A model of a problem described in terms of balls being drawn from an urn. An example of a problem where a hypergeometric distribution is appropriate is as follows. Three people are to be chosen, at random, from a group of twenty. In the group of twenty are four named Smith. The event of interest is that all those chosen are named Smith. The corresponding urn model has an urn containing four black balls and sixteen white balls. Three balls are to be selected at random and without replacement. The event of interest is that all are black.

A trivial example of a problem where a binomial distribution is appropriate concerns the chance of getting three heads when tossing a fair coin four times. The corresponding urn model involves an urn containing one black ball and one white ball. A ball is selected at random and replaced on four occasions. The event of interest is that a black ball is chosen on three occasions.

The phrases without replacement and with replacement can best be understood by imagining that the balls are selected one at a time. In the first case, after each selection the ball is placed on one side, so that the number in the urn has reduced by one. In the second case the ball is replaced in the urn, so that before each draw the urn contains the same mixture of balls as before.