Let $\{ B\_i \}\_{i=1}^n$ be a set of $n$ ball in the unit cube $C$ of dimension $d$.


If I want to estimate
$$
\frac{ \lambda \left( \cup B\_i \right) }{\lambda\left( C \right) }, \tag{1}
$$
where $\lambda$ is Lebesgue Measure, is it ok to sample uniformly in the cube then count the number of point which fall in any ball? 


I ask, because I saw a [question](https://mathoverflow.net/q/979/24451) saying that estimating the volume of a convex body is hard, and one needs special algorithm. 


Is the estimation of (1) similar to the Dyer, Frieze, Kannan context, and a naive Monte Carlo is doomed to failed? If so why? 


From some numerical tests, it seems to work.  

Especially that I know in advance the volume of $\cup B\_i$ is at least half the volume of the cube. 



Estimating the volume of a union of balls
A naive Monte Carlo will always work, probabilistically, by the Law of Large Numbers. The problem only arises if you want guaranteed correctness, 100% chance as opposed to say 99.999%.


