[extropy-chat] what is probability?

Benjamin Goertzel ben at goertzel.org
Tue Jan 16 17:23:35 UTC 2007


The classic work on defining randomness is:

P. Martin-Lof " On the concept of a random sequence " , Theory
Probability Appl. 11: 177-179, (1966)

Roughly speaking: No computable function betting on the bits of a
Martin-Lof random sequence can make arbitrarily large amounts of

This is known to be equivalent to Chaitin's definition of randomness
as incompressibility, see


and see


for more recent work.

An extension to quantum computing (random qubit series) is here:


Note that all these definitions are about the randomness of infinitely
long sequences, hence not too useful for finite real-world situations.
 It seems that the notion of randomness is definable objectively only
for infinite entities.  To define randomness for finite entities one
needs to introduce an observer and define X as random if "observer O
can see no patterns in X."

How does this help with the PI and so forth?  Not at all as far as I
can tell, it's just the only work I know that formally addresses the
issue of defining randomness ;-)

-- Ben G

On 1/16/07, gts <gts_2000 at yahoo.com> wrote:
> On Mon, 15 Jan 2007 18:44:23 -0500, Rafal Smigrodzki
> <rafal.smigrodzki at gmail.com> wrote:
> > ### I would see it this way: The meaning of "random" is "obeying the
> > principle of indifference, where the sample space is unambiguously
> > described". If the sample space is exactly two outcomes, then each one
> > must occur 50% of the time, or else the coin is weighted, and the
> > tosses are not quite random anymore.
> Hmm, I think this is not at all what is or should be the definition of
> "random".
> As the word is normally defined, a series of tosses of an unfair coin
> still result in a completely random sequence! How can you suggest
> otherwise?
> In objectivist terms, all that matters for the sake of randomness is that
> the sequence satisfy what von Mises called "The Law of Excluded Gambling
> Systems", which is just to say (roughly) that the sequence must contain no
> predictable sub-sequences, i.e., that the result of each toss is
> independent of the others. (Subjectivists have a different way of saying
> essentially the same thing.)
> -gts
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