Georges-Louis Leclerc Comte de Buffon
(1707 - 1788)
An elegant result
If a needle of length *2a* is dropped on a parquet formed of floorboard of
width *2b*,
the probability that the needle cuts one of the lines of this parquet is
Reference from the Encyclopedie of Integer Sequences
: A060294
Slices of life
Georges Louis Leclerc was born in 1707. Naturally,
his main work is that of a naturalist. So, *general and particular natural History* (15 volumes!), the *natural History of birds* (9 vol.), *Supplement in the
natural history* (7 vol.) and other *natural History of minerals and treaty
of the magnet* (5 vol.) will be worth him the greatest celebrity... Especially
since his style is very pleasant... But, philosopher also , this excellent administrator
was member of all the great European academies and was interested in mathematics...
Oh, oh!.. In the *Attempt of moral arithmetic* published in 1777, an entitled
volume *Memory on the game of the "franc carreau" *presents indeed
the famous problem of the needle...
About
This problem is one of the first to bring in probability
and *Pi*.
Another proof of the omnipresence of this number in mathematics!!! Obviously, this
presence is not foreign to the original geometrical definition of *Pi* as perimeter of the
circle of diameter *1*... But the subtle mixture with the analysis is very pleasant! So
pleasant moreover that two proofs are told below! Naturally, the floor is supposed
flat (let us stay in euclidian spaces!).
In fact, it is not necessary to hope to obtain a good
estimation of *Pi* by going only to buy a package of needles on the corner of the street!!!
A preciseness of *10 *^{-3}^{ }is obtained with a probability of *95 %* from *888 697* throws!
Proofs
1) A little
of geometry...
With hypotheses above (length a needle *2a*, width of floorboard *2b*) :
Let us appoint by y the distance of the middle of the needle () to the line of
bottom and by F the angle between the needle and the line .
If or then respectively, the needle cuts the line of the height or the one of the bottom...
So,
there will be intersection if the point *P* defined by its coordinates *(ß,
y)* belongs to the zone hatched of the graph
opposite:
Now, the distribution of *y* on *[0,2b]* and *ß* on *[0,/2]* is uniform and the looked probability represent the ratio of the area of the surface
hatched to the area of the rectangle *[0,2b]*[0,/2]* (which contains all the possible cases). Now the area of
the rectangle is A=/2*2b=b.
and the area of the hatched surface is B=2 (the *2* hatched surfaces have the same area because they are symmetric
with regard to the line of equation *y=b*)
So the looked probability is *p===.*
If we have a great patience, it is enough to throw a great number of time n the needle
on the ground and to count the number of intersections. By setting b=2 and a=1, the
law of the great numbers allows us to conclude that .
2) Emile
Borel also found a crafty and fast demonstration of this Buffon's result... Whatever
is the form, the number of intersections of a needle with the edge of slats is proportional
to its length *2a* and inversely proportional to the slats of width *2b*. So he can write
in the form of *Ka/b*. Remain to find constant *K*...
And there, craftiness! Let us take a circular needle of diameter *2b*. The perimeter, so
the length, is obviously 2b.Whatever
the way of which it falls, it cuts exactly twice lines so we deduct from it 2=Ka/b=Kb/b=K,
where from K=2/ and
the looked probability is !
Trials
Some patient individuals tempted their chance in
the trow of needles... Notably Wolf in *1850* which arms itself with *5000* needles with *a=0,8b* and observes *2532* intersections that brings him to the estimation *=**3,1**596*.
For my part, I have not yet tried. In my opinion, principle is especially interesting,
but experiment is long and little effective, that limits the interest. But nothing
prevents from automating process!
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