# Why are random numbers necessary for a Monte Carlo simulation?

This may be somewhat of a question with an obvious answer, but I can not seem to understand the necessity of "truly" random numbers to make a Monte Carlo simulation a good one.

I understand that not all simulations require "truly" random numbers...

Does it have something to do with the probability that such a number/value may be generated multiple times, offsetting the approximation?

I found this, but I have a bit of a hard time wrapping my head around it; could anyone could explain/elaborate more simply?

• They don't have to be "truly" random. Look up quasi-Monte Carlo methods, for instance... – J. M. is a poor mathematician Feb 9 '12 at 2:51

When you have a choice, it is typically better to use systematic sampling. For example, suppose we didn't know how to do $\int_{-1/2}^{1/2} x d x$ in closed form. With Monte Carlo simulation, we'd generate $N$ random numbers between $-1/2$ and 1/2 and average them. We would get an average close to zero, but the average would only converge to zero in proportion to $1/\sqrt{N}$, which is pathetic. Even the most simpleminded numerical integration method would give an error falling off like $1/N$.
But sometimes you don't have a choice. You might have 1000 random variables, and it's not practical to do an integral of the form $\int\int\int\ldots dx_1 dx_2 dx_3 \ldots$ nested 1000 deep by systematic sampling, because if your number of grid points along each axis is $N$, the number of samples you have to integrate is $N^{1000}$.