{"success":true,"version":"v1","request":{"tool":"get_calculator_schema","calculator_id":"dice-probability"},"result":{"entity_type":"calculator","id":"dice-probability","calculator_id":"dice-probability","canonical_url":"https://tttkmbb.com/statistics/dice-probability","name":"Dice Probability Calculator","title":"Dice Probability Calculator – Sum Equals, At Least or At Most a Target, or Rolling a Specific Face","category":"statistics","category_name":"Statistics & Probability","tool_name":"calculate_dice_probability","featured_mcp_tool":false,"description":"Computes the exact probability that the sum of several dice equals, is at least or at most a target value (by convolution of the face distributions), or that at least one die shows a specific face, with favourable and total outcomes and the odds.","use_when":"You need probabilities for rolling dice of any number of sides (d4 to d100): totals in board games and role-playing games, or the chance of at least one six in several rolls.","do_not_use_when":"The dice are weighted or you want the probability of a number of successes among trials with a given per-trial probability (use binomial-distribution).","inputs":[{"name":"dice","label":"Number of dice","type":"integer","required":true,"min":1,"max":50,"description":"How many dice are rolled together.","example":2},{"name":"sides","label":"Sides per die","type":"integer","required":true,"min":2,"max":100,"description":"Faces on each die, numbered 1 to sides (6 for a standard die).","example":6},{"name":"target","label":"Event","type":"enum","required":false,"default":"sum_equals","values":[{"value":"sum_equals","label":"Sum equals target_value","aliases":["equals","sum","exactly"]},{"value":"sum_at_least","label":"Sum is at least target_value","aliases":["at_least","sum_greater_or_equal","minimum"]},{"value":"sum_at_most","label":"Sum is at most target_value","aliases":["at_most","sum_less_or_equal","maximum"]},{"value":"at_least_one_specific","label":"At least one die shows target_value","aliases":["at_least_one","specific_face","any_die_shows"]}],"description":"Which event to evaluate.","example":"sum_equals"},{"name":"target_value","label":"Target value","type":"integer","required":true,"min":1,"description":"The sum of interest, or the face value (1 … sides) for at_least_one_specific.","example":7}],"outputs":[{"name":"probability","label":"Probability","type":"number","decimals":6,"description":"Probability of the event, 0 to 1."},{"name":"probability_percent","label":"Probability","type":"number","unit":"%","decimals":2,"description":"Same probability in percent."},{"name":"odds","label":"Odds","type":"string","decimals":4,"description":"Probability expressed as '1 in N'."},{"name":"favorable_outcomes","label":"Favourable outcomes","type":"number","decimals":0,"description":"Number of ordered dice results producing the event."},{"name":"total_outcomes","label":"Total outcomes","type":"number","decimals":0,"description":"sides^dice ordered results."},{"name":"expected_sum","label":"Expected sum","type":"number","decimals":2,"description":"dice × (sides + 1) / 2."},{"name":"sum_std_dev","label":"SD of the sum","type":"number","decimals":4,"description":"√(dice × (sides² − 1) / 12)."}],"input_schema":{"type":"object","properties":{"dice":{"description":"How many dice are rolled together.","type":"integer","minimum":1,"maximum":50,"examples":[2]},"sides":{"description":"Faces on each die, numbered 1 to sides (6 for a standard die).","type":"integer","minimum":2,"maximum":100,"examples":[6]},"target":{"description":"Which event to evaluate.","type":"string","enum":["sum_equals","sum_at_least","sum_at_most","at_least_one_specific"],"default":"sum_equals","examples":["sum_equals"]},"target_value":{"description":"The sum of interest, or the face value (1 … sides) for at_least_one_specific.","type":"integer","minimum":1,"examples":[7]}},"additionalProperties":false,"required":["dice","sides","target_value"]},"output_schema":{"type":"object","properties":{"probability":{"description":"Probability of the event, 0 to 1.","type":"number"},"probability_percent":{"description":"Same probability in percent. Unit: %.","type":"number","x-unit":"%"},"odds":{"description":"Probability expressed as '1 in N'.","type":"string"},"favorable_outcomes":{"description":"Number of ordered dice results producing the event.","type":"number"},"total_outcomes":{"description":"sides^dice ordered results.","type":"number"},"expected_sum":{"description":"dice × (sides + 1) / 2.","type":"number"},"sum_std_dev":{"description":"√(dice × (sides² − 1) / 12).","type":"number"}}},"formula":"P(sum = s) from the convolution of dice uniform distributions on 1…sides; P(sum ≥ s) = Σ_{j ≥ s} P(sum = j); P(at least one die shows f) = 1 − ((sides − 1)/sides)^dice; expected sum = dice(sides + 1)/2; variance = dice(sides² − 1)/12","method":"Exact enumeration by iterated convolution (no simulation); sums outside dice … dice × sides have probability 0 and trigger a note. All dice are fair and independent.","sources":[{"name":"Wolfram MathWorld – Dice","url":"https://mathworld.wolfram.com/Dice.html","type":"reference","retrieved_at":"2026-09-24"},{"name":"Wikipedia – Dice","url":"https://en.wikipedia.org/wiki/Dice","type":"reference","retrieved_at":"2026-09-24"},{"name":"Wolfram MathWorld – de Méré's Problem","url":"https://mathworld.wolfram.com/deMeresProblem.html","type":"reference","retrieved_at":"2026-09-24"}],"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":"Sum of two dice equals 7","inputs":{"dice":2,"sides":6,"target":"sum_equals","target_value":7},"expected":{"probability":0.166667,"probability_percent":16.67,"odds":"1 in 6","favorable_outcomes":6,"total_outcomes":36,"expected_sum":7,"sum_std_dev":2.4152},"url":"https://tttkmbb.com/api/v1/calculate/dice-probability?dice=2&sides=6&target=sum_equals&target_value=7"},{"name":"At least one six in four dice (de Méré)","inputs":{"dice":4,"sides":6,"target":"at_least_one_specific","target_value":6},"expected":{"probability":0.517747,"probability_percent":51.77,"favorable_outcomes":671,"total_outcomes":1296,"expected_sum":14},"url":"https://tttkmbb.com/api/v1/calculate/dice-probability?dice=4&sides=6&target=at_least_one_specific&target_value=6"}],"faq":[{"q":"Why is 7 the most likely total of two dice?","a":"Six of the 36 ordered outcomes (1+6, 2+5, 3+4, 4+3, 5+2, 6+1) sum to 7, more than any other total; the distribution of a sum of dice is triangular for two dice and approaches a bell curve for more."},{"q":"What is de Méré's problem?","a":"Whether at least one six in four rolls (51.77 %) is more likely than at least one double six in 24 rolls of two dice (49.14 %); the calculator gives the first directly (dice = 4, target 6) and the second with dice = 24, sides = 36, target 36."}],"tags":["dice probability","probability of rolling","sum of dice","at least one six","d20 probability"],"related":[{"calculator_id":"binomial-distribution","reason":"Number of dice showing a given face."},{"calculator_id":"expected-value","reason":"Expected payoff of a dice game."},{"calculator_id":"probability-of-events","reason":"Combine two independent dice events."}],"links":{"html":"https://tttkmbb.com/statistics/dice-probability","markdown":"https://tttkmbb.com/statistics/dice-probability.md","json":"https://tttkmbb.com/statistics/dice-probability.json","api":"https://tttkmbb.com/api/v1/calculate/dice-probability","schema":"https://tttkmbb.com/api/v1/calculators/dice-probability","openapi":"https://tttkmbb.com/openapi.json","mcp":"https://tttkmbb.com/mcp"},"version":"v1","updated_at":"2026-09-24"},"timestamp":"2026-09-24T01:58:52Z","next_actions":[{"tool":"run_calculator","calculator_id":"dice-probability","reason":"Run Dice Probability Calculator with the inputs above."}],"links":{"markdown":"https://tttkmbb.com/statistics/dice-probability.md"}}