What is the probability that a randomly drawn chord will be longer than the radius of the circle?
I wish I would have read the other comments more closely before posting my solution. I'm really amazed at this paradox.
Well, I created a simple program in QBasic that creates two absolutely random points on a circle of radius 10 and calculated how many times out of 100000 that the length of the chord is greater than the radius. I was surprised. Unless my algorithm is off (which I don't believe it is), the computer returned a value of 63.2 (and I ran multiple tests). I randomized the random number generator with the computer's timer before getting any value for the points to ensure absolute randomness and calculated values for 100,000 iterations. This method seems the closest to the cosmic ray method and I really can't see the values that I got for this approaching 66.66 even though they are close. If I had more time, I think I'd make a program that works for a single fixed point and the second point totally random, and see how close they compare.