://www.zero.xyz/host/dice.openverbs.com/llms.txt)