Needle

by Pawel Strzelecki


Consider the following problem: a needle of length d is dropped onto a floor made of equal, parallel battens of width s>d (Fig. 1). What is the probability of the event Z consisting in the needle intersecting one of the joints between the battens?

Fig. 1.

The problem being symmetric, the position of the needle is determined by two numbers: the distance x of its centre from the nearest joint and the angle between the needle and the direction perpendicular to all the joints. All the positions co

,

which in the sequel will be our space of elementary events.

The so-called geometric probability will be defined by the formula , where |X| represents the area of X. No particular position of the needle being thus distinguished, the definition seems to be quite reasonable.

When one of the needle endpoints just touches the joint, we have . Hence the needle will intersect the joint if and only if . Therefore the event Z is the subset of the rectangle included under the diagram of the function (Fig. 2).

Fig. 2

This implies

If we assume s=2d, we get P(Z)=1/p . Now the law of large numbers implies that if the total number n of throws is big enough, the quotient of n over the number k of those throws after which the needle cuts a joint will be quite close to p .

Thus it turns out that throwing a needle onto the floor (or onto a lined sheet of paper) may be a way for determining an approximate value of p . The man who made this surprising invention in 1777 was Georges Leclerc de Buffon, a natural scientist, philosopher and bon-vivant, much better remembered as co-author, together with Daubenton and Lacépède, of the Histoire Naturelle in 44 volumes. Each of the three men has his own street in Paris today, in the Quartier Latin in the proximities of the Jardin Botanique.