Fun problem from the new book Inside Interesting Integrals: given a circle, pick three points inside it at random (uniformly). These points define another circle. What is the probability that this circle lies entirely inside the original circle? The theoretical answer is 2pi/15 = 0.418879... But Monte-Carlo simulations strongly suggest that the answer is in fact exactly 2/5. So what gives? Original derivation from A Treatise on the Integral Calculus, Edwards, 1922: https://dl.dropboxusercontent.com/u/70818776/edwards-circles.png My MC source code (Nahin, author of Inside Interesting Integrals, also did MC simulations, which give the same result, and pointed out the discrepancy): https://dl.dropboxusercontent.com/u/70818776/interesting-circles.cpp I should say that a bunch of us debated this over Facebook for a while, and the problem was eventually resolved to my satisfaction by Zachary Abel. But it's still something you may enjoy puzzling over. Bob Hearn