Editing Urn problem (section)

==Examples of urn problems==

* [[binomial distribution]]: the distribution of the number of successful draws (trials), i.e. extraction of white balls, given ''n'' draws with replacement in an urn with black and white balls.<ref name="StatisticsHowTo" />
* [[multinomial distribution]]: there are balls of more than two colors. Each time a ball is extracted, it is returned before drawing another ball.<ref name="StatisticsHowTo" /> This is also known as '[[Balls into bins problem|Balls into bins]]'.
* [http://probabilityandstats.wordpress.com/2010/03/27/the-occupancy-problem/ Occupancy problem]: the distribution of the number of occupied urns after the random assignment of ''k'' balls into ''n'' urns, related to the [[coupon collector's problem]] and [[birthday problem]].
* [[negative binomial distribution]]: number of draws before a certain number of failures (incorrectly colored draws) occurs.
* [[geometric distribution]]: number of draws before the first successful (correctly colored) draw.<ref name="StatisticsHowTo" />
* [[hypergeometric distribution]]: the balls are not returned to the urn once extracted. Hence, the number of total marbles in the urn decreases. This is referred to as "drawing without replacement", by opposition to "drawing with replacement".
* [[Hypergeometric distribution#Multivariate hypergeometric distribution|multivariate hypergeometric distribution]]: the balls are not returned to the urn once extracted, but with balls of more than two colors.<ref name="StatisticsHowTo" />
* Mixed replacement/non-replacement: the urn contains ''x'' white and ''y'' black balls. While black balls are set aside after a draw (non-replacement), white balls are returned to the urn after a draw (replacement).  The probability ''P(m,k)'' that ''k'' black balls will be drawn after ''m'' draws can be calculated recursively using the formula <math>P(m,k)=\frac{y+1-k}{x+y+1-k}P(m-1,k-1)+\frac{x}{x+y-k}P(m-1,k)</math>.<ref>[https://matheplanet.de/matheplanet/nuke/html/article.php?sid=2008/ Matheplanet: Ein Urnenproblem - reloaded]</ref>
* [[Pólya urn model|Pólya urn]]/[[beta-binomial model|beta-binomial distribution]]: each time a ball is drawn, it is replaced along with an additional ball of the same colour. Hence, the number of total balls in the urn grows.
* [[Hoppe urn]]: a Pólya urn with an additional ball called the '''mutator'''. When the mutator is drawn it is replaced along with an additional ball of an entirely new colour.
* [[Statistical physics]]: derivation of energy and velocity distributions.
* The [[Ellsberg paradox]].