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The needle problem was not setup to generate complex numbers. It was setup as a question. What is the probability of the needle landing on a line? The analysis of that question yields pi.
No you didn't.
You stated random mechanisms always yield complexity.
Clue for the confused #1: An Introduction to Kolmogorov Complexity and Its Applications
A random mechanism yielded pi which you said was not complex.
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).
Clue for the confused #3: Tallying up the "hits" and dividing by the total number of drops is a deterministic conversion which drastically reduces the amount of complexity in the original sequence of drop results.
Clue for the confused #4: Consider the similar case of flipping a fair coin 1023 times in a row. The resulting sequence has a Kolmogorov Complexity near 1023 bits. Now do the Buffon's Needle method in order to approximate the probability of a "hit" (a head) on any given flip, which is exactly 0.5 -- divide the number of heads (0-1023) by the number of flips (constant 1023). Due to the "lumping" effect of extracting only the total number of heads (while ignoring their sequence), the numeric result of (Nhits/1023) has a Kolmogorov Complexity of only around 10 bits (fewer, actually, since the distribution of the number of heads is not uniform across 0-1023), resulting in over a 99% reduction in complexity.
[The sequence of crossings (or non-crossings) in a large number of throws may be complex.]
That "probability" does not equal one then.
Clue for the confused #5: "May" in this context is not used in the sense of "may or may not". It is used in the sense of, "you may have a brain, but you're not using it." Doctor Stochastic was saying, "Although the sequence of throws may be complex, a deterministic description of throws is not".
[Only a randomly generated sequence can yield complexity.]
How many times do I have to flip a coin.....
How much complexity do you want?
Oh never mind the "probability" is still not one.
Sure it is.
Needles follow
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The needle problem was not setup to generate complex numbers. It was setup as a question. What is the probability of the needle landing on a line? The analysis of that question yields pi.