 Pólya urn model

Begin with positive counts of two colours. At each step select one ball uniformly and replace it together with one new ball of its colour. If the total initial count is $m$, the blue proportion satisfies $X_{n+1}-X_n=(I_{n+1}-X_n)/(m+n+1)$, where the next blue indicator has conditional mean $X_n$. Thus $X_n$ is a bounded martingale and converges almost surely and in $L^2$ to a random proportion. If uniforms generate the draws, the update from time $n$ to $n+1$ must use the fresh $U_{n+1}$ relative to ${\mathcal{F}}_n=\sigma (U_1,\dots ,U_n)$.