Urn problem

In probability and statistics, an urn problem is an idealized mental exercise 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.

An urn model is either a set of probabilities that describe events within an urn problem, or it is a probability distribution, or a family of such distributions, of random variables associated with urn problems.

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

• binomial distribution: the distribution of the number of successful draws (trials), eg. extraction of white ball, given n draws with replacement in an urn with black and white balls.
• beta-binomial distribution: the distribution of the number of successful draws (trials), eg. extraction of white ball, given n draws with replacement in an urn with black and white balls when every time a white ball is observed, an additional white ball is added to the urn and every time a black ball is observed, an additional black ball is added to the urn.
• Multinomial distribution: the urn contains balls in several separate colours (agents).
• Hypergeometric distribution: the balls are not returned to the urn once extracted (without replacement).
• Multivariate hypergeometric distribution: the multiple coloured balls are not returned.
• geometric distribution: number of draws before the first successful (correctly colored) draw.
• negative binomial distribution: number of draws before a certain number of failures (incorrectly colored draws) occurs.
• Polya distribution - Polya urn model: 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 of the same colour. Hence, the number of total marbles in the urn grows. Let X_n be the number of red marbles in the urn after n iterations of this procedure, and let Y_n=X_n/(n+r+b). Then the sequence { Y_n : n = 1, 2, 3, ... } is a martingale and converges to the beta distribution.
• Statistical physics: derivation of energy and velocity distributions
• The Ellsberg paradox

Historical remarks

In Ars conjectandi (1713), Bernoulli considered the problem of determining, given a number of pebbles drawn from an urn, the proportions of different colored pebbles with the urn. 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.

Bernoulli's inspiration may have been lotteries, elections, or games of chance which involved drawing balls from a container, and 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.

See also

• Coin-tossing problems
• Coupon collector's problem
• Noncentral hypergeometric distributions
• Multivariate Pólya distribution

Further reading

• Johnson, N.L.; Kotz, S. (1977) Urn Models and Their Application: An Approach to Modern Discrete Probability Theory, Wiley ISBN 0471446300
• Mahmoud , Hosam M.(2008) Polya Urn Models, Chapman & Hall/CRC. ISBN 1420059831