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Someone confused wrote: You stated random mechanisms always yield complexity. A random mechanism yielded pi which you said was not complex.

I responded: Clue for the confused #2: This random process does *not* yield pi (nor did Doctor Stochastic say that it did). It yields a random rational number of rather high complexity which is statistically constrained to likely be in the close neighborhood of pi, because the probability of a "hit" is related to pi itself. But the result itself is guaranteed to *not* actually be the non-complex constant "pi" (because pi is irrational, whereas the result of the Buffon needle-throwing experiment will be rational since it is the quotient of two integers).

The confused person responded:

The astute reader will note that Mr. Confused's rudimentary ability to cut-and-paste an image of a formula from the pi formulas page of mathworld.wolfram.com hardly constitutes any sort of actual rebuttal, especially when it in no way contradicts any of the points I made concerning Mr. Confused's errors.

It's most likely that, lacking any sort of actual defense for his mistakes, and being unwilling to admit them, he decided to go for a "baffle-'em-with-BS" approach and to just fling impressive looking formulae (borrowed from elsewhere) in an attempt to look erudite.

But on the off-chance that he actually thought that he was making some sort of point by posting a well-known formula for pi, the only conceivable point may have been that he was under the mistaken impression that the above formula demonstrates that pi must be a rational number after all, since all the terms are quotients of integer values. If that be the case, then:

Clue for the confused #6: Infinite series of rational numbers do not necessarily produce rational results, and indeed this one does not. One of the most elementary facts about pi, known to almost anyone with even a rudimentary education in geometry or basic math, is that pi is, indeed, an irrational number.

Mr. Confused went on to cut-and-paste thusly:

Needles follow

Why yes, yes they do. Mr. Confused can cut-and-paste from this webpage. Here's a cookie.

But yet again, this does nothing whatsoever to defend Mr. Confused's error-prone post from my various refutations. This appears nothing more than further "baffle-'em-with-BS". Further support for that hypothesis is the fact that this same poster has engaged in such time-wasting tactics numerous times in the past as well whenever one of his points began to crumble.

The needle problem was not setup to generate complex numbers.

That's nice. Again, though, this doesn't repair any of the errors that Mr. Confused has made in this thread, many of which I've pointed out.

It was setup as a question.

Oh, it was a "setup" all right -- Mr. Confused is often in the habit of asking "innocent" questions, waiting for someone to respond and fall into his "trap" (so he imagines) so that he can spring forth in response and nail the responder with a "gotcha". More often than not, though, his carefully laid traps are not the clever bit of mental outmaneuvering he thinks them to be, they are transparent cheap tricks which don't stand up to examination. Rather like this one. He thought he could "catch" Doctor Stochastic by getting him to commit to a response to the Buffon's Needle question which would contradict Doctor Stochastic's earlier comment about random processes and complexity -- but instead Mr. Confused fell on his butt by revealing that he misunderstood the manner in which his own example related to complexity theory.

What is the probability of the needle landing on a line? The analysis of that question yields pi.

Yup, the *analysis* does indeed. But Mr. Confused was, well, confused enough to think that this meant that the needle-dropping and tallying *process* actually yields pi. As I pointed out, it does not. He wrote:

Yes it is 2/pi, but you can take that and divide it into one and multiply by two and bingo you have pi. Which you said is not complex. You have a contradiction. [...] You stated random mechanisms always yield complexity. A random mechanism yielded pi which you said was not complex.

Mr. Confused was and still is wrong -- the random mechanism of the Buffon's Needle drop does *not* "yield pi", as I've already explained in detail, and which Mr. Confused's response fails to rebut in any way -- it's just hand-waving, an attempt to divert from a) the point he tried to make, and b) the fact that his point was a complete dud.