A Pólya urn is a sampling model with reinforcement. In the classical two-color version, an urn initially contains 
red and 
black balls, where 
and 
are positive integers. At each step, a ball is
chosen uniformly from the urn, returned, and accompanied by 
additional balls of the same color, where 
is a fixed positive integer.
The successive colors form a stochastic process
(Siegrist).
Let 
be the number of red draws among the
first 
draws, where 
is a nonnegative integer.
The conditional probability of red on
the next draw is
 |
The probability mass function of 
is
 |
for integer 
with 
,
where 
is the beta
function and 
is a binomial coefficient. This is the beta binomial distribution with parameters
, 
, and 
(Siegrist).
The probability of any specified color sequence depends only on its numbers of red and black draws, not their order. Multiplying the successive
conditional probabilities gives one factor
for each red draw and one factor
for each black draw, where 
counts preceding draws of that color. Consequently, when 
, 
has a discrete
uniform distribution on 0, 1, ..., 
(Siegrist).
