{
    "name": "Expected Value Calculator",
    "url": "https://calculators.geekmidst.com/calculators/expected-value-calculator",
    "markdown": "https://calculators.geekmidst.com/calculators/expected-value-calculator.md",
    "category": "Statistics & probability",
    "description": "Find the expected value, variance and standard deviation of a discrete random variable from its outcomes and probabilities, percentages or weights.",
    "summary": "The expected value is the probability-weighted average outcome: what you would average per play over many repetitions.",
    "also_called": [
        "EV calculator",
        "Discrete distribution mean 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": "outcomes",
            "label": "Outcomes (values)",
            "type": "textarea",
            "default": "1, 2, 3, 4, 5, 6"
        },
        {
            "name": "probabilities",
            "label": "Probabilities",
            "type": "textarea",
            "default": "1/6, 1/6, 1/6, 1/6, 1/6, 1/6"
        },
        {
            "name": "format",
            "label": "Probabilities are",
            "type": "select",
            "default": "decimal",
            "options": {
                "decimal": "Decimals or fractions (sum to 1)",
                "percent": "Percentages (sum to 100)",
                "weights": "Relative weights (normalised)"
            }
        }
    ],
    "outputs": [
        {
            "key": "expected_value",
            "label": "Expected value E[X]",
            "format": "number"
        },
        {
            "key": "variance",
            "label": "Variance",
            "format": "number"
        },
        {
            "key": "sd",
            "label": "Standard deviation",
            "format": "number"
        },
        {
            "key": "outcomes",
            "label": "Number of outcomes",
            "format": "integer"
        }
    ],
    "formula": {
        "text": "E[X] = Σ xᵢpᵢ;  Var(X) = Σ pᵢ(xᵢ − E[X])²;  SD = √Var(X)",
        "variables": {
            "xᵢ": "each outcome",
            "pᵢ": "its probability"
        }
    },
    "worked_example": {
        "inputs": {
            "outcomes": "1, 2, 3, 4, 5, 6",
            "probabilities": "1/6, 1/6, 1/6, 1/6, 1/6, 1/6",
            "format": "decimal"
        },
        "steps": [
            "E[X] = (1 + 2 + 3 + 4 + 5 + 6) ÷ 6 = 21 ÷ 6 = 3.5.",
            "Var = (6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25) ÷ 6 = 17.5 ÷ 6 = 2.916667; SD = 1.7078."
        ],
        "result": "A fair die has an expected value of 3.5 with a standard deviation of about 1.71."
    },
    "interpretation": "A negative expected value for a bet means you lose on average in the long run. The standard deviation shows how far single results typically stray from that average.",
    "assumptions": [
        "Outcomes are mutually exclusive and the probabilities cover every possibility."
    ],
    "limitations": [
        "Expected value ignores risk preferences: two choices with the same EV can differ greatly in how much you could lose."
    ],
    "rules": null,
    "faqs": [
        {
            "question": "Does the expected value have to be a possible outcome?",
            "answer": "No. A die never shows 3.5; it is a long-run average."
        },
        {
            "question": "How do I value a lottery ticket?",
            "answer": "List each prize (net of the ticket price) with its probability, including the “lose the stake” outcome; the EV is almost always negative."
        }
    ],
    "sources": [],
    "related": [
        {
            "name": "Probability Calculator",
            "url": "https://calculators.geekmidst.com/calculators/probability-calculator"
        },
        {
            "name": "Conditional Probability Calculator",
            "url": "https://calculators.geekmidst.com/calculators/conditional-probability-calculator"
        },
        {
            "name": "Hypergeometric Probability Calculator",
            "url": "https://calculators.geekmidst.com/calculators/hypergeometric-probability-calculator"
        }
    ],
    "disclaimer": null,
    "publisher": "GM¹X CalcVerse"
}