I ran a simulation of 100,000,000 trials using bonanova's method (I think) on a unit circumference. I pick two random numbers, P1 and P2, between 0 and 1. If P1 < P2, then my three lengths are L1 = P1, L2 = P2 - P1, and L3 = 1 - P2. If P2 < P1, L1 = P2, L2 = P1 - P2, and L3 = 1 - P1. I sort the three lengths and put them into a matrix where the first column consists of all the shortest lengths, the second column is the middle length and the third column is the longest length. I then calculate the mean of the various columns. I get this result: mean of longest arcs = 0.611101750112 mean of middle arcs = 0.277769774084 mean of shortest arcs = 0.111128475804 This matches bonanova's results pretty closely and those lengths add up to 1, as expected. It would seem to me that the probability that a particular point falls in the longest arc should be 0.6111... And the middle arc 0.277... and shortest arc 0111... Which means that the average length of the arc that a random point (or fixed point) falls in should be the sum of the squares of the mean arc lengths. However, the sum of the squares is only 0.462950934519. I would expect it to be .5, given the analysis from the previous problem. I'm a bit confused. Thoughts?