simulate-monte-carlo

Actually draws random samples from real distributions and counts outcomes, instead of a model guess about a probability. Declare named random variables (uniform, normal, bernoulli, binomial, poisson, exponential, discrete), an "event" boolean expression over those variable names (e.g. "a > 0.5 && b == 1"), and an optional "condition" expression to estimate a conditional probability P(event | condition) by rejection sampling. Event/condition expressions are parsed and evaluated by a small built-in interpreter (arithmetic, comparisons, &&/||/!, min/max/abs) — no arbitrary code execution. Returns the estimated probability, a 95% confidence interval, and the seed used (pass the same seed back to reproduce the exact result). Costs $0.03 USDC (Base) per call.

{
  "type": "object",
  "properties": {
    "variables": {
      "minItems": 1,
      "maxItems": 10,
      "type": "array",
      "items": {
        "type": "object",
        "properties": {
          "name": {
            "type": "string",
            "pattern": "^[a-zA-Z_][a-zA-Z0-9_]*$",
            "description": "Variable name, referenced by \"event\"/\"condition\" (e.g. \"a\", \"wait_time\")."
          },
          "distribution": {
            "oneOf": [
              {
                "type": "object",
                "properties": {
                  "type": {
                    "type": "string",
                    "const": "uniform"
                  },
                  "min": {
                    "type": "number"
                  },
                  "max": {
                    "type": "number"
                  }
                },
                "required": [
                  "type",
                  "min",
                  "max"
                ],
                "description": "Continuous, equally likely between min and max."
              },
              {
                "type": "object",
                "properties": {
                  "type": {
                    "type": "string",
                    "const": "normal"
                  },
                  "mean": {
                    "type": "number"
                  },
                  "stdDev": {
                    "type": "number",
                    "minimum": 0
                  }
                },
                "required": [
                  "type",
                  "mean",
                  "stdDev"
                ],
                "description": "Gaussian/bell curve, via Box-Muller sampling."
              },
              {
                "type": "object",
                "properties": {
                  "type": {
                    "type": "string",
                    "const": "bernoulli"
                  },
                  "p": {
                    "type": "number",
                    "minimum": 0,
                    "maximum": 1
                  }
                },
                "required": [
                  "type",
                  "p"
                ],
                "description": "Single 0/1 trial with success probability p."
              },
              {
                "type": "object",
                "properties": {
                  "type": {
                    "type": "string",
                    "const": "binomial"
                  },
                  "n": {
                    "type": "integer",
                    "exclusiveMinimum": 0,
                    "maximum": 1000
                  },
                  "p": {
                    "type": "number",
                    "minimum": 0,
                    "maximum": 1
                  }
                },
                "required": [
                  "type",
                  "n",
                  "p"
                ],
                "description": "Count of successes in n independent Bernoulli(p) trials. n capped at 1000."
              },
              {
                "type": "object",
                "properties": {
                  "type": {
                    "type": "string",
                    "const": "poisson"
                  },
                  "lambda": {
                    "type": "number",
                    "exclusiveMinimum": 0,
                    "maximum": 1000
                  }
                },
                "required": [
                  "type",
                  "lambda"
                ],
                "description": "Event count with mean rate lambda, via Knuth's algorithm. lambda capped at 1000."
              },
              {
                "type": "object",
                "properties": {
                  "type": {
                    "type": "string",
                    "const": "exponential"
                  },
                  "rate": {
                    "type": "number",
                    "exclusiveMinimum": 0
                  }
                },
                "required": [
                  "type",
                  "rate"
                ],
                "description": "Time between events at the given rate."
              },
              {
                "type": "object",
                "properties": {
                  "type": {
                    "type": "string",
                    "const": "discrete"
                  },
                  "values": {
                    "minItems": 1,
                    "maxItems": 20,
                    "type": "array",
                    "items": {
                      "type": "number"
                    }
                  },
                  "weights": {
                    "minItems": 1,
                    "maxItems": 20,
                    "type": "array",
                    "items": {
                      "type": "number",
                      "minimum": 0
                    }
                  }
                },
                "required": [
                  "type",
                  "values",
                  "weights"
                ],
                "description": "Weighted pick from a custom list of outcomes (e.g. a die). Up to 20 outcomes."
              }
            ]
          }
        },
        "required": [
          "name",
          "distribution"
        ]
      },
      "description": "Random variables to sample each trial. Max 10."
    },
    "event": {
      "type": "string",
      "maxLength": 500,
      "description": "Boolean expression over the variable names, evaluated each trial (e.g. \"a + b > 10\", \"x == 1 && y < 0.2\")."
    },
    "condition": {
      "description": "Optional boolean expression; if given, the result is P(event | condition), estimated only over trials where this is true.",
      "type": "string",
      "maxLength": 500
    },
    "trials": {
      "description": "Number of trials to run. Default 10000, between 100 and 100000.",
      "type": "integer",
      "minimum": 100,
      "maximum": 100000
    },
    "seed": {
      "description": "Optional PRNG seed for a reproducible run. If omitted, a random seed is generated and returned in the output.",
      "type": "integer",
      "minimum": -9007199254740991,
      "maximum": 9007199254740991
    }
  },
  "required": [
    "variables",
    "event"
  ],
  "$schema": "https://json-schema.org/draft/2020-12/schema"
}

{
  "type": "object",
  "properties": {
    "valid": {
      "type": "boolean",
      "description": "false if the input (variables, expressions, limits) was invalid."
    },
    "error": {
      "anyOf": [
        {
          "type": "string"
        },
        {
          "type": "null"
        }
      ],
      "description": "Error or warning message. null if none."
    },
    "trials_run": {
      "type": "number",
      "description": "Number of trials actually simulated (0 if valid is false)."
    },
    "seed_used": {
      "type": "number",
      "description": "The PRNG seed used — pass it back as `seed` to reproduce this exact result."
    },
    "probability": {
      "anyOf": [
        {
          "type": "number"
        },
        {
          "type": "null"
        }
      ],
      "description": "Estimated P(event) or P(event | condition). null if invalid, or if condition matched zero trials."
    },
    "event_successes": {
      "type": "number",
      "description": "Number of trials (or condition-matching trials) where event was true."
    },
    "condition_successes": {
      "anyOf": [
        {
          "type": "number"
        },
        {
          "type": "null"
        }
      ],
      "description": "Number of trials where condition was true. null if no condition was given."
    },
    "standard_error": {
      "anyOf": [
        {
          "type": "number"
        },
        {
          "type": "null"
        }
      ],
      "description": "Estimated standard error of the probability estimate."
    },
    "confidence_interval_95": {
      "anyOf": [
        {
          "type": "array",
          "prefixItems": [
            {
              "type": "number"
            },
            {
              "type": "number"
            }
          ]
        },
        {
          "type": "null"
        }
      ],
      "description": "Approximate 95% confidence interval [low, high] via the normal approximation."
    }
  },
  "required": [
    "valid",
    "error",
    "trials_run",
    "seed_used",
    "probability",
    "event_successes",
    "condition_successes",
    "standard_error",
    "confidence_interval_95"
  ],
  "$schema": "https://json-schema.org/draft/2020-12/schema",
  "additionalProperties": false
}