Two individuals are walking around a cylindrical tower. What is the probability that they can see each other?

Question

It'd be of the greatest interest to have not only a rigorous solution, but also an intuitive insight onto this simple yet very difficult problem: Let there exist some tower which has the shape of a cylinder and whose radius is A. Further, let this tower be surrounded by a walking lane whose width is B. Now, there are two individuals who are on the walk; what is the probability that they're able to see each other?

Answer 1

Consider a tangent

Answer 2

Consider a tangent connecting 2 points a + b from the tower (this is the point that one person just sees the other). The midpoint is a distance a from the center. Thus, a/(a+b) = cos(1/2 theta). We need to double the angle. But the other person could be on either side.

Thus, the solution is 4 arccos(a/(a+b))

This is the angle, but the entire circle is 2 pi, so the probability is 4 arccos(a/(a+b))/2 pi =

2 arccos(a/(a+b))/pi