The comment has a point. It is possible that the particular deterministic rule that chooses points could interact with the shape (quarter circle).

In this case, the error would not necessarily go to zero as the number of points increases without bound.

In the particular case of a grid based deterministic probe, and a quarter-circle target, it seems clear that this would not happen.

But consider another example where the underlying target was "all points with rational coordinates". All the probes in a grid sampling scheme would hit the target, but the target has measure zero.

Incidentally, the idea of using a deterministic, low variability sequence for sampling is called quasi Monte Carlo (http://en.wikipedia.org/wiki/Quasi-Monte_Carlo_method). It can give almost order 1/n convergence, much better than the 1/sqrt(n) convergence possible with ordinary Monte Carlo.

> But consider another example where the underlying target was "all points with rational coordinates". All the probes in a grid sampling scheme would hit the target, but the target has measure zero.

And how exactly would the Monte Carlo version differ here? Every single random number generated is going to be rational isn't it?

Every single random number being generated will be rational, but that is much finer granularity than any fixed grid that you might be using.

Unless your fixed grid includes every single rational point that could be expressed by the (P)RNG.

It's just a smart-aleck comment pointing out that there is a "bias" because we're using a finite number of points and we get the same exact answer every time -- and since that answer isn't exactly equal to pi, the procedure is "biased".

Which no one cares about because (1) this isn't a probabilistic process in the first place, so "bias" itself is meaningless, and (2) there is now an exact bound on the error, unlike the previous case (I guess you can think of this as the "variance" if you want).

Imagine if you were sampling the amount of black area in a checker board.