Pólya Urn Model: Reinforcement Dynamics

Rich-get-richer: each draw reinforces itself — convergence to a random Beta-distributed proportion

0.50
red proportion
0.50
blue proportion
2
total balls
0
draws
1
1
1
10/frame
Pólya urn (1923): Start with a red + b blue balls. Draw one; return it plus k new balls of same color. Repeat.
De Finetti's theorem: The limiting red fraction X converges a.s. to a Beta(a,b) random variable. E[X] = a/(a+b), Var[X] = ab/((a+b)²(a+b+1)).
Exchangeability: The sequence is exchangeable — the order doesn't matter, only the counts. This characterizes Pólya urns completely.
Power laws: Multiple-color Pólya urns give Dirichlet distributions; with fitness: Yule process → power-law degree distributions (preferential attachment).