Solution 3: Buffon's Needle Problem - II.

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(a)

We can restrict our attention to events in B. What then is the probability that such an event also lies in A? By the Venn diagram we can see that this probability must be

(b)

Now we can restrict our attention to events in . Since the events in lie in both A and B, then the quantity counts such events twice. Thus,

More formally, since and are both disjoint unions, then by the axiom of additivity we must have that and . Similarly is a disjoint union and by the axiom of countable additivity and our formulas above for P(A) and P(B), we can write

(c)

If we consder tossing a circle of radius a onto the plane, then the expectation value for the number of intersections is 4. The Java applet below (a = 1.76) will let you move a circle of radius a around the plane. You will see that "almost all" circles generate four intersection points. [Can you find the circles that have only three intersection points?]

Please enable Java for an interactive construction (with Cinderella).

Created with Cinderella

Recall that E is linear, so that for some real constant r, where l is the length of a curve. Thus we must have that since our circle has length , and so . Now for any needle of length l < a, we know by the defintion of expectation value that the expected number of intersections of our needle with the square grid is given by E(l). [Note that since the probabilities are all equal to zero for n > 2, as our needle can never have three or more intersections.] Thus,



EXERCISES:

  1. Broken Segment Buffon Problem: Let the plane be ruled by equidistant parallel lines d units apart. Suppose that we drop an "elbow" consisting of a segment X1 of length a attached by a rotating joint to a segment X2 of length b. Let c denote the distance from one endpoint of the elbow to the other, and assume that c < d. What is the probability that this elbow will intersect a line exactly once? Exactly twice? Can the elbow intersect three or more times? Explain. The elbow intersects a line once in Figure 1 and twice in Figure 2. You may use the fact that , where p0, p1, and p2 denote the probability that the elbow will intersect zero, once, or twice.





    Solutions

Problem 3

References: Santalo; Kain & Rota