Is the probability actually more interesting though? I find the symmetry of this type of result extremely compelling, beautiful even. Buffon himself restricted his attention to the case where the needle was short enough that "probability" and "expectation" had the same answer. Put simply, math is best when complicated-seeming things suddenly become simple.
Why is the simulation always exactly 2 for a closed polygon, but has error for the almost-full circle?
I think it's because only the closed polygon is totationally symmetric, so you don't get errors from the edge case at the edge of the finite sample space. But I'm not sure.
The illustration is missing the more interesting visualization of how linearity of expectation applies to all possible rotations and translations of all segments of the needle/noodle. Each noodle is equivalent to a curve of discs, like a string of pearls. And those pearls do not even need to be connected!
The simulation is always 2 for the closed circle of width W. Actually if you run it enough, the simulation will occasionally end up not perfectly equal to 2 because it's a 50 sided polygon, not a true circle. ;)
But for a perfect circle of diameter W, it will alway hit exactly two vertical lines.
I am guessing that service returns the XML file as a directory listing; the file called winbridge.html does exist in that directory (and contains a JavaScript code to redirect to a different URL). (Another comment said they shortened the URL to remove PII (which I am guessing was in the fragment part of the original URL; the JavaScript code makes a new URL from randomly selecting a domain name (even though the list has only one) and appending the fragment part as the path), so I suppose the file name was removed and then this directory listing is the result.)
The root issue here is that your government does not consider translational cyberterrorism against civilians to be worthy of national security protection or retaliation.
(But if you are in the US, your government considers foreign human laborers doing useful work as "invaders".)
What I do is, I'm always responding to output N while the AI is working on prompt N+1. So we are both always responding to each other's question/answer before last.
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