The hypergeometric distribution is much like the binomial distribution except the probabilities involved with a hypergeometric distribution are not constant from experiment to experiment. Instead, after each observation, it is not assumed that the state of the population is left untouched; rather, observations are not replaced and are removed from the set of possibilities. Take a card deck for example. If you draw a card from a deck, if a hypergeometric distribution is assumed, that card is not replaced to the deck for the next draw. If it were, we would have to look to the binomial distribution. That is the only difference.
The probability of observing x occurrences of some type from a population N that has k possible outcomes that could result in x where a sample of size n is taken is represented by the probability density function (pdf) P(X = x) = kCx * N-kCn-x / NCn.
The mean of such a distribution is n * k / N and the variance is n * k * (N - k) * (N - n) / ((N2 * (N - 1)). You can think of the mean of this distribution as the expected number of successes that occur out of your sample (n).
You can use the calculator below to calculate the probabilities under your own scenario using the assumptions you input. The probability density function yields probabilities for a specified number of successes out of your sample, whereas the cumulative density function yields probabilities for each and every number of successes out of your sample up to a specified number. The graphs generated are shaped in accordance with the assumptions you enter.
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