Imagine a drunk person flipping a coin to decide the direction of every step. It is completely random, but can they eventually return to where they started? Surprisingly, the answer depends on the number of dimensions. Let them take a walk and see.
Pólya's theorem: In one dimension (a line) or two dimensions (a plane), a random walk will, given enough time, return to the origin with probability 1. But in three dimensions (space), the probability of returning is less than 100% — in three or more dimensions, there is a real chance of never returning.
Steps 0Current position 0
Steps 0Distance from origin 0
💡 When the number of steps is n, the average distance from the origin is proportional to √n. With 100 steps, it is about 10 units on average; with 10,000 steps, about 100 units — even if you increase the number of steps 100-fold, the distance grows only 10-fold.