{"success":true,"version":"v1","request":{"tool":"get_calculator_schema","calculator_id":"binomial-distribution"},"result":{"entity_type":"calculator","id":"binomial-distribution","calculator_id":"binomial-distribution","canonical_url":"https://tttkmbb.com/statistics/binomial-distribution","name":"Binomial Distribution Calculator","title":"Binomial Distribution Calculator – P(X = k), P(X ≤ k), P(X ≥ k) for n Trials","category":"statistics","category_name":"Statistics & Probability","tool_name":"calculate_binomial_probability","featured_mcp_tool":false,"description":"Computes the probability of exactly, at most, at least, fewer than or more than k successes in n independent trials with success probability p, plus the mean, variance and standard deviation of the distribution.","use_when":"You have a fixed number of independent yes/no trials with a constant success probability (coin flips, defect counts in a batch, conversions among n visitors) and need the probability of a number of successes.","do_not_use_when":"Trials are not independent or p changes, the number of trials is not fixed, or you count events over time or space at an average rate (use poisson-distribution).","inputs":[{"name":"trials","label":"Number of trials (n)","type":"integer","required":true,"min":1,"max":100000,"description":"Total number of independent trials.","example":10},{"name":"successes","label":"Number of successes (k)","type":"integer","required":true,"min":0,"max":100000,"description":"Number of successes of interest (0 ≤ k ≤ n).","example":3},{"name":"probability_of_success","label":"Probability of success (p)","type":"number","required":true,"min":0,"max":1,"description":"Probability of success on a single trial, as a number between 0 and 1 (0.5 for 50 %), not a percentage.","example":0.5}],"outputs":[{"name":"probability_exact","label":"P(X = k)","type":"number","decimals":6,"description":"Probability of exactly k successes."},{"name":"probability_at_most","label":"P(X ≤ k)","type":"number","decimals":6,"description":"Cumulative probability of k or fewer successes."},{"name":"probability_at_least","label":"P(X ≥ k)","type":"number","decimals":6,"description":"Probability of k or more successes."},{"name":"probability_less_than","label":"P(X < k)","type":"number","decimals":6,"description":"Probability of fewer than k successes."},{"name":"probability_more_than","label":"P(X > k)","type":"number","decimals":6,"description":"Probability of more than k successes."},{"name":"mean","label":"Mean","type":"number","decimals":4,"description":"Expected number of successes n·p."},{"name":"variance","label":"Variance","type":"number","decimals":4,"description":"n·p·(1 − p)."},{"name":"std_dev","label":"Standard deviation","type":"number","decimals":4,"description":"√(n·p·(1 − p))."}],"input_schema":{"type":"object","properties":{"trials":{"description":"Total number of independent trials.","type":"integer","minimum":1,"maximum":100000,"examples":[10]},"successes":{"description":"Number of successes of interest (0 ≤ k ≤ n).","type":"integer","minimum":0,"maximum":100000,"examples":[3]},"probability_of_success":{"description":"Probability of success on a single trial, as a number between 0 and 1 (0.5 for 50 %), not a percentage.","type":"number","minimum":0,"maximum":1,"examples":[0.5]}},"additionalProperties":false,"required":["trials","successes","probability_of_success"]},"output_schema":{"type":"object","properties":{"probability_exact":{"description":"Probability of exactly k successes.","type":"number"},"probability_at_most":{"description":"Cumulative probability of k or fewer successes.","type":"number"},"probability_at_least":{"description":"Probability of k or more successes.","type":"number"},"probability_less_than":{"description":"Probability of fewer than k successes.","type":"number"},"probability_more_than":{"description":"Probability of more than k successes.","type":"number"},"mean":{"description":"Expected number of successes n·p.","type":"number"},"variance":{"description":"n·p·(1 − p).","type":"number"},"std_dev":{"description":"√(n·p·(1 − p)).","type":"number"}}},"formula":"P(X = k) = C(n, k) · p^k · (1 − p)^(n − k); P(X ≤ k) = Σ_{i=0..k} P(X = i); P(X ≥ k) = Σ_{i=k..n} P(X = i); mean = n·p; variance = n·p·(1 − p)","method":"Probabilities are computed term by term in log space, so large n does not overflow the binomial coefficient. Results are exact (no normal approximation).","sources":[{"name":"NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.6.18 Binomial Distribution","url":"https://www.itl.nist.gov/div898/handbook/eda/section3/eda366i.htm","type":"government","retrieved_at":"2026-09-23"},{"name":"Wikipedia – Binomial distribution","url":"https://en.wikipedia.org/wiki/Binomial_distribution","type":"reference","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":"3 heads in 10 fair coin flips","inputs":{"trials":10,"successes":3,"probability_of_success":0.5},"expected":{"probability_exact":0.117188,"probability_at_most":0.171875,"probability_at_least":0.945313,"probability_less_than":0.054688,"probability_more_than":0.828125,"mean":5,"variance":2.5,"std_dev":1.5811},"url":"https://tttkmbb.com/api/v1/calculate/binomial-distribution?trials=10&successes=3&probability_of_success=0.5"},{"name":"2 defects in 20 items at 10 %","inputs":{"trials":20,"successes":2,"probability_of_success":0.1},"expected":{"probability_exact":0.28518,"probability_at_most":0.676927,"probability_at_least":0.608253,"mean":2,"variance":1.8,"std_dev":1.3416},"url":"https://tttkmbb.com/api/v1/calculate/binomial-distribution?trials=20&successes=2&probability_of_success=0.1"}],"faq":[{"q":"Is p a percentage?","a":"No. Enter the single-trial success probability as a fraction between 0 and 1: 0.25 for 25 %."},{"q":"When can the normal approximation be used instead?","a":"When n·p ≥ 10 and n·(1 − p) ≥ 10 the binomial is close to a normal with mean n·p and SD √(n·p·(1 − p)); this calculator does not need the approximation because it computes the exact sum."}],"tags":["binomial distribution","binomial probability","probability of k successes","coin flip probability","bernoulli trials"],"related":[{"calculator_id":"poisson-distribution","reason":"Counts of rare events at an average rate (limit of the binomial for large n, small p)."},{"calculator_id":"probability-of-events","reason":"Combine the probabilities of two events."},{"calculator_id":"combinations-permutations","reason":"The binomial coefficient C(n, k) used in the formula."}],"links":{"html":"https://tttkmbb.com/statistics/binomial-distribution","markdown":"https://tttkmbb.com/statistics/binomial-distribution.md","json":"https://tttkmbb.com/statistics/binomial-distribution.json","api":"https://tttkmbb.com/api/v1/calculate/binomial-distribution","schema":"https://tttkmbb.com/api/v1/calculators/binomial-distribution","openapi":"https://tttkmbb.com/openapi.json","mcp":"https://tttkmbb.com/mcp"},"version":"v1","updated_at":"2026-09-23"},"timestamp":"2026-09-23T23:25:00Z","next_actions":[{"tool":"run_calculator","calculator_id":"binomial-distribution","reason":"Run Binomial Distribution Calculator with the inputs above."}],"links":{"markdown":"https://tttkmbb.com/statistics/binomial-distribution.md"}}