Randomness and the Appearance of Pattern
Or: Wolfgang Köhler Meets Degrees of Freedom
  "How often things occur by the merest chance."
  —Terence, Phormio
Below this text, to the left, is a matrix of 4x4=16 cells. When you click the button labeled "Click!", some of these cells will become blue and some will become white. Each time you click the line, any particular one of the 16 cells has a 50% chance of becoming blue and a 50% chance of becoming white. Any pattern that seems to emerge is therefore not the result of design, but derives instead from nothing other than "the merest chance." Yet, if your eye is anything like mine, you will find "organized" patterns emerging on almost every click.


I suspect this effect—"organized" patterns emerging from "the merest chance"—derives in part from what the Gestalt psychologists described in the early decades of the 20th century as the tendency of the mind to impose order upon its perceptions, and in part from what statisticians fuss over under the heading of "degrees of freedom." Meanwhile, to keep this site "computational," here are the relevant calculations.
Total number of possible blue/white combinations:
   216 = 65,536
Probability of any particular blue/white combination:
   0.516 = 0.000,015

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