{
    "name": "Poisson Probability Calculator",
    "url": "https://calculators.geekmidst.com/calculators/poisson-probability-calculator",
    "markdown": "https://calculators.geekmidst.com/calculators/poisson-probability-calculator.md",
    "category": "Statistics & probability",
    "description": "Find Poisson probabilities for k events when the average count is λ: exactly k, at most k, fewer than, at least and more than k, with mean and SD.",
    "summary": "The Poisson distribution models how many times an event happens in a fixed interval when events occur independently at a steady average rate λ, such as calls per hour or defects per batch.",
    "also_called": [
        "Poisson distribution calculator",
        "Poisson CDF calculator"
    ],
    "applies_to": "Worldwide",
    "last_updated": "2026-10-29",
    "computation": "Results are computed in the visitor’s browser on the HTML page. This API describes the calculator; it does not compute results.",
    "inputs": [
        {
            "name": "lambda",
            "label": "Average number of events (λ)",
            "type": "number",
            "default": "2"
        },
        {
            "name": "k",
            "label": "Number of events (k)",
            "type": "number",
            "default": "3"
        }
    ],
    "outputs": [
        {
            "key": "p_exact",
            "label": "P(X = k)",
            "format": "number"
        },
        {
            "key": "p_exact_text",
            "label": "P(X = k), full precision",
            "format": "text"
        },
        {
            "key": "p_at_most",
            "label": "P(X ≤ k), at most k",
            "format": "number"
        },
        {
            "key": "p_less",
            "label": "P(X < k), fewer than k",
            "format": "number"
        },
        {
            "key": "p_at_least",
            "label": "P(X ≥ k), at least k",
            "format": "number"
        },
        {
            "key": "p_more",
            "label": "P(X > k), more than k",
            "format": "number"
        },
        {
            "key": "mean",
            "label": "Mean (λ)",
            "format": "number"
        },
        {
            "key": "sd",
            "label": "Standard deviation (√λ)",
            "format": "number"
        }
    ],
    "formula": {
        "text": "P(X = k) = e^(−λ)·λᵏ ÷ k!;  P(X ≤ k) = Q(k + 1, λ);  P(X ≥ k) = P(k, λ)",
        "variables": {
            "λ": "average number of events in the interval",
            "k": "number of events",
            "P, Q": "regularised lower and upper incomplete gamma functions (exact cumulative sums)"
        }
    },
    "worked_example": {
        "inputs": {
            "lambda": "2",
            "k": "3"
        },
        "steps": [
            "P(X = 3) = e⁻² × 2³ ÷ 3! = 0.135335 × 8 ÷ 6 = 0.180447.",
            "P(X ≤ 3) = e⁻² × (1 + 2 + 2 + 4/3) = 0.857123.",
            "P(X ≥ 3) = 1 − P(X ≤ 2) = 1 − e⁻² × 5 = 0.323324."
        ],
        "result": "If a help desk averages 2 calls an hour, the chance of exactly 3 calls in an hour is 18.0% and of 3 or more is 32.3%."
    },
    "interpretation": "Use “at least” for capacity questions (how likely is a busy hour) and “at most” for service levels (how often can we handle everything). Variance equals the mean in a Poisson process; if your real counts vary much more than that, the data are over-dispersed and a negative binomial model fits better.",
    "assumptions": [
        "Events occur independently, one at a time, at a constant average rate over the interval."
    ],
    "limitations": [
        "Rates that change through the day (rush hours, seasonality) break the constant-rate assumption; split the interval instead."
    ],
    "rules": null,
    "faqs": [
        {
            "question": "How do I change the interval?",
            "answer": "Scale λ. If you average 2 calls per hour, use λ = 1 for half an hour and λ = 16 for an 8-hour shift."
        },
        {
            "question": "How does Poisson relate to the binomial?",
            "answer": "For many trials with a small success probability, binomial(n, p) is close to Poisson with λ = np."
        }
    ],
    "sources": [],
    "related": [
        {
            "name": "Binomial Probability Calculator",
            "url": "https://calculators.geekmidst.com/calculators/binomial-probability-calculator"
        },
        {
            "name": "Exponential Distribution Calculator",
            "url": "https://calculators.geekmidst.com/calculators/exponential-distribution-probability-calculator"
        },
        {
            "name": "Probability Calculator",
            "url": "https://calculators.geekmidst.com/calculators/probability-calculator"
        }
    ],
    "disclaimer": null,
    "publisher": "GM¹X CalcVerse"
}