[Ether]

Originally Posted by Ether

What's the average length of all the line segments which can be drawn inside a 1 inch square?

(accurate to 5 decimal digits)

(show your work)

OK I'm going to drop a bombshell here.

The correct answer to the OP, as worded, is 0.33634

Aren was the first to articulate the key principle when he wrote:

Originally Posted by Aren Siekmeier

Randomly sample the set of all line segments, and find the mean length of only the segments in the sample (a finite number chosen to suit your computational resources, rather than the uncountably infinite number in the entire set). If the sampling matches the probability distribution/weighting of the set, then the law of large numbers says your mean will approach the true mean as sample size increases.

Sampling segments by choosing the segment endpoint coordinates from a random uniform distribution does not match the probability density of the segment lengths in the infinite set.