Two points on a circle of radius 1 are chosen at random. Find the probability that the distance between the two points is at most 1.
The arc measure of a central angle is defined as the ratio of the length of the arc to the circumference of the circle. The arc measure of a central angle is always between 0 and 2π.
Consider the central angle AOC. The arc measure of ∠AOC is 200 degrees, which is 910π radians. The probability that the line segment AB lies within the circle is the same as the probability that the central angle AOB measures at most 200 degrees.
Since the central angle AOB is chosen at random, all central angles between 0 and 2π are equally likely. Therefore, the probability that the central angle AOB measures at most 200 degrees is (10/9π)/2π = 5/9.