Urn problem

In statistics, an urn problem is an idealized thought experiment in which some objects of real interest (such as atoms, people, cars, etc.) are represented as colored balls in an urn or other container. One pretends to draw (remove) one or more balls from the urn; the goal is to determine the probability of drawing one color or another, or some other properties.

Basic urn model
In this basic urn model in probability theory, the urn contains x white and y black balls; one ball is drawn randomly from the urn and its color observed; it is then placed back in the urn, and the selection process is repeated.

Possible questions that can be answered in this model are:
 * can I infer the proportion of white and black balls from n observations ? With what degree of confidence ?
 * knowing x and y, what is the probability of drawing a specific sequence (e.g. one white followed by one black)?
 * if I only observe n white balls, how sure can I be that there are no black balls?

Other models
Many other variations exist:
 * the urn could have numbered balls instead of colored ones
 * balls may not be returned to the urns once drawn.

Examples of urn problems

 * Derivation of the binomial distribution
 * Derivation of the hypergeometric distribution
 * Statistical physics: derivation of energy and velocity distributions
 * The Ellsberg paradox
 * Pólya's urn: an urn initially contains r red and b blue marbles. One marble is chosen randomly from the urn.  The marble is then put back into the urn together with another marble (presumably from a collection stored elsewhere) of the same colour.  Hence, the number of total marbles in the urn grows.  Let Xn be the number of red marbles in the urn after n iterations of this procedure, and let Yn=Xn/(n+r+b).  Then the sequence { Yn : n = 1, 2, 3, ... } is a martingale.

Historical remarks
Urn problems have been a part of the theory of probability since at least the publication of the Ars conjectandi by Jakob Bernoulli (1713). Bernoulli's inspiration may have been lotteries, elections, or games of chance which involved drawing balls from a container. It has been asserted  that


 * Elections in medieval and renaissance Venice, including that of the doge, often included the choice of electors by lot, using balls of different colors drawn from an urn.

Bernoulli himself, in Ars conjectandi, considered the problem of determining, from a number of pebbles drawn from an urn, the proportions of different colors. This problem was known as the inverse probability problem, and was a topic of research in the eighteenth century, attracting the attention of Abraham de Moivre and Thomas Bayes.