In probability theory and 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, and the same 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.
Basic results Questions of interest are the evolution of the urn population and the sequence of colors of the balls drawn out. After n {\displaystyle n} draws, the probability that the urn contains ( x + n 1 ) {\displaystyle (x+n_{1})} white balls and ( y + n 2 ) {\displaystyle (y+n_{2})} black balls (for 0 ≤ n 1 , n 2 ≤ n , n 1 + n 2 = n {\displaystyle 0\leq n_{1},n_{2}\leq n\,\,,n_{1}+n_{2}=n} ) is
( n n 1 ) x n ¯ 1 y n ¯ 2 ( x + y ) n ¯ {\displaystyle {\binom {n}{n_{1}}}{\frac {x^{{\bar {n}}_{1}}y^{{\bar {n}}_{2}}}{(x+y)^{\bar {n}}}}}
where the overbar denotes rising factorial. This can be proved by drawing the Pascal's triangle of all possible configurations. In particular, starting with one white and one black ball (i.e., x = y = 1 {\displaystyle x=y=1} ) the probability to have any number 1 ≤ n 1 + 1 ≤ n + 1 {\displaystyle 1\leq n_{1}+1\leq n+1} of white balls in the urn after n {\displaystyle n} draws is the same,
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