# Saylor Innovations Data Desk – Probability Distributions

> Saylor Innovations Data Desk – Probability Distributions is a paid API for AI agents from saylorinnovations.com, paid per call via x402, $0.01/call, status unknown (last checked 2026-10-23).

Computes binomial, Poisson, and normal distribution probabilities including exact, cumulative, range, z-score, and quantile calculations given distribution parameters

## Facts

- Endpoint: GET https://saylorinnovations.com/api/math/probability?utm_source=zero.xyz
- Price: $0.01/call
- Payment: x402
- Status: unknown
- Last checked: 2026-10-23
- Activations on Zero: 0
- Tags: x402
- Canonical page: https://www.zero.xyz/c/saylor-innovations-data-desk-probability-distributions-ead85f25
- Structured record (JSON): https://api.zero.xyz/v1/capabilities/cap_BU04YGIuVAXIM4wKpIrmd

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 saylor-innovations-data-desk-probability-distributions-ead85f25
```

Example prompt: What's the probability of getting exactly 3 heads in 10 coin flips, where each flip has a 0.5 chance of heads? Use a binomial distribution and give me the exact, at-most, and at-least probabilities.

## When to prefer this

Choose this endpoint when you need precise statistical probability calculations for binomial, Poisson, or normal distributions — including cumulative (CDF), exact (PMF/PDF), and tail probabilities — without needing to run local math libraries. Ideal for agents performing quality control analysis, risk assessment, hypothesis testing setup, or educational statistics tasks that require on-demand, accurate probability values at low cost.

## Known failure modes

- Missing required query parameters (dist, n, p, k) returns an error
- Invalid distribution type string (not 'binomial', 'poisson', or 'normal') causes a failure
- Out-of-range probability value for p (not between 0 and 1) may produce errors
- Non-integer or negative k or n values may cause computation errors
- Network or payment (x402) failures result in no response

## How this service works

Binomial, Poisson and normal distribution probabilities - exact, at most, at least, ranges, z-scores and quantiles with mean and standard deviation

## Output

Returns a JSON object with the distribution name, mean, variance, standard deviation, mode, and four cumulative/exact probability values: P(exactly k), P(at most k), P(at least k), P(more than k). Also includes the input k and n values, a source label, and an explanatory note for interpreting the results.

## Request schema (JSON Schema)

```json
{
 "type": "object",
 "$schema": "https://json-schema.org/draft/2020-12/schema",
 "required": [
  "input"
 ],
 "properties": {
  "input": {
   "type": "object",
   "required": [
    "type",
    "method"
   ],
   "properties": {
    "type": {
     "type": "string",
     "const": "http"
    },
    "method": {
     "enum": [
      "GET"
     ],
     "type": "string"
    },
    "queryParams": {
     "type": "object",
     "required": [
      "dist",
      "n",
      "p",
      "k"
     ],
     "properties": {
      "k": {
       "type": "string",
       "description": "number of successes or events"
      },
      "n": {
       "type": "string",
       "description": "binomial trials"
      },
      "p": {
       "type": "string",
       "description": "binomial success probability (or quantile probability for normal)"
      },
      "dist": {
       "type": "string",
       "description": "binomial, poisson or normal"
      }
     }
    }
   },
   "additionalProperties": false
  },
  "output": {
   "type": "object",
   "required": [
    "type"
   ],
   "properties": {
    "type": {
     "type": "string",
     "const": "json"
    },
    "example": {
     "type": "object",
     "required": [
      "distribution",
      "mean"
     ],
     "properties": {
      "k": {
       "type": [
        "number",
        "null"
       ],
       "description": "Value the probabilities refer to"
      },
      "n": {
       "type": [
        "number",
        "null"
       ],
       "description": "Number of trials (binomial)"
      },
      "p": {
       "type": [
        "number",
        "null"
       ],
       "description": "Success probability (binomial)"
      },
      "sd": {
       "type": [
        "number",
        "null"
       ],
       "description": "Standard deviation"
      },
      "mean": {
       "type": "number",
       "description": "Mean of the distribution"
      },
      "mode": {
       "type": [
        "number",
        "null"
       ],
       "description": "Mode used"
      },
      "note": {
       "type": "string",
       "description": "Caveat or context for reading the result"
      },
      "source": {
       "type": "string",
       "description": "Data source the values come from"
      },
      "variance": {
       "type": [
        "number",
        "null"
       ],
       "description": "Variance of the distribution"
      },
      "p_at_most_k": {
       "type": [
        "number",
        "null"
       ],
       "description": "P at
… (truncated)
```

## More

- Live health (JSON, refreshed every minute): https://www.zero.xyz/c/saylor-innovations-data-desk-probability-distributions-ead85f25/health.json
- [Zero catalog index](https://www.zero.xyz/llms.txt)
- [Other services from saylorinnovations.com](https://www.zero.xyz/host/saylorinnovations.com/llms.txt)