Comment by bonzini

> What's the intuition here? Law of large numbers?

For unit vectors the cosine of the angle between them is a1*b1+a2*b2+...+an*bn.

Each of the terms has mean 0 and when you sum many of them the sum concentrates closer and closer to 0 (intuitively the positive and negative terms will tend to cancel out, and in fact the standard deviation is 1/√n).