# probability – Uniform distribution on the unit sphere

Let $$vec uinmathbb{R}^d$$ be a random unit vector, with uniform distribution on the surface of the unit sphere. For a fixed unit vector $$vec v$$, what is the following probability?

$$Prleft( left| langle vec u, vec v rangle right| geq xi right)$$

My guess is that it would be something like $$1-xi$$, since (presumably, I’m not sure about that either) $$langle vec u, vec v rangle$$ also has a uniform distribution for fixed unit vector $$vec v$$ and uniformly-distributed-on-surface-of-unit-sphere $$vec u$$, but I’m not aware of distributions on the surface of spheres at all, so if someone could enlighten me, please do.