Problem 4: Invariant Measures

Problem 4: Invariant Measures - A First Look

We are going to take a more advanced look at Buffon's Needle Problem and in so doing develop an approach to geometric probability that will guide us for the rest of the course. When we looked at Bertrand's Paradox, we encountered a difficulty because the word "random" was not well defined. So when we drop a needle "randomly" on a floor tiled with equidisdant parallel lines, what exactly do we mean? Intuitively we mean that if A and B are both regions on the floor of equal area, then the center of gravity of the needle is equally likely to land in region A as it is in region B.

A fancier way of saying the same thing is that, under transformations that preserve area, the probability of the needle intersecting a line is invariant. Of course if were interested in our probabilty remaining invariant by another class of transformations, then we would have to change our notion of "random". This apporch to geometric probability was first developed by the great mathematician Poincaré.

Note: As you may recall from linear algebra, the transformations of the plane that preserve area must be affine transformations of the form

and so preserve lines! [Here the angle is the angle of rotaion and the vector (a,b) is the translation vector.] This neat little fact will come in handy later.

First, let us coordinize the space of dropped needles using the ordered pair where x is the vertical distance from the midpoint (or center of gravity) of the needle to the nearest line above it, and is the angle made by the needle with the x-axis (we may assume that the parallel lines are horizontal without loss of generality):

Here d is the distance between consecutive parallel lines, and l < d is the length of the needle. So

and the space of all possible needle drops, that is, the sample space, has area . More formally, this area equals

Let's quickly review double integrals. This is the same as . Evaluate the integral inside the brackets with respect to x and then integrate the resulting equation with respect to .

Now we need to find the area of the event space, that is, the area of the region in the - plane corresponding to those needles which intersect one of the parallel lines. We must have x < in order for the needle to intersect a line. (Very similary to Problem 1). This area is given by

By evaluating the double integral,can you see why this is so? The probability of a needle interesecting one of the parallel lines is equal to

PROBLEM 4:

(a) Let us tile the plane with unit squares (that is, squares of unit side length). If we drop a needle of length l < 1, what is the expected value of the number of intersections?

(b) What is the probability P that a convex n-gon with perimeter L and diameter less than d, when dropped onto a plane tiled with parallel lines d units apart, will interesect a line?

Hint: Exaplain the following: First note that the polygon, if it intersects a line at all, must have exactly two intersections points (the subset of the sample space consisting of those cases where the polygon intersect a line only once, as indicated in red below, has area zero and so such cases can be ignored). And of course it is impossible for the polygon to intersect two different lines.

Now let be the probability that side intersects a line, and if , then let be the probability that both sides and intersect a line. Now explain why the following formulas are true (use the axioms of probability if needed):

so that

Finally, show that

Solution.

References: Solomon.