Problem 1: Buffon's Needle Problem

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These web pages were modified from Kyle Siegrist's pages on Buffon's Coin Problem as can be found in his Virtual Laboratories in Probability and Statistics.

Consider the following problem, analogous to ones you solved in Problem 0:

Problem. Throw a coin at random on a floor tiled with congruent squares. What is the probability that the coin will not cross the edge of any square?

See Buffon's Coin Experiment. (View in Microsoft Internet Explorer).

Georges Louis Leclerc, Comte de Buffon, first considered this kind of problem in his 1777 paper, Sur le jeu de franc-carreau. Buffon is widely considered to be the first mathematician to study the field of geometric probability. Buffon's coin experiment is a very old and famous random experiment. The experiment consists of dropping a coin randomly on a floor covered with identically shaped tiles. The event of interest (that is, a successful outcome) is that the coin does not cross a crack between tiles. We will model Buffon's coin problem first with square tiles of side length L = 1. (Assuming the side length is 1 is equivalent to measuring distance in units of side length.)

First, let us define the experiment mathematically. As usual, we will idealize the physical objects by assuming that the coin is a perfect circle with radius r and that the cracks between tiles are line segments. A natural way to describe the outcome of the experiment is to record the center of the coin relative to the center of the tile where the coin happens to fall. More precisely, we will construct coordinate axes so that the tile where the coin falls occupies the square created by: -1/2 < x < 1/2 and -1/2 < y < 1/2. A toss is successful if the center of the coin lands in the blue square in the figure below.

Now, when the coin is tossed, we will denote the center of the coin by (X, Y) S so that S is our sample space and X and Y are our basic random variables. Finally, we will assume that r < 1/2, so that it is at least possible for the coin to fall inside the square without touching a crack.

Next, we need to define an appropriate probability measure that describes our basic random vector (X, Y). If the coin falls "randomly" on the floor, then it is natural to assume that (X, Y) is uniformly distributed on S.

So the probability that the coin will not cross a crack is given by:

where

The probabiltiy that the coin will cross a crack is:

(why?)

Use calculus (or your knowledge of parabolas) to show that the probability that the coin will cross a crack, expressed as a function of r, is given by the graph given below:

Probability of a crack crossing

Now let's consider the following infamous problem:

Buffon's Needle Problem. Throw a needle at random on a floor covered by equidisdant parallel lines. What is the probability that the needle will cross at least one of the lines?

See Buffon's Needle Experiment. (View in Microsoft Internet Explorer.)

Our first step is to define the experiment mathematically. Again we idealize the physical objects by assuming that the floorboards have a uniform width. Let L be the length of the needle and d be the width of the floorboard. We will also assume that the needle has length L < d so that the needle cannot cross more than one crack. Finally, we assume that the cracks between the floorboards and the needle are line segments.

We can assume that one of the parallel lines is the x-axis, and that all the lines are horizontal (why?). Furthermore, we need only consider what can happen when the needle lands between a single "strip" as indicated in the picture below.

Here, y is the distance from the lowest point of the needle to the nearest line above it. If the needle happens to fall horizontally, then y is simply the distance from the needle to the nearest line above it. So 0 < y < d. Let represent the angle between the needle and the positive x-axis, so that 0 < < . Then the ordered pair ( , y) uniquely determines the position of the needle up to vertical translations by integer multiples of d, and by any horizontal translation. So the square:

forms the sample space. The quantity y = L sin () is then the vertical height of the needle. Now the needle will intersect one of the lines if and only if

So the event space is given by the region that lies between the curves and . Thus the probability that the needle intersects one of the lines is given by:

or

Let d = 3 and L = 2. The region that lies between the curves and is shaded red below.

If both L and d are known, Buffon's Needle experiment can be used to estimate the value of . There are several Java applets that do this, and you may find links to them on the links page. Suppose that we run Buffon's needle experiment a large number of times. By the law of large numbers, the proportion of crack crossings should be about the same as the probability of a crack crossing. More precisely, we will denote the number of crack crossings in the first n runs by Nn. Note that Nn is a random variable for the compound experiment that consists of n replications of the basic needle experiment. Thus, if n is large, we should have

Nn / n ~ 2L / and hence ~ 2Ln / Nn.

Finally, we should note that as a practical matter, Buffon's needle experiment is not a very efficient method of approximating . According to Richard Durrett, to estimate to four decimal places with L = 1 / 2 would require about 100 million tosses!

PROBLEM 1 - Buffon's Needle Problem: Throw a needle of length L at random on a floor covered by equidisdant parallel lines d units apart. What is the probability that the needle will cross at least one of the lines? (Note that in this case, L is not necessarily less than d.)

Other References: De-lin, Santalo.