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.
Let's consider two points chosen at random on a circle of radius 1.
Without loss of generality, we can fix one point and consider the distribution of the other point with respect to the first one. Suppose we fix one point at the top of the circle. Then the second point can lie anywhere on the circle except for a segment of length 2 centered at the bottom of the circle.
To see this, imagine drawing a circle of radius 1 centered at the first point. The second point must lie somewhere on this circle. The distance between the two points is at most 1 if and only if the second point lies within a distance of 1 from the first point. This corresponds to a segment of length 2 centered at the bottom of the circle, as shown in the figure below.
The length of this segment is equal to the length of the diameter of the circle, which is 2. Therefore, the probability that the distance between the two points is at most 1 is equal to the ratio of the length of the segment of the circle that is allowed for the second point to the length of the circumference of the circle.
The length of the segment of the circle that is allowed for the second point is equal to the length of the circumference of the circle minus the length of the segment of length 2 centered at the bottom of the circle. The length of the circumference of the circle is 2π, and the length of the segment of length 2 is 2. Therefore, the length of the allowed segment is 2π - 2.
So the probability that the distance between the two points is at most 1 is:
(length of the allowed segment) / (length of the circumference of the circle) = (2π - 2) / (2π) = (π - 1)/π.
