On the probability of union in the n-space

In this paper, our sets are orthants in $R^n$ and $N$, the number of them, is large ($N>n$). We introduce the modified inclusion–exclusion formula in order to efficiently calculate the probability of a union of such events. The new formula works in the bivariate case, and can also be used in $R^n,n\ge 3$ with a condition on the projected sets onto lower dimensional spaces. Numerical examples are presented.