Buffon's Needle Problem Revisted
Let's go back and consider again the problem above involving a convex set K1 that is contained in a bounded convex set K, where we wanted to find the probability that a random strip of breadth a that intersects K also intersects K1. Let us assume now that K is a set of constant breadth D and K1 is any convex set that is contained in K, as in the diagram below. Recall that if K1 is any convex set with diameter  , then K1 can be contained in K. We saw that the probability that a strip B having breadth a that intersects K will also intersect K1 is given by equation (5). But, instead of thinking that K's position is fixed and the strip B is thrown down at random, let us now assume that the plane is covered with parallel strips B at a distance D from one another, and the set K, along with the set K1, are thown at random onto this plane. When the event is done in this fashion, observe that K will intersect exactly one strip, except, of course, when it is tangent to two strips, which has a measure of zero. Thus, the probability that K1 will intersect a strip is again given by equation (5). Hence, knowing that  we have the generalized form of Buffon's Needle Problem
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| (1) |
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| Problem 18:
Explain how this equation relates to the solution of Buffon's Needle Problem given in the Introduction section.
Solution
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Further Generalizations of Buffon's Needle
Let us now consider Buffon's needle problem without the restriction that the length of the needle is less than the distance between the parallel strips. In other words, the diameter of K1 need not be less than the diameter of K.
- Lemma: Let B be a strip of breadth D and K1 be a bounded convex domain with width function
 , and let function  be defined by
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- Thus, the measure of the set that K1 is contained within the strip B is given by
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(3) |
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- Proof: Observe that the density of B is
- Theorem: Suppose the plane is covered by parallel strips of breadth a and are separated by a distance D, and that K1 is a bounded convex domain with width function . If K1 is dropped onto the plane at random, then the probability that K1 intersects at least one strip is given by
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| (4) |
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- where is defined as before.
Proof: Let us assume that K1 is fixed in the plane, and the parallel strips are dropped at random. First, we want to find the measure of all positions of the parallel strips. Suppose we randomly drop a disk K of diameter D onto the plane. It follows that the measure of all positions of the parallel strips is equal to the measure of all positions that the disk K intersects a strip B of breadth a, which we found out in Problem 17 (b), is given by
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(5) |
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- Now, let us consider the region between two strips, which can be viewed as a strip of breadth D, which we will denote B1. By the lemma above, we find that the measure of the set that K1 is contained inside B1 is given by
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(6) |
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- Thus, the measure of the set that the parallel strips of breadth a intersect K1 is equal to the difference between equation (5) and equation (6); that is,
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(7) |
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- Equation (7) is the event space, and since equation (5) is the sample space, the Theorem follows.
- Corollary: Suppose that K1 reduces to a line segment, and that a=0. Thus, the breadth function of the line segment is given by
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(8) |
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- (1) If
 , then equation (4) along with the breadth function above give the solution to Buffon's needle problem
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(9) |
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- (2) If
 , then we have
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| (10) |
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- Thus, with equation (4), we have the solution of Buffon's long needle problem
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| (11) |
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- Now, let us consider the probability that the convex set K1 intersects a particular number of strips. Consider the diagram below.
We want to define a function, call it  , that we can integrate to give us the measure of the set that K1 intersects exactly h strips. Let us define the following:
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(12) |
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- Thus, if you want the probability that K1 intersects h strips, then the breadth a1 of K1, in the direction perpendicular to the strips, must satisfy
 in order to possible intersect h strips. In order for K1 to meet two strips, then its breadth must be at least D. To meet three strips, the breadth of K1 must be at least 2D + a, and so on. Let us define .
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| (13) |
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- Thus, we can make the following statement:
- Theorem: Let the plane be covered in parallel strips of breadth a, separated from each other by a distance D, and suppose K1 is a bounded convex set with breadth function
 . If K1 is dropped at random onto the plane, then the probability that it intersects exactly h strips is given by
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| (14) |
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