Hypergeometric Probability

This is the binomial's honest sibling for sampling without replacement, where every draw changes the odds for the next. Split the choices in two: pick k of the K successes, pick the remaining n − k from the N − K non-successes, and divide by all the ways to pick n from N. A five-card poker hand containing exactly two aces: C(4,2) × C(48,3) / C(52,5) = 6 × 17,296 / 2,598,960 ≈ 0.0399, about one deal in twenty-five.

The distribution prices lottery tiers, acceptance sampling on a production lot, and the capture–recapture estimates ecologists use to count fish: tag K animals, catch n later, and the number recaptured pins down N. Matching exactly 4 of 6 numbers in a 6/49 draw is C(6,4) × C(43,2) / C(49,6) = 15 × 903 / 13,983,816 ≈ 0.000969, roughly 1 in 1032. The trap is reaching for the binomial: it is a good approximation only when the sample is a small slice of the population, under about 5%, because then removing items barely shifts the odds. Sample 20 items from a lot of 50 and the two answers diverge badly. A value of k outside the support (more successes than the sample holds, or more than exist in the population) returns exactly 0 rather than an error, because those draws genuinely never happen. There is no closed form for N, K, n or k given P, so this calculator solves for the probability.