TITLE: Are all Monte Carlo algorithms to approximate $\pi$ equivalent?
QUESTION [5 upvotes]: There are several ways one can approximate $\pi$ using a Monte-Carlo type algorithm. For example, one can draw random points in the unit square, and approximate $\pi$ via the ratio of points that fall in the unit disk. Another way would be to use an algorithm based on Buffon's needle problem: from the proportion of needles that fall across the parallel lines, one can approximate $\pi$. One can presumably devise many other random algorithms to estimate $\pi$. 
My question is: in terms of speed of convergence, are all these algorithms equivalent? Or are there some that will yield correct decimals of $\pi$ faster? What is the Monte Carlo method to estimate $\pi$ that has the fastest convergence?

REPLY [2 votes]: All random algorithms that approximate $\pi$ by trying something that succeeds with probability $f(\pi)$ lots of times, getting an estimate of $f(\pi)$, and using that to estimate $\pi$ will be approximately equivalent. (Whether the thing we try is "drop a needle that interects a line" or "pick a point that lands inside a circle" or anything else.)
The reason is that their convergence rate will have nothing to do with the method at all. If the probability we're trying to estimate is $p$, then after $n$ trials the number of successes will follow a binomial distribution, and the typical number of successes will be $np \pm C \sqrt{np(1-p)}$. This gives an estimate of $p$ to within $O(\frac1{\sqrt n})$ error, which means millions of trials before we can estimate $p$ to three decimal places.
I only say "approximately" equivalent because the probability we're estimating can depend on $\pi$ in lots of ways. If the probability of success is $\frac12 + \frac{\pi}{1000}$, then it will take many trials before our estimate of this probability becomes significantly better than "it's about $\frac12$". On the other hand, if the probability of success is $10000\pi - 31415$, then estimating that the probability is approximately $0.9$ is enough to give use $\pi$ to five digits after the decimal: $\pi \approx 3.14159$.