Given two random unit vectors in $\mathbb R^n$, one can consider the probability that the absolute value of their dot product is $x$, and thus form a probability density function. Supposing that the random vectors were drawn from a uniform distribution with respect to the surface of the sphere that they live in, what is the probability that two vectors will have dot product less than $T$?
My intuition is that this should be proportional to the surface area of the "cap" of the cone containing all angles less than $\arccos(T)$. But how to compute this?
Extra question: if we choose the vectors by drawing each component of each vector from a mean zero, unit variance Gaussian distribution, does the answer change?