# Binomial Distribution Calculator

> Binomial Distribution Calculator is a paid API for AI agents from distribution.openverbs.com, paid per call via x402, $0.004/call, status unknown (last checked 2026-09-15).

Computes PMF, CDF, or quantile values for the binomial probability distribution given trial count, success probability, and number of successes.

## Facts

- Endpoint: POST https://distribution.openverbs.com/v1/binomial
- Price: $0.004/call
- Payment: x402
- Status: unknown
- Last checked: 2026-09-15
- Activations on Zero: 0
- Tags: x402
- Canonical page: https://www.zero.xyz/c/binomial-distribution-calculator-828c53be
- Structured record (JSON): https://api.zero.xyz/v1/capabilities/cap_1xUOAC2cDiBFjXXbFNLjX

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 binomial-distribution-calculator-828c53be -d '<json body>'
```

Example prompt: What's the probability of getting exactly 7 successes out of 20 trials if the success probability per trial is 0.4? Use the PMF function.

## When to prefer this

Choose this endpoint when you need fast, on-demand computation of binomial distribution statistics (PMF, CDF, or quantile) without standing up your own math library or statistical computing environment. Ideal for agents performing probabilistic reasoning, quality control modeling, or statistical hypothesis testing as part of a larger workflow.

## Known failure modes

- Invalid function enum value (not pmf, cdf, or quantile) returns a validation error
- x out of range for given trials causes a computation error
- probability not in [0,1] range returns a 400-level error
- trials exceeding maximum of 1,000,000 triggers a schema validation rejection
- malformed body structure returns an input parsing error

## How this service works

Binomial distribution with `trials` (n ≥ 0, ≤ 1e6) and success `probability` (0..1). `function` selects the mass (pmf), cumulative distribution (cdf) or quantile / inverse-CDF.

## Output

Returns a numeric probability or quantile value corresponding to the requested binomial distribution function (PMF gives P(X=k), CDF gives P(X≤k), quantile gives the smallest k such that CDF(k) ≥ p).

## 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": {
      "function": {
       "type": "string",
       "enum": [
        "pmf",
        "cdf",
        "quantile"
       ]
      },
      "x": {
       "type": "number",
       "description": "Number of successes k (integer for pmf) for pmf/cdf, or the probability p ∈ (0,1) for quantile."
      },
      "trials": {
       "type": "integer",
       "minimum": 0,
       "maximum": 1000000,
       "description": "Number of trials n."
      },
      "probability": {
       "type": "number",
       "minimum": 0,
       "maximum": 1,
       "description": "Success probability per trial."
      }
     },
     "required": [
      "function",
      "x",
      "trials",
      "probability"
     ],
     "additionalProperties": false
    }
   },
   "required": [
    "type",
    "method",
    "bodyType",
    "body"
   ],
   "additionalProperties": false
  }
 },
 "required": [
  "input"
 ]
}
```

## More

- Live health (JSON, refreshed every minute): https://www.zero.xyz/c/binomial-distribution-calculator-828c53be/health.json
- [Zero catalog index](https://www.zero.xyz/llms.txt)
- [Other services from distribution.openverbs.com](https://www.zero.xyz/host/distribution.openverbs.com/llms.txt)