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In statistics, a Pólya urn model , named after George Pólya, is a type of statistical model used as an idealized mental exercise framework, unifying many treatments.
In an urn model, objects of real interest are represented as colored balls in an urn or other container. In the basic Pólya urn model, the urn contains x white and y black balls; one ball is drawn randomly from the urn and its color observed; it is then returned in the urn, and an additional ball of the same color is added to the urn, and the selection process is repeated. Questions of interest are the evolution of the urn population and the sequence of colors of the balls drawn out.
This endows the urn with a self-reinforcing property sometimes expressed as the rich get richer.
Note that in some sense, the Pólya urn model is the "opposite" of the model of sampling without replacement, where every time a particular value is observed, it is less likely to be observed again, whereas in a Pólya urn model, an observed value is more likely to be observed again. In both of these models, the act of measurement has an effect on the outcome of future measurements. In a Pólya urn model, successive acts of measurement over time have less and less effect on future measurements, whereas in sampling without replacement, the opposite is true: After a certain number of measurements of a particular value, that value will never be seen again.