Why use Monte Carlo method instead of a simple grid? when integrating a function or in complex simulations, I have seen the Monte Carlo method is widely used. I'm asking myself why one doesn't generate a grid of points to integrate a function instead of drawing random points. Wouldn't that bring more exact results?
 A: Previous comments are right in that simulation is easier to use in multidimensional problems. However, there are ways to address your concern - take a look at http://en.wikipedia.org/wiki/Halton_sequence and 
http://en.wikipedia.org/wiki/Sparse_grid.
A: I found chapter 1 and 2 of these lecture notes helpful when I asked the same question myself a few years ago. A short summary: A grid with $N$ points in 20 dimensional space will demand $N^{20}$ function evaluations. That is a lot. By using Monte Carlo simulation, we dodge the curse of dimensionality to some extent. The convergence of a Monte Carlo simulation is $O(N^{-1/2})$ which is, albeit pretty slow, dimensionally independent.
A: Sure it does; however it comes with much larger CPU usage. The problem increases especially in many dimensions, where grids become effectively unusable.
A: While one typically things of rejection sampling when considering Monte Carlo, Markov Chain Monte Carlo allows one to explore a multi-dimensional parameter space more efficiently than with a grid (or rejection sampling for that matter). How MCMC can be used for integration is clearly stated in this tutorial- http://bioinformatics.med.utah.edu/~alun/teach/stats/week09.pdf 
A: Two things - 


*

*Faster convergence by avoiding curse of dimensionality. Because most points in a grid lie on the same hyper plane without contributing significantly extra information. Random points fill the N-dimensional space evenly. LDS is even better.

*Sometimes for Monte carlo methods we need statistically random points in no particular order. An ordered sequence of grid points will result in poor statistical properties.  
