Mathematical game: is the number Pi the result of chance?

This type of method is widely used today in mathematics. We talk about Monte-Carlo methods, in homage to this high place of chance.

Question from Buffon

In 1777, buffon raises the following problem: if we throw a needle of length l on a wooden floor whose slats are of width a (with l ≤ a), what is the probability p so that the needle falls astride two slats? “.

Surprising as it may seem, the answer involves the Pi number : the probability is equal to 2 I divided by pi has. This result gives an amazing way to calculate Pi by throwing needles on a wooden floor.

Needle jets

Carrying out Buffon’s experiment is not very complicated, just a bit tedious. Before reporting some historical results, note that the phenomenon does not depend on the scale used. The result is only a function of the ratio the. It turns out that several experimenters used separate ratios, here are their results:

• in 1850, Wolf threw 5,000 needles 8 units long on a floor whose slats were 10 units wide (ratio 0.8). He found 2,532 intersections;
• in 1855, Smith threw 3,204 needles (ratio 0.6) and found 1,218 intersections;
• in 1860, de Morgan threw 600 needles (ratio 1) and found 382 intersections;
• in 1864, Captain Fox threw 1,030 needles (ratio 0.75) and found 489 intersections;
• in 1901, Lozzerini threw 3,408 needles (ratio 0.83) and found 1,808 intersections;
• in 1925, Reina threw 2,520 needles (ratio 0.5419) and found 859 intersections.

We won’t name all the other brave and patient needle throwers but we get Pi values ​​between 3.13 and 3.17; which corresponds well to reality but shows that it takes a very large number of jets to obtain a very low precision.

Monte Carlo methods

The calculation methods thus relying on the chance are called Monte Carlo methods. John von Neumann (1903 – 1957) recommended their use within the framework of the Manhattan project of construction of the first atomic bomb.