Buffon's Needle
Buffon's Needle is a geometric probability problem asking for the chance that a needle, thrown randomly onto a floor with parallel lines, crosses a line. Its solution uses additivity of expected value to derive that the probability is 2/π for a needle as long as the line spacing.
THE IDEA
Buffon's Needle, Buffon's Noodle, Buffon's Circle
Georges-Louis LeClerc, Comte de Buffon, introduced the problem in 1733 as part of his work on geometric probability. The classic version: given a floor with parallel lines spaced 1 unit apart, and a needle of length 1, what is the probability that the needle crosses a line? Assume the position relative to the lines and the needle’s orientation are uniformly random.
Buffon solved this using calculus, but later Joseph-Émile Barbier discovered a simpler proof relying on linearity of expectation. The key insight: represent the expected number of crossings E for a needle of length L as L × p, where p is the unknown probability for a unit-length needle. By bending the needle into a circle of diameter 1 (circumference π), the expected crossings become exactly 2 (apart from exact tangencies, which have probability zero under the random-placement assumption). Thus, πp = 2, so p = 2/π.
This reasoning extends to any curve (polygonal or smooth), leading to "Buffon's Noodle" problem: the expected number of crossings of a curve of length L is L × (2/π).
When to use it
Estimating π by repeatedly throwing a needle and recording the fraction of crossings.
Calculating the expected number of times a thrown wire crosses fixed lines (e.g., spaghetti on a tiled floor).
Modeling random intersections in fields like materials science or traffic flow.
Demonstrating the power of linearity of expectation in a non-intuitive geometric probability setting.
FROM THEORY TO EVERYDAY LIFE
In practice
Classic Needle Throw
Throw a needle of length 1 onto a floor with parallel lines 1 unit apart. The probability it crosses a line is 2/π ≈ 0.6366. This can be verified by simulation or actual needle-throwing experiments.
Bent Needle (Triangle)
A bent needle forming a triangle with sides 1, 2, 2 has total length 5. Its expected number of crossings is 5 × (2/π) ≈ 3.183, which equals the sum of expected crossings of each straight segment separately, demonstrating additivity.
A MOMENT TO REFLECT
Put it to work.
How does bending the needle into a circle simplify the problem?
Why does additivity of expected value allow us to handle needles of arbitrary lengths and shapes?
Can you think of other physical systems where linearity of expectation gives a simple solution?
References
- Sur le jeu du franc-carreau — Georges-Louis LeClerc, Comte de Buffon (1733)
- Note sur les aiguilles de Buffon — Joseph-Émile Barbier (1860)
CONTINUE THE CONVERSATION
Source texts
- How Not to Be Wrong The Power of Mathematical Thinking (Ellenberg, Jordan)