Consider two nonparallel planes $X,Y$ that go through to the center of a sphere of radius one. This cuts the sphere into 4 parts $A,B,C,D$. The two parts $A,B$ are on the same side of $X$, such that the volume of $A$ is greater than the volume of $B$. If the expected value of the volume of $A$ can be expressed as $\frac{a}b\cdot\pi$, where $\gcd(a,b)=1$ find $a+b$.

Guest Feb 2, 2021