Problem 3: Buffon's Needle Problem - II

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Let's begin with a defintion of probability. If S is a sample space (S can be any finite or infinite set) and Ei is an event in S, that is, Ei is a subset of S, then P(Ei) will denote the probability of Ei : P(Ei) is a real number such that 0 < P(Ei) < 1 and such that the following probability axioms are satisfied for all events Ei.

  1. for all events .
  2. .
  3. Additivity: if events E1 and E2 are mutually disjoint, that is, if .
  4. Countable additivity: if the events Ei, i = 1, . . . , n, are pairwise mutually exclusive, that is, if .

Given two events A and B, we will let denote the conditional probability that event A happens given that event B did happen. Then you can show that

If A and B are any two events (so A and B may not be mutually exclusive, that is ), then one may show that

Using this basic understanding of probability theory, we will give a different proof of Buffon's Needle Problem from that given in Problem 1. Again, let there be drawn on the plane parallel lines d units apart. If we drop a needle of length l, the needle may intersect one or more of the lines: note that in order for there to be a possibility of two or more intersections, we must have l > d. Let pn be the probability that the needle intersects exactly n of the straight lines, and X1 the number of lines intersected by dropping a needle of length l1. Then the expected value, E(X1) of the function X1, is the average value of the outcomes. In this case, E(X1) is the average number of lines intersected by a needle of length l1. The expected value is given by the equation:

So if l < d for example, then pn = 0 for all integer values of n greater than 2, and we must have that

If we now drop independently a second needle of length l2, then we must have that . In fact, even if both needles are connected at the ends by a rotating joint, then we still have that . In general, if we have n needles of lengths l1, l2, . . . ln, all connected together end-to-end via rotating joings to form a polygonal segment, then

This is true even if the needles form a loop.

In fact, since the value of E(X) is only going to depend on the length l of X, we may write E(X) = E(l). This implies that E is a linear function, so that for some real number r. In order to determine the value of r, let's consider a circle C of diameter d, and approximate it with an inscribed regular n-sided polygon .

The circle of diameter d inscribed by a regular 11-gon

Note that E(d) = 2, as any circle C of diameter d thrown onto the plane will intersect exactly two of the parallel lines. As , , so that . So if l < d, then E(l) = p1 = rl = , as shown previously.

PROBLEM 3:

(a) Prove, using the probability axioms and the Venn diagram below, that

for any events A and B.

(b) Prove, using the Venn diagram above, that

for any events A and B.

(c) The Buffon-Laplace Needle Problem: Given a rectangular grid of parallel lines distances a and b apart, what is the probability that a needle of length l, where l < a and l < b, will intersect one of the lines?

Using the techniques demonstrated above, if a = b (so the plane is tiled by squares of side length a), show that the expected value of dropping a needle of length l < a is



We will show later on that the probability is equal to

Solutions. (View in Microsoft Internet Explorer)


References: Klain & Rota.