Random....random.....
Nice word, very difficult to define coherently I've found.
Interesting Ian posed a nice question that led to some question about the nature of randomness, specifically wether the digits of PI are randomly distributed (at least I think that was implied in the question).
Take a very long string of digits that have an appearance of randomness. PI will do. Now, take a small string of digits, such as 12345. Will 12345 occur in the string which is PI? Well, it does, at position 49702 (
http://www.angio.net/pi/piquery). Now, how about 123456? yes. But, you get to 123456789, now, that doesn't occur in the first 100 million digits of PI, but it's a pretty near certainty that it does occur somewhere within the first x digits.
OK.
Now, I can construct a number which is PI, but with all occurrences of 12345 removed. Therefore, there's a string that cannot by definition occur there. And, now the question: is the depleted-PI number random?
Well, probably not, because there is now a zero probability of the digit 5 occurring after the string 1234.
Could you discover this deficiency by analyzing depleted-PI?
In other words, is there an algorithmic process for determining whether an (infinite) string of digits is random or not?
Feel free to join the search for the truth over here:
http://host.randi.org/vbulletin/showthread.php?s=&threadid=17828
I've pretty much reached the point of flailing about but others might have something to contribute.
Why bring this up here? Well, if we have a hard time deciding whether something as cut-and-dried as PI is random, how much more challenging the question of the nature of the universe as a whole?