An urn problem is a probability problem about drawing colored balls from a container according to specified sampling and replacement rules. The rules determine the resulting statistical distribution.
For an urn containing balls of two colors, the count of one color in a feasible fixed number of draws has a binomial distribution under uniform sampling with replacement and unchanged composition. Uniform sampling without replacement gives a hypergeometric distribution. A Pólya urn instead returns each drawn ball with additional balls of the same color, giving a beta binomial distribution when the added number is fixed and positive (Siegrist).
