{"success":true,"version":"v1","request":{"tool":"get_calculator_schema","calculator_id":"hypergeometric-distribution"},"result":{"entity_type":"calculator","id":"hypergeometric-distribution","calculator_id":"hypergeometric-distribution","canonical_url":"https://tttkmbb.com/statistics/hypergeometric-distribution","name":"Hypergeometric Distribution Calculator","title":"Hypergeometric Distribution Calculator – Probability of k Successes when Drawing without Replacement","category":"statistics","category_name":"Statistics & Probability","tool_name":"calculate_hypergeometric_probability","featured_mcp_tool":false,"description":"Computes the probability of exactly, at most, at least, fewer than or more than k successes in n draws without replacement from a population of N items containing K successes, plus the mean, variance and standard deviation.","use_when":"Items are drawn without replacement from a finite population (cards from a deck, defective units in a sampled lot, lottery numbers, capture–recapture) and you need the probability of a given number of successes.","do_not_use_when":"Draws are with replacement or the population is effectively infinite (use binomial-distribution), or you want lottery prize-tier odds directly (use lottery-odds).","inputs":[{"name":"population_size","label":"Population size (N)","type":"integer","required":true,"min":1,"max":10000000,"description":"Total number of items.","example":52},{"name":"successes_in_population","label":"Successes in population (K)","type":"integer","required":true,"min":0,"description":"Number of items in the population that count as successes (0 ≤ K ≤ N).","example":4},{"name":"draws","label":"Number of draws (n)","type":"integer","required":true,"min":1,"max":100000,"description":"Number of items drawn without replacement (1 ≤ n ≤ N).","example":5},{"name":"successes_in_sample","label":"Successes of interest (k)","type":"integer","required":true,"min":0,"description":"Number of successes among the draws whose probability you want (0 ≤ k ≤ n).","example":2}],"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":"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":"n·K/N."},{"name":"variance","label":"Variance","type":"number","decimals":4,"description":"n·(K/N)·(1 − K/N)·(N − n)/(N − 1)."},{"name":"std_dev","label":"Standard deviation","type":"number","decimals":4,"description":"√variance."}],"input_schema":{"type":"object","properties":{"population_size":{"description":"Total number of items.","type":"integer","minimum":1,"maximum":10000000,"examples":[52]},"successes_in_population":{"description":"Number of items in the population that count as successes (0 ≤ K ≤ N).","type":"integer","minimum":0,"examples":[4]},"draws":{"description":"Number of items drawn without replacement (1 ≤ n ≤ N).","type":"integer","minimum":1,"maximum":100000,"examples":[5]},"successes_in_sample":{"description":"Number of successes among the draws whose probability you want (0 ≤ k ≤ n).","type":"integer","minimum":0,"examples":[2]}},"additionalProperties":false,"required":["population_size","successes_in_population","draws","successes_in_sample"]},"output_schema":{"type":"object","properties":{"probability_exact":{"description":"Probability of exactly k successes.","type":"number"},"probability_at_most":{"description":"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":"n·K/N.","type":"number"},"variance":{"description":"n·(K/N)·(1 − K/N)·(N − n)/(N − 1).","type":"number"},"std_dev":{"description":"√variance.","type":"number"}}},"formula":"P(X = k) = C(K, k) · C(N − K, n − k) / C(N, n) for max(0, n − N + K) ≤ k ≤ min(n, K); mean = n·K/N; variance = n·(K/N)·(1 − K/N)·(N − n)/(N − 1)","method":"Binomial coefficients are evaluated in log space with the log-gamma function, so large populations do not overflow; results are exact to the displayed precision.","sources":[{"name":"Wikipedia – Hypergeometric distribution","url":"https://en.wikipedia.org/wiki/Hypergeometric_distribution","type":"reference","retrieved_at":"2026-09-24"},{"name":"Wolfram MathWorld – Hypergeometric Distribution","url":"https://mathworld.wolfram.com/HypergeometricDistribution.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":"Exactly 2 aces in a 5-card poker hand","inputs":{"population_size":52,"successes_in_population":4,"draws":5,"successes_in_sample":2},"expected":{"probability_exact":0.03993,"probability_at_most":0.998246,"probability_at_least":0.041684,"probability_less_than":0.958316,"probability_more_than":0.001754,"mean":0.3846,"variance":0.3272,"std_dev":0.572},"url":"https://tttkmbb.com/api/v1/calculate/hypergeometric-distribution?population_size=52&successes_in_population=4&draws=5&successes_in_sample=2"},{"name":"4 green marbles in 10 draws from 50 marbles with 5 green (Wikipedia example)","inputs":{"population_size":50,"successes_in_population":5,"draws":10,"successes_in_sample":4},"expected":{"probability_exact":0.003965,"probability_at_least":0.004083,"mean":1,"variance":0.7347},"url":"https://tttkmbb.com/api/v1/calculate/hypergeometric-distribution?population_size=50&successes_in_population=5&draws=10&successes_in_sample=4"}],"faq":[{"q":"How does this differ from the binomial distribution?","a":"The binomial assumes a constant success probability (sampling with replacement); the hypergeometric accounts for each draw changing the remaining population. They converge when n is small relative to N (below about 5 %)."},{"q":"What if k is impossible?","a":"Values of k outside max(0, n − (N − K)) … min(n, K) have probability 0; the cumulative probabilities are still returned."}],"tags":["hypergeometric distribution","without replacement","probability of drawing","cards probability","acceptance sampling"],"related":[{"calculator_id":"binomial-distribution","reason":"Sampling with replacement or from a very large population."},{"calculator_id":"lottery-odds","reason":"Prize-tier odds of a lottery, a hypergeometric application."},{"calculator_id":"combinations-permutations","reason":"The binomial coefficients used here."}],"links":{"html":"https://tttkmbb.com/statistics/hypergeometric-distribution","markdown":"https://tttkmbb.com/statistics/hypergeometric-distribution.md","json":"https://tttkmbb.com/statistics/hypergeometric-distribution.json","api":"https://tttkmbb.com/api/v1/calculate/hypergeometric-distribution","schema":"https://tttkmbb.com/api/v1/calculators/hypergeometric-distribution","openapi":"https://tttkmbb.com/openapi.json","mcp":"https://tttkmbb.com/mcp"},"version":"v1","updated_at":"2026-09-24"},"timestamp":"2026-09-24T01:45:03Z"}