{
  "slug": "probability",
  "title": "Probability Calculator",
  "heading": "Probability Calculator",
  "category": "math",
  "url": "https://www.revenuelab.fyi/toolbox/probability",
  "summary": "Independent events, unions, intersections, and repeated trials.",
  "description": "Computes the standard combinations of two independent event probabilities, plus the odds of an event happening at least once across repeated trials — the calculation that explains why unlikely things happen constantly at scale.",
  "formula": "P(A and B) = P(A)P(B). P(A or B) = P(A) + P(B) − P(A)P(B). At least once = 1 − (1−P)ⁿ.",
  "dateModified": "2026-09-30",
  "run_url": "https://www.revenuelab.fyi/api/public/calc?tool=probability",
  "inputs": [
    {
      "id": "pa",
      "label": "Probability of A",
      "kind": "number",
      "hint": null,
      "default": 25,
      "unit": "%",
      "min": 0,
      "max": 100
    },
    {
      "id": "pb",
      "label": "Probability of B",
      "kind": "number",
      "hint": null,
      "default": 40,
      "unit": "%",
      "min": 0,
      "max": 100
    },
    {
      "id": "trials",
      "label": "Repeated trials of A",
      "kind": "number",
      "hint": null,
      "default": 10,
      "unit": null,
      "min": 1,
      "max": null
    }
  ],
  "outputs": [
    {
      "id": "atLeastOnce",
      "label": "A happens at least once",
      "format": "percent",
      "hint": null,
      "primary": true
    },
    {
      "id": "both",
      "label": "Both A and B",
      "format": "percent",
      "hint": null,
      "primary": false
    },
    {
      "id": "either",
      "label": "A or B",
      "format": "percent",
      "hint": null,
      "primary": false
    },
    {
      "id": "neither",
      "label": "Neither",
      "format": "percent",
      "hint": null,
      "primary": false
    },
    {
      "id": "onlyA",
      "label": "A but not B",
      "format": "percent",
      "hint": null,
      "primary": false
    }
  ],
  "worked_example": {
    "inputs": [
      "Probability of A: 25 %",
      "Probability of B: 40 %",
      "Repeated trials of A: 10"
    ],
    "outputs": [
      "A happens at least once: 94.369%",
      "Both A and B: 10.000%",
      "A or B: 55.000%",
      "Neither: 45.000%",
      "A but not B: 15.000%"
    ]
  },
  "how_to": {
    "title": "How to use this",
    "steps": [
      "Enter probability of a (%).",
      "Enter probability of b (%).",
      "Enter repeated trials of a.",
      "Read your a happens at least once on the right — it updates as you type.",
      "Hit Share to keep the scenario or send it to someone."
    ]
  },
  "scenarios": [
    {
      "name": "Conservative",
      "description": "Lower-end numbers — what if things land soft?",
      "values": {
        "pa": 15,
        "pb": 24,
        "trials": 6
      }
    },
    {
      "name": "Typical",
      "description": "Defaults — the most common real-world setup.",
      "values": {
        "pa": 25,
        "pb": 40,
        "trials": 10
      }
    },
    {
      "name": "Ambitious",
      "description": "Higher-end numbers — what if things really pop?",
      "values": {
        "pa": 40,
        "pb": 64,
        "trials": 16
      }
    }
  ],
  "limitations": [
    "Results are rounded for display; the underlying calculation keeps full precision.",
    "Very large or very small inputs may hit floating-point limits in the browser.",
    "Inputs outside the accepted range are clamped rather than rejected."
  ],
  "faq": [
    {
      "q": "How does the probability calculator work?",
      "a": "Computes the standard combinations of two independent event probabilities, plus the odds of an event happening at least once across repeated trials — the calculation that explains why unlikely things happen constantly at scale. The underlying maths is: P(A and B) = P(A)P(B). P(A or B) = P(A) + P(B) − P(A)P(B). At least once = 1 − (1−P)ⁿ."
    },
    {
      "q": "What do I need to enter?",
      "a": "3 values: probability of a, probability of b, and repeated trials of a. Each field starts with a sensible default, so you can change one number at a time and watch the result move."
    },
    {
      "q": "What does the a happens at least once result mean?",
      "a": "It is returned as a percentage and updates live as you edit the inputs, so you can compare two or three versions of a scenario in a few seconds."
    },
    {
      "q": "Is this calculator free, and do I need an account?",
      "a": "Yes, it's free, and no account is required. Nothing you type is stored on our servers — the maths runs entirely in your browser."
    },
    {
      "q": "How accurate is the result?",
      "a": "It applies the standard formula exactly, so the arithmetic is precise. Results are rounded for display; the underlying calculation keeps full precision."
    },
    {
      "q": "Who is this tool for?",
      "a": "It's built for students, developers, and anyone who needs the answer right now without setting up a spreadsheet."
    }
  ],
  "related": [
    "https://www.revenuelab.fyi/toolbox/permutation-combination",
    "https://www.revenuelab.fyi/toolbox/z-score"
  ],
  "license": "CC-BY-4.0",
  "citation": "RevenueLab — Probability Calculator (https://www.revenuelab.fyi/toolbox/probability)"
}