In 1923, the mathematician George Pólya described an urn that starts fair and ends lopsided. The model helps explain why small advantages tend to become permanent ones.
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It contains two balls — one red, one blue. A perfectly symmetric starting point.
Draw a ball at random. Note its color. Put it back, then add a new ball of the same color.
Whichever color gets drawn becomes slightly more likely to be drawn again.
The first draw is a fair coin flip. But the second draw already isn't — whichever color won the first round now holds a 2-to-1 edge.
Early draws carry outsized influence.
A pattern has formed. The proportion of red to blue has drifted away from 50-50, pushed by the accumulation of those early draws.
The proportion barely moves. What started as a random walk has settled into a fixed ratio. Mathematicians call this path dependence — the system's long-run behavior was determined by its early history.
Plotted step by step, the red fraction swings wildly at first, then converges. The first ~20 draws contain most of the information about where it will end up.
Same urn. Same rule. What happens?
The mechanism is identical, but the result is not. Each run of the urn locks into a different proportion, determined by the particular sequence of early draws.
We simulated the urn 20 separate times, each starting from the same initial state. The chart below shows the red proportion in each simulation over 500 draws.
Each line represents an independent urn. They all begin at 50% and diverge.
20 independent simulations, 500 steps each. Initial conditions: 1 red, 1 blue.
You might expect the final proportions to cluster near 50%. They don't. With one red and one blue ball to start, the long-run red fraction follows a uniform distribution — every value between 0% and 100% is equally likely.
This is a known result. The limiting proportion of a Pólya urn starting with a red and b blue balls follows a Beta(a, b) distribution. When a = b = 1, that's Uniform(0, 1).
2,000 simulated urns, 500 draws each. The histogram is roughly flat, consistent with a uniform distribution.
Green dashed line shows the expected count per bin under a uniform distribution.
Pólya's urn formalizes a pattern called preferential attachment: things that are popular become more popular. It has been used to model how cities grow, how scientific citations accumulate, how network effects entrench dominant platforms, and how competing technologies converge on a single standard.
The lesson is consistent: in systems where success breeds success, small early differences can produce large, lasting asymmetries — even when the starting conditions are perfectly fair.
Adjust the initial number of red and blue balls. With larger starting values, the urn's proportion is more constrained; with smaller values, early draws have more leverage.