[extropy-chat] what is probability?

[Benjamin Goertzel]

> I think that these results
>
> http://citeseer.ist.psu.edu/calude94borel.html
>
> imply that Chaitin randomness implies exchangeability for infinite
> sequences.    Also, they show that for finite sequences, almost all
> sequences are exchangeable (so in that sense, a 'randomly chosen'
> sequence is very likely to be exchangeable).

Sorry: I meant in the last sentence that almost all RANDOM (i.e.
incompressible) finite sequences are exchangeable [for any specific
observer, and as sequence length gets long enough...]

I don't know whether anyone has proved that, conversely, almost all
finite exchangeable sequences are Chaitin-random.  But I would bet
that they are.

For instance, tossing a coin according to the bits of the binary
Champernowne sequence [scroll down in

http://en.wikipedia.org/wiki/Champernowne_constant

] may lead to an exchangeable but not Chaitin random series.  But,
this sort of case is probably a rare one statistically...

-- Ben G