[extropy-chat] what is probability?

[gts]

I have a possible philosophical problem with the idea that  
non-compressibility is a true measure of randomness, as seems to be an  
assumption in Chaitin-randomness.

Every gambler knows that some very unlikely-looking sub-sequences can  
appear in any long random sequence. Flip a coin a thousand times, for  
example, and there is a good chance you'll see some extraordinarily long  
runs of heads and/or tails, along with other very non-random looking  
sub-sequences.

If I understand the compressibility principle correctly (and I very well  
may not!) those apparently non-random sub-sequences are compressible and  
so would count against the sequence in any compressibility measure of its  
randomness.

But of course we all know those remarkable sub-sequences are to be  
expected even if they cannot be predicted. Their absence in an extremely  
non-compressible and thus supposedly extremely random sequence might even  
be cause for suspicion that the sequence was artificially constructed and  
thus in some sense not truly random at all.

To put this another way: although entropy is associated with randomness, I  
question whether it is a good measure of it.

-gts