In mathematics, Buffon's needle problem is a question first posed in the 18th century by Georges-Louis Leclerc, Comte de Buffon: suppose we have a floor made of parallel strips of wood, each the same width, and we drop a needle onto the floor. What is the probability that the needle will lie across a line between two strips? Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... (17th century - 18th century - 19th century - more centuries) As a means of recording the passage of time, the 18th century refers to the century that lasted from 1701 through 1800. ... Georges-Louis Leclerc, Comte de Buffon, by François-Hubert Drouais (1727-1775). ... A hardwood floor (parquetry) is a popular feature in many houses. ... Parallel is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. ... For other uses, see Wood (disambiguation). ... Needles used for sewing A sewing needle is a long slender object with a pointed tip. ... Probability is the likelihood that something is the case or will happen. ...

Using integral geometry, the problem can be solved to get a Monte Carlo method to approximate π. In mathematics, the term integral geometry in is used in two ways, which, although related, imply different views of the content of the subject. ... Monte Carlo methods are a widely used class of computational algorithms for simulating the behavior of various physical and mathematical systems, and for other computations. ... When a circles diameter is 1, its circumference is Ï€. Pi or Ï€ is the ratio of a circles circumference to its diameter in Euclidean geometry, approximately 3. ...

Solution

The a needle lies across a line, while the b needle does not.

The problem in more mathematical terms is: Given a needle of length dropped on a plane ruled with parallel lines t units apart, what is the probability that the needle will cross a line? Image File history File links No higher resolution available. ...

Let x be the distance from the center of the needle to the closest line, let θ be the acute angle between the needle and the lines, and let .

The probability density function of x between 0 and t/2 is In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ...

The probability density function of θ between 0 and π/2 is

The two random variables, x and θ, are independent, so the joint probability density function is the product A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. ...

The needle crosses a line if

Integrating the joint probability density function gives the probability that the needle will cross a line:

For n needles dropped with h of the needles crossing lines, the probability is

which can be solved for π to get

Now suppose . In this case the probability that the needle will cross a line is

Lazzarini's estimate

Mario Lazzarini, an Italian mathematician, performed the Buffon's needle experiment in 1901. Tossing a needle 3408 times, he attained the well-known estimate 355/113 for π, which is a very accurate value, differing from π by no more than 3×10⁻⁷. This is an impressive result, but is something of a cheat. Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... Year 1901 (MCMI) was a common year starting on Tuesday (link will display calendar) of the Gregorian calendar (or a common year starting on Monday [1] of the 13-day-slower Julian calendar). ...

Lazzarini chose needles whose length was 5/6 of the width of the strips of wood. In this case, the probability that the needles will cross the lines is 5/3π. Thus if one were to drop n needles and get x crossings, one would estimate π as

π ≈ 5/3 · n/x

π is very nearly 355/113; in fact, there is no better rational approximation with fewer than 5 digits in the numerator and denominator. So if one had n and x such that: When a circles diameter is 1, its circumference is Ï€. Pi or Ï€ is the ratio of a circles circumference to its diameter in Euclidean geometry, approximately 3. ...

355/113 = 5/3 · n/x

or equivalently,

x = 113n/213

one would derive an unexpectedly accurate approximation to π, simply because the fraction 355/113 happens to be so close to the correct value. But this is easily arranged. To do this, one should pick n as a multiple of 213, because then 113n/213 is an integer; one then drops n needles, and hopes for exactly x = 113n/213 successes.

If one drops 213 needles and happens to get 113 successes, then one can triumphantly report an estimate of π accurate to six decimal places. If not, one can just do 213 more trials and hope for a total of 226 successes; if not, just repeat as necessary. Lazzarini performed 3408 = 213 · 16 trials, making it seem likely that this is the strategy he used to obtain his "estimate".

See also

• Buffon's noodle

In geometric probability, the problem of Buffons noodle is a variation on the well-known problem of Buffons needle, named after Georges-Louis Leclerc, Comte de Buffon who lived in the 18th century. ...

External links

• Buffon's Needle at cut-the-knot
• Math Surprises: Buffon's Noodle at cut-the-knot
• MSTE: Buffon's Needle
• Buffon's Needle Java Applet
• Estimating PI Visualization (Flash)
• Ramaley, J. F. (Oct 1969). "Buffon's Noodle Problem". The American Mathematical Monthly 76 (8): 916-918.