Solution 3: Invariant Measures - A First Look.

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(a)

All we need do here is consider two independent cases separately: one where the plane is tiled by vertical parallel lines one unit apart, and the other where the plane is tiled by horizontal parallel lines one unit apart. For each case the expected value of intersections is given by . So the expected value we seek is given by .

(b)

Because the n-gon is convex, any line in the plane must have 0, 1, or 2 intersections with it. The only lines that intersect the polygon at precisely one point are those lines that pass through a vertex yet line on one side of the polygon. In the sample space of all possible lies of the polygon on the plane, such happenings will form a region of area zero, and so are of no consequence when we calculate

Thus the value of P will only be determined by the probability of the polygon intersecting a line twice. If , then the event that both sides and intersect a line is independent from the event that both sides and intersect a line. So the the axiom of countable additivity we must have that or equivalently . Similarly, if , then the event that both sides and intersect a line is independent from the event that both sides and intersect a line. Again, by the axiom of countable additivity we can write

.

Thus

Now the event that side intersects a line has probability by our solutions to Buffon's Needle Problem, where is the length of side . Since we can write

so that

Exercise: Suppose that the plane is tiled with parallel lines d units apart. Show that the probability of a closed convex curve of diameter d < l intersecing a line is where L is the length of the convex curve. Hint: Approximate the convex curve with a suitably chosen n- gon and take the limit as n goes to infinity. Solution