Pi and Simulation

Quite interestingly, Pi appears in some probabilistic settings where geometry plays a role.

A well-known example is Buffon's problem. In this experiment, one throws a needle of length a on a grid of parallel lines at distances k (>a). The probability of the needle cutting a line is 2a/kPi. Click here for a proof.

Conversely, and in an ideal setting, Pi can be approximated by 2a/kn_h where n_h is the proportion of "hits" (when the needle cuts a line). ... Needles to say, even under such ideal conditions, this method would not be particularly efficient to achieve any descent accuracy. Buffon himself is said to have thrown quite a lot of needles, only to claim that he verified a few Pi digits.

A much more modern approach comes from simulation techniques. Computers can simulate random throws, one-dimensional needles and lines, and they can “perceive” whether the “needle” touches the “line” or not to a high accuracy. They are also very fast. It seems therefore reasonable to simulate the experiment. There is a catch though: random computer numbers are not really random. They are produced by complicated recursive formulas (pseudo-random generators) designed and tested so as not to produce patterns in the series “as far as it can be tested”. Such random series are very good for normal uses, but it is highly questionable whether they do not affect the outcome when it comes to calculating many digits of Pi.

Anyway, the subject has attracted a lot of attention and it is good for educational purposes. The reason is in the really interesting connection of randomness with one of the most fundamental constants of cosmos (not in the calculation of Pi digits themselves). Incidentally, in the context, it is more like "gambling and the cosmos" and such methods are appropriately called "Monte-Carlo" methods (don't take this to mean "light" or not serious, Monte-Carlo methods are more typically used to find solutions to very hard problems which are intractable with other more "conventional" methods).

Some relevant places in the Web:

The Virtual Laboratories in Probability and Statistics by Kyle Siegrist. This is a project aiming at "providing interactive, web-based modules for students and teachers of probability and statistics". There is a special page for Buffon's Needle Problem (under Geometric Models) with lots of exercises
Buffon's Needle/ An Analysis and Simulation, by George Reese - another good educational site
Buffon's Needle in Paul Kunkel's Mathematics Lessons

A similar but less impressive experiment (because the connection with Pi is obvious) is with darts.

In this figure, if the circle has radius R, its area is PiR² and the area of the circumscribing square is (2R)².
If you throw darts to the square at random, the proportion of them hitting the circle will tend to be
(area of circle)/(area of square) = Pi/4. This means again that Pi can be approximated by 4(no. of hits).