Probability Distribution

The Polya process can be presented as follows: Assume an urn initially containing equal quantities of black and red balls. You are to guess each time which color you will pull out before you make the draw. Here the game is rigged. Unlike a conventional urn, the probability of guessing correctly depends on past success,

Taleb, Nassim Nicholas. Fooled by Randomness: The Hidden Role of Chance in Life and in the Markets (Incerto Book 1) (pp. 213-214). Random House Publishing Group. Kindle Edition.

In statistics, a Pólya urn model (also known as a Pólya urn scheme or simply as Pólya's urn), named after George Pólya, is a family of urn models that can be used to interpret many commonly used statistical models.

The model represents objects of interest (such as atoms, people, cars, etc.) as colored balls in an urn. In the basic Pólya urn model, the experimenter puts x white and y black balls into an urn. At each step, one ball is drawn uniformly at random 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.

If by random chance, more black balls are drawn than white balls in the initial few draws, it would make it more likely for more black balls to be drawn later. Similarly for the white balls. Thus the urn has a self-reinforcing property ("the rich get richer"). It is the opposite 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 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.

It is also different from sampling with replacement, where the ball is returned to the urn but without adding new balls. In this case, there is neither self-reinforcing nor anti-self-reinforcing.