BUFFON’S NEEDLE PROBLEM
Abby Yinger
Probability theory and stochastic processes for additional applications
• Geometric probability and stochastic geometry
BRIEF HISTORY
 Buffon’s Needle problem is one of the oldest problems in the field of geometrical
probability.
 It was first proposed in 1733 by Georges-Louis Leclerc, Comte de Buffon and later
solved by Buffon in 1777.
WHAT IS THIS PROBLEM?
 Find the probability that a needle of length t will land on a line, given a floor with
equally spaced parallel lines a distance d apart.
THE SOLUTION
 To solve this problem we need to look at three different cases:
 Case One: Where the object being thrown (a needle in Buffon's case) is equal to the
distance between the lines
 Case Two: Where the object being thrown is smaller than the distance between the
lines.
 Case Three: Where the object being thrown is larger than the distance between the
lines.
CASE ONE
 Let L be the length of the needle and therefore the distance between the lines in the floor.
Suppose we throw the needle at an angle 𝜃 considered to be between 0 and 𝜋 by symmetry (if
the angle is greater than 𝜋 , such as an angle 𝜋 +𝜃, the needle will be in the same position as
when thrown with an angle 𝜃 itself). We can draw a line parallel to the lines on the floor, passing
through the center of the needle. Let d be the distance from the center of the needle to the
closest line. As shown in Figure 1, if 𝑑 ≤ 𝑥 the needle hits the line, but if this is not true the
needle does not hit the line.
CASE ONE CONT.
 By trigonometry we can see that 𝑥 = 𝑙/2𝑠𝑖𝑛𝜃. We need to find the probability of
that occurring. If we plot the graph of 𝑑 = 𝑙/2 𝑠𝑖𝑛𝜃, for the specified values of 𝜃, we
can see that for the values under the sin curve d is smaller than the function and
therefore the needle hits the curve. From this we can find the probability of the
needle hitting the line by taking the ratio of the area under the curve with respect
to the rectangle 𝑙/2 × π, which represents the total amount of possible cases. We
find this to be:
𝑙
2
 𝑃 (𝑑 ≤ 𝑠𝑖𝑛𝜃) =
𝜋 𝑙 𝑠𝑖𝑛𝜃𝑑𝜃
0 2
𝑙
×
2
𝜋
=
𝑙
2
(−cos𝜃) 𝜋0 −ܿ‫ 𝜋𝑠݋‬−(−ܿ‫𝑠݋‬0) 1+1
=
=
= 2/𝜋.
𝑙
𝜋
𝜋
×𝜋
2
From
this relation we find the probability that the needle touches the line to be 0.63662
or 63.662%.
SOURCES
 http://mste.illinois.edu/reese/buffon/buffon.html
 http://mathworld.wolfram.com/BuffonsNeedleProblem.html
 http://www.math.dartmouth.edu/~m20f11/RezhdoProj.pdf