{"success":true,"version":"v1","request":{"tool":"get_calculator_schema","calculator_id":"expected-value"},"result":{"entity_type":"calculator","id":"expected-value","calculator_id":"expected-value","canonical_url":"https://tttkmbb.com/statistics/expected-value","name":"Expected Value Calculator","title":"Expected Value Calculator – Mean, Variance and Standard Deviation of Discrete Outcomes","category":"statistics","category_name":"Statistics & Probability","tool_name":"calculate_expected_value","featured_mcp_tool":false,"description":"Computes the expected value E[X] = Σ x·p(x), the variance and the standard deviation of a discrete random variable from its list of outcomes and their probabilities.","use_when":"You have a set of possible outcomes (payoffs, scores, counts) with known probabilities and need the long-run average and its spread: bets, lotteries, insurance, decision analysis.","do_not_use_when":"The probabilities are unknown and you have observed data instead (use descriptive-statistics), or the weights are not probabilities (use weighted-average).","inputs":[{"name":"values","label":"Outcome values","type":"number_list","required":true,"description":"Numeric value of each possible outcome (negative for losses).","example":[0,10,100]},{"name":"probabilities","label":"Probabilities","type":"number_list","required":true,"description":"Probability of each outcome as a fraction between 0 and 1, in the same order as values; they must sum to 1 (within 0.001).","example":[0.7,0.25,0.05]}],"outputs":[{"name":"expected_value","label":"Expected value E[X]","type":"number","decimals":4,"description":"Σ values × probabilities: the long-run average outcome."},{"name":"variance","label":"Variance","type":"number","decimals":4,"description":"Σ probabilities × (values − E[X])²."},{"name":"std_dev","label":"Standard deviation","type":"number","decimals":4,"description":"√variance."},{"name":"count","label":"Number of outcomes","type":"integer","decimals":4,"description":"Number of outcome/probability pairs."}],"input_schema":{"type":"object","properties":{"values":{"description":"Numeric value of each possible outcome (negative for losses).","type":"array","items":{"type":"number"},"minItems":1,"examples":[[0,10,100]]},"probabilities":{"description":"Probability of each outcome as a fraction between 0 and 1, in the same order as values; they must sum to 1 (within 0.001).","type":"array","items":{"type":"number"},"minItems":1,"examples":[[0.7,0.25,0.05]]}},"additionalProperties":false,"required":["values","probabilities"]},"output_schema":{"type":"object","properties":{"expected_value":{"description":"Σ values × probabilities: the long-run average outcome.","type":"number"},"variance":{"description":"Σ probabilities × (values − E[X])².","type":"number"},"std_dev":{"description":"√variance.","type":"number"},"count":{"description":"Number of outcome/probability pairs.","type":"integer"}}},"formula":"E[X] = Σ values_i × probabilities_i; variance = Σ probabilities_i × (values_i − E[X])²; std_dev = √variance","method":"Probabilities are checked to sum to 1 within 0.001 (to tolerate rounded inputs such as 0.1667) and rescaled to sum exactly to 1 before computing; a note is added when rescaling changed them by more than 10⁻⁶.","sources":[{"name":"Wikipedia – Expected value","url":"https://en.wikipedia.org/wiki/Expected_value","type":"reference","retrieved_at":"2026-09-23"},{"name":"NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.5.1 Measures of Location","url":"https://www.itl.nist.gov/div898/handbook/eda/section3/eda351.htm","type":"government","retrieved_at":"2026-09-23"}],"freshness":{"type":"static","max_age_seconds":null,"note":"Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output."},"examples":[{"name":"Payouts 0, 10, 100 with probabilities 0.7, 0.25, 0.05","inputs":{"values":[0,10,100],"probabilities":[0.7,0.25,0.05]},"expected":{"expected_value":7.5,"variance":468.75,"std_dev":21.6506,"count":3},"url":"https://tttkmbb.com/api/v1/calculate/expected-value?values=0%2C10%2C100&probabilities=0.7%2C0.25%2C0.05"},{"name":"Fair six-sided die","inputs":{"values":[1,2,3,4,5,6],"probabilities":[0.1666667,0.1666667,0.1666667,0.1666667,0.1666667,0.1666667]},"expected":{"expected_value":3.5,"variance":2.9167,"std_dev":1.7078},"url":"https://tttkmbb.com/api/v1/calculate/expected-value?values=1%2C2%2C3%2C4%2C5%2C6&probabilities=0.1666667%2C0.1666667%2C0.1666667%2C0.1666667%2C0.1666667%2C0.1666667"}],"faq":[{"q":"What does a negative expected value mean?","a":"On average you lose that amount per play; for a lottery ticket or casino bet the expected value is negative because the house keeps a margin. It says nothing about any single outcome."},{"q":"Can I enter percentages?","a":"No, probabilities are fractions: 0.25 for 25 %. Values can be any numbers, including negative amounts for losses."}],"tags":["expected value","expected payoff","mean of a probability distribution","variance of discrete random variable","ev calculator"],"related":[{"calculator_id":"probability-of-events","reason":"Combine probabilities of events before computing payoffs."},{"calculator_id":"weighted-average","reason":"Weighted mean when the weights are not probabilities."},{"calculator_id":"descriptive-statistics","reason":"Mean and variance of observed data instead of a known distribution."}],"links":{"html":"https://tttkmbb.com/statistics/expected-value","markdown":"https://tttkmbb.com/statistics/expected-value.md","json":"https://tttkmbb.com/statistics/expected-value.json","api":"https://tttkmbb.com/api/v1/calculate/expected-value","schema":"https://tttkmbb.com/api/v1/calculators/expected-value","openapi":"https://tttkmbb.com/openapi.json","mcp":"https://tttkmbb.com/mcp"},"version":"v1","updated_at":"2026-09-23"},"timestamp":"2026-09-23T23:23:56Z","next_actions":[{"tool":"run_calculator","calculator_id":"expected-value","reason":"Run Expected Value Calculator with the inputs above."}],"links":{"markdown":"https://tttkmbb.com/statistics/expected-value.md"}}