Pólya's Theorem (1921)
George Pólya proved that a random walker on a 2D grid will always return to the origin—given enough time. The probability is exactly 1.
In 3D, this breaks down. A random walker has only a ~34% chance of ever returning. This is why mosquitoes in a flat field are inescapable, but a fly in a room might never land on you again.
Jordan Ellenberg uses this in Shape to explain malaria transmission: if mosquitoes move randomly, infected ones will inevitably return to bite again. Spatial probability governs disease spread.