There are two major problems with your idea:

1. Your answer can be "no" when the true answer is "yes". Consider this process with a circle of perimeter "21":

    ---------------------      (unwrap the circle)
    ----------+----------      (bisect the line)
    -**-------+--------**      (drop four points)

2. Your answer of "1/2, because it's divided into two equal lengths" is completely wrong for the scenario that you specify.

Consider the case where we drop a single point. We can do the same procedure:

A. Unwrap the circle;

B. Bisect the line;

C. Drop one point.

But even though the line is still divided into two equal lengths, our one point has a 100% chance of falling either on one side of the bisection point, or on the other side.

For the case where we drop four points, the article already gives the correct answer for your method, which is 1/2^3 (because there are 3+1 points).