MATH 3033 based on Dekking et al MATH 3033 based on Dekking et al. A Modern Introduction to Probability and Statistics. 2007 Slides by Longin Jan Latecki C5:

5.1 Darts Example Suppose we want to make a probability model for an experiment: A dart hits a disc of radius r in a completely arbitrary way. We are interested in the distance X from the hitting point to the center of the disc. Since distances cannot be negative, we have F(b) = P(X ≤ b) = 0 when b < 0. Since the object hits the disc, we have F(b) = 1 when b > r.

Compute for the darts example the probability thatn0 < X ≤ r/2, and the probability that r/2< X ≤ r. We have P(0 < X ≤ r/2) = F(r/2) − F(0) = (1/2)2 − 02 = 1/4, and P(r/2 < X ≤ r) = F(r)−F(r/2) = 1−1/4 = 3/4, no matter what the radius of the disc is.