Two random directions in space

Idea. Fix the first direction at the north pole. The angle is less than $60°$ when the second point lies in the cap above height $\cos 60°=\frac{1}{2}$. For a sphere, the height $z$ of a uniform point is uniform on $[-1,1]$, so the answer is a length ratio.

Step 1: fix the first direction. The distribution of the second point is the same whatever the first one is, so take the first to be $(0,0,1)$. The angle $\theta$ between them satisfies $\cos \theta =z$, the height of the second point.

Step 2: the height is uniform. The band of the unit sphere between heights $z$ and $z+dz$ has area $2\pi dz$, whatever $z$ (the band gets narrower near the poles but also more slanted, and the two effects cancel exactly). So $z$ is uniform on $[-1,1]$.

Step 3. $$\mathbb{P}(\theta <60°)=\mathbb{P}\left(z>\frac{1}{2}\right)=\frac{1-1/2}{2}=\frac{1}{4}.$$

In other dimensions. The cosine of the angle is uniform only in three dimensions. In $n$ dimensions its density is proportional to $(1-c^2{)}^{(n-3)/2}$, so for large $n$ random directions are almost always nearly orthogonal: $\mathbb{E}[{\cos }^2\theta ]=\frac{1}{n}$.