This question came up in my research: What is the probability that $Ax\geq0$ where $x$ is a vector of iid gaussians and $A$ is matrix of $1$s and $0$s?


So far I only figured out that I can do Monte Carlo or assume inequality encoded by each row of $A$ is independent for all the other inequalities; then there's an explicit formula *for and upper bound using Boole inquality.*



Computing probability that $Ax\geq0$ where $x$ is a vector of iid gaussians and $A$ is matrix of $1$s and $0$s
You can determine the exact distribution of the Gaussian vector Ax but there is no closed formula for estimating its cumulative distribution function (see <http://en.wikipedia.org/wiki/Multivariate_normal_distribution#Cumulative_distribution_function>). 


You will have to estimate it numerically. Alan Genz's worked a lot about this subject it seems. I found a related question <https://stackoverflow.com/questions/11109465/multivariate-normal-cdf-in-c-c-or-fortran>.
Implementation exists in R, Matlab and Fortran (!) at least. 


Hope it helps. 


