[extropy-chat] Coin Flip Paradox
gts
gts_2000 at yahoo.com
Mon Jan 29 17:02:48 UTC 2007
On Sun, 28 Jan 2007 23:59:59 -0500, The Avantguardian
<avantguardian2020 at yahoo.com> wrote:
> The *actual* frequency of any real random
> sequence would be more accurately described to
> chaotically orbit the probability, like a strange
> attractor, rather than approach it as any kind of
> deterministic limit in a classical calculus sense.
I totally disagree, and wonder where you came up with the unusual idea
that frequencies "chaotically orbit the probability like a strange
attractor". Do you have mathematical or empirical evidence to support that
claim?
Frequentists have plenty of evidence, both empirical and mathematical, to
support their much more boring claim that frequencies converge in an
ordinary way as n increases.
But let's talk a bit about the meaning of randomness.
I surmise that you see an ambiguity in the conventional view of randomness
that I also see, but that you are expressing your displeasure about it in
ways that make no sense to me.
As I mentioned and you agreed, randomness and entropy are closely related
ideas, but the ideas should (perhaps) be kept apart.
Rafal objected, for example, when I wrote that a sequence of flips of a
heavily weighted coin is still a completely random sequence. It seems his
intuition was telling him that a weighted coin should produce a sequence
less random than a fair coin.
I think Rafal really meant that such a heavily weighted sequence has lower
*entropy*, not lower *randomness*. I think people are sometimes confused
about the two terms because of their close meanings.
As probability theorists normally use the word (at least in my experience)
randomness is mainly about the independence (or exchangeability) of
individual trials/observations, not about the measure of disorder in the
sequence of trials/observations.
The situation is made more cloudy (or perhaps more clear, depending on
your perspective) by algorithmic definitions of randomness.
Consider a binary sequence generated by an idealized perfectly random fair
coin, where Heads=1 and Tails=0. What if this unlikely sequence came up?
11111111111111111111
20 heads in a row! Is this freaky sequence still random? It certainly
doesn't *look* random, but how could it not still *be* random? After all
we stipulated in advance that it was generated by an idealized perfectly
random coin-flip process.
Well, according to the algorithmic definition of randomness, randomness is
a property of the *sequence*, not a property of the *process*. So this
sequence of 20 heads is extremely un-random by that definition even though
it was obtained via a purely random process. This is a sort of marriage of
entropy to randomness, for better or worse.
-gts
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