Let $\mathbb{S}^{d-1}=\{v\in\mathbb{R}^d:\|v\|\_2=1\}$, namely $d-$dimensional sphere. It is well-known that if a random vector $X$ is distributed uniformly on $\mathbb{S}^{d-1}$, then there exists i.i.d. standard normal random variables $N\_1,\dots,N\_d$ such that
$$
X \stackrel{d}{=}(N\_1/N,\cdots,N\_d/N),
$$
where $N=\sqrt{N\_1^2+\cdots+N\_d^2}$. 


Now, my question is the following. Suppose that I am interested in understanding the size of a certain subset $S$ of $\mathbb{S}^{d-1}$ (with respect to $(d-1)-$dimensional Lebesgue measure). One can then set
$$
\mathbb{P}(X\in S) = {\rm Size}(S)/{\rm Size}(\mathbb{S}^{d-1}).
$$
Here, the presence of Gaussianity often makes the computation simple, thereby suggesting a potential approach for computing the size of $S$. Is this type of an approach well-known? Namely, is there any work on calculating the sizes of complicated sets using this type of probabilistic reasoning?



Volume computation using probabilistic approach
This approach is of course well known. Clearly, it just says that 
$$P(X\in A)=P((N\_1,\dots,N\_d)\in C\_A),$$
where $A$ is a Borel subset of the unit sphere $S^{d-1}$ and $C\_A:=\mathbb R\_+A$ is the corresponding cone. 


The hard part is to compute the Gaussian measure, $P((N\_1,\dots,N\_d)\in C\_A)$, of the cone $C\_A$. 


This is hard even when $A$ is a spherical simplex. The work on this begins with Schläfli (1858), whose paper (I think) is hard to read. A more general result was obtained by [Plackett (1954)](https://www.jstor.org/stable/2332716?seq=1), who gave a recursive formula (see formulas (7) and (6) there) for the centered Gaussian measure of a simplicial cone in $\mathbb R^d$ (not necessarily with the vertex at the origin) in terms of a certain integral functional of certain partial derivatives of the Gaussian measure of a varying simplicial cone in $\mathbb R^{d-2}$. See also references in Plackett's paper, including the one to Schläfli. (Plackett is actually dealing with the obviously equivalent problem, where the simplicial cone is just $c+\mathbb R\_+^d$ for some $c\in\mathbb R^d$, but the Gaussian distribution is not necessarily the standard one.)


