Random Walk

A mosquito starts at the center of a 100×100 grid. Each step it moves up, down, left, or right at random. Where does it end up after 100 steps?

Steps

(50, 50)

Position

Distance from start

Spatial Probability

The green circles show probability zones. After 100 random steps, 68% of walks end within 10 units of the start, 95% within 20 units.

The grey zigzag shows the "optimal" escape path—71 steps of pure up-right-up-right to reach the outer circle.

P(optimal path) = (1/4)^71 ≈ 10^-43

That's a 1 in 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000 chance.

The Square Root Law

After n random steps, the expected distance from start is √n, not n.

100 steps → ~10 units away
10,000 steps → ~100 units away

Randomness cancels itself out. Steps in opposite directions undo each other.

From Jordan Ellenberg's "Shape"

This is how mosquitoes fly, molecules diffuse, and stock prices fluctuate. The random walk is nature's favorite path.