Buffon's Coin and Needle Problems
Origin
George-Lois
Leclerc, Comte de Buffon (1707-1788) introduced his coin and
needle problems along with other geometric probability problems
and their solutions in a paper presented to the Royal Academy of
Science in 1733. Little is known regarding how he created these
problems or how he solved them, other than that he used ratios of
areas. However, since he first presented them, many variations
of the problems have appeared, and this work inspired
probabilists to begin to study geometric probabilities.
(Note that on this page, the word geometric
refers to
probabilities calculated using geometry, specifically ratios of
areas, rather than probabilities calculated from the geometric
distribution discussed on the
random variables page.)
Problems and Solutions
Buffon's Coin Problem
Consider randomly tossing a coin onto a square grid. What is the probability that when the coin is tossed, it falls entirely in one square without crossing any grid line? Assume that the diameter of the coin is smaller than the side length of each square in the grid.
Notice that a success
occurs exactly when the center
of the coin is at least one radius length away from any line
(see the figure below). This means that for a given square on
the grid, the success
area consists of the area of an
inner square that is smaller on each side by one radius length.
Then if the side lengths of the squares in the grid are $b$ and
the radius of the circle is $a$, the area of each square in the
grid is $b^2$, and the area of the blue square showing
successes is $(b-2a)^2$. Thus the probability of success is
$\frac{(b-2a)^2}{b^2}$.
Simulate the coin problem and a few of its variations using the applet below.
Buffon's Needle Problem
Buffon's needle problem is a bit more complicated.
Consider randomly tossing a needle onto a set of horizontal parallel lines. Calculate the probability that when the needle is tossed, it does not cross any of the lines. Assume that the length of the needle is less than the distance between parallel lines.
Using the applet below, notice that a success occurs exactly when the vertical distance between the midpoint of the needle and the parallel line is less than $\frac{1}{2}a\cdot cos(\theta)$ where $a$ is the length of the needle and $\theta$ is the acute angle between the needle and the vertical line connecting the midpoint of the needle vertically to the parallel line.No matter where the needle lands, its midpoint is at most $\frac{1}{2}b$ from a parallel lines that are $b$ apart. This means the sample space area can be limited to the area where the midpoint of the needle is between $0$ and $\frac{1}{b}$. Then the area that corresponds to a value for which the vertical distance, $x$, is less than $\frac{1}{2}a\cdot cos(\theta)$ is represented by the area under the curve shown below. The area representing success is $\int_{-\pi/2}^{\pi/2} \frac{1}{2}a\cdot cos(\theta)= a$. The whole area is $\frac{1}{2}b*\pi$. This means that the probability of success is $\frac{a}{\frac{1}{2}b\cdot \pi}=\frac{2a}{\pi b}$.
Finding an empirical probability by simulating the random experiment using an applet, such as the one found at https://mste.illinois.edu/activity/buffon/, gives a value that can be set equal to $\frac{2a}{\pi b}$. When the equation is solved for $\pi$, it will be an approximation for $\pi$.
Both Buffon's coin problem and Buffon's needle problem helped introduce geometric probability as a branch of probability theory. See the section on Buffon for biographical information as well as his other contributions to statistics.